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Simulation of wave penetration—involving diffraction and refraction—into Tedious Creek, Maryland, using CGWAVE (which solves the mild-slope equation).

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting geomorphology changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.

A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972.^[1]^[2]^[3] Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.

In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.

Formulation for monochromatic wave motion

For monochromatic waves according to linear theory—with the free surface elevation given as $ζ (x, y, t) = ℜ {η (x, y) e^{- i ω t}}$ and the waves propagating on a fluid layer of mean water depth $h (x, y)$ —the mild-slope equation is:^[4]

\nabla \cdot (c_{p} c_{g} \nabla η) + k^{2} c_{p} c_{g} η = 0,

where:

$η (x, y)$ is the complex-valued amplitude of the free-surface elevation $ζ (x, y, t);$
$(x, y)$ is the horizontal position;
$ω$ is the angular frequency of the monochromatic wave motion;
$i$ is the imaginary unit;
$ℜ {\cdot}$ means taking the real part of the quantity between braces;
$\nabla$ is the horizontal gradient operator;
$\nabla \cdot$ is the divergence operator;
$k$ is the wavenumber;
$c_{p}$ is the phase speed of the waves and
$c_{g}$ is the group speed of the waves.

The phase and group speed depend on the dispersion relation, and are derived from Airy wave theory as:^[5]

\begin{aligned} ω^{2} & = g k \tanh (k h), \\ c_{p} & = \frac{ω}{k} and \\ c_{g} & = \frac{1}{2} c_{p} [1 + k h \frac{1 - \tanh^{2} (k h)}{\tanh (k h)}] \end{aligned}

where

$g$ is Earth's gravity and
$\tanh$ is the hyperbolic tangent.

For a given angular frequency $ω$ , the wavenumber $k$ has to be solved from the dispersion equation, which relates these two quantities to the water depth $h$ .

Transformation to an inhomogeneous Helmholtz equation

Through the transformation

ψ = η \sqrt{c_{p} c_{g}},

the mild slope equation can be cast in the form of an inhomogeneous Helmholtz equation:^[4]^[6]

Δ ψ + k_{c}^{2} ψ = 0 with k_{c}^{2} = k^{2} - \frac{Δ (\sqrt{c_{p} c_{g}})}{\sqrt{c_{p} c_{g}}},

where $Δ$ is the Laplace operator.

Propagating waves

In spatially coherent fields of propagating waves, it is useful to split the complex amplitude $η (x, y)$ in its amplitude and phase, both real valued:^[7]

η (x, y) = a (x, y) e^{i θ (x, y)},

where

$a = | η |$ is the amplitude or absolute value of $η$ and
$θ = \arg {η}$ is the wave phase, which is the argument of $η .$

This transforms the mild-slope equation in the following set of equations (apart from locations for which $\nabla θ$ is singular):^[7]

\begin{aligned} \frac{\partial κ_{y}}{\partial x} - \frac{\partial κ_{x}}{\partial y} = 0 & with κ_{x} = \frac{\partial θ}{\partial x} and κ_{y} = \frac{\partial θ}{\partial y}, \\ κ^{2} = k^{2} + \frac{\nabla \cdot (c_{p} c_{g} \nabla a)}{c_{p} c_{g} a} & with κ = \sqrt{κ_{x}^{2} + κ_{y}^{2}} and \\ \nabla \cdot (v_{g} E) = 0 & with E = \frac{1}{2} ρ g a^{2} and v_{g} = c_{g} \frac{κ}{k}, \end{aligned}

where

$E$ is the average wave-energy density per unit horizontal area (the sum of the kinetic and potential energy densities),
$κ$ is the effective wavenumber vector, with components $(κ_{x}, κ_{y}),$
$v_{g}$ is the effective group velocity vector,
$ρ$ is the fluid density, and
$g$ is the acceleration by the Earth's gravity.

The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy $E$ is transported in the $κ$ -direction normal to the wave crests (in this case of pure wave motion without mean currents).^[7] The effective group speed $| v_{g} |$ is different from the group speed $c_{g} .$

The first equation states that the effective wavenumber $κ$ is irrotational, a direct consequence of the fact it is the derivative of the wave phase $θ$ , a scalar field. The second equation is the eikonal equation. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with $\nabla \cdot (c_{p} c_{g} \nabla a) ≪ k^{2} c_{p} c_{g} a,$ the splitting into amplitude $a$ and phase $θ$ leads to consistent-varying and meaningful fields of $a$ and $κ$ . Otherwise, κ² can even become negative. When the diffraction effects are totally neglected, the effective wavenumber κ is equal to $k$ , and the geometric optics approximation for wave refraction can be used.^[7]

Template:Hidden begin When $η = a \exp (i θ)$ is used in the mild-slope equation, the result is, apart from a factor $\exp (i θ)$ :

c_{p} c_{g} (Δ a + 2 i \nabla a \cdot \nabla θ - a \nabla θ \cdot \nabla θ + i a Δ θ) + \nabla (c_{p} c_{g}) \cdot (\nabla a + i a \nabla θ) + k^{2} c_{p} c_{g} a = 0 .

Now both the real part and the imaginary part of this equation have to be equal to zero:

\begin{aligned} c_{p} c_{g} Δ a - c_{p} c_{g} a \nabla θ \cdot \nabla θ + \nabla (c_{p} c_{g}) \cdot \nabla a + k^{2} c_{p} c_{g} a = 0 and \\ 2 c_{p} c_{g} \nabla a \cdot \nabla θ + c_{p} c_{g} a Δ θ + \nabla (c_{p} c_{g}) \cdot (a \nabla θ) = 0 . \end{aligned}

The effective wavenumber vector $κ$ is defined as the gradient of the wave phase:

κ = \nabla θ

and its vector length is

κ = | κ | .

Note that $κ$ is an irrotational field, since the curl of the gradient is zero:

\nabla \times κ = 0 .

Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by $a$ :

\begin{aligned} κ^{2} = k^{2} + \frac{\nabla (c_{p} c_{g})}{c_{p} c_{g}} \cdot \frac{\nabla a}{a} + \frac{Δ a}{a} and \\ c_{p} c_{g} \nabla (a^{2}) \cdot κ + c_{p} c_{g} \nabla \cdot κ + a^{2} κ \cdot \nabla (c_{p} c_{g}) = 0 . \end{aligned}

The first equation directly leads to the eikonal equation above for $κ$ , while the second gives:

\nabla \cdot (κ c_{p} c_{g} a^{2}) = 0,

which—by noting that $c_{p} = ω / k$ in which the angular frequency $ω$ is a constant for time-harmonic motion—leads to the wave-energy conservation equation. Template:Hidden end

Derivation of the mild-slope equation

The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach.^[4]^[8] The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. These assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using potential flow theory.

Template:Hidden begin

Luke's variational principle

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Luke's Lagrangian formulation gives a variational formulation for non-linear surface gravity waves.^[9] For the case of a horizontally unbounded domain with a constant density $ρ$ , a free fluid surface at $z = ζ (x, y, t)$ and a fixed sea bed at $z = - h (x, y),$ Luke's variational principle $δ ℒ = 0$ uses the Lagrangian

ℒ = \int_{t_{0}}^{t_{1}} \iint L d x d y d t,

where $L$ is the horizontal Lagrangian density, given by:

L = - ρ {\int_{- h (x, y)}^{ζ (x, y, t)} [\frac{\partial Φ}{\partial t} + \frac{1}{2} ({(\frac{\partial Φ}{\partial x})}^{2} + {(\frac{\partial Φ}{\partial y})}^{2} + {(\frac{\partial Φ}{\partial z})}^{2})] d z + \frac{1}{2} g (ζ^{2} - h^{2})},

where $Φ (x, y, z, t)$ is the velocity potential, with the flow velocity components being $\partial Φ / \partial x,$ $\partial Φ / \partial y$ and $\partial Φ / \partial z$ in the $x$ , $y$ and $z$ directions, respectively. Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.^[10] Taking the variations of $ℒ (Φ, ζ)$ with respect to the potential $Φ (x, y, z, t)$ and surface elevation $ζ (x, y, t)$ leads to the Laplace equation for $Φ$ in the fluid interior, as well as all the boundary conditions both on the free surface $z = ζ (x, y, t)$ as at the bed at $z = - h (x, y) .$

Linear wave theory

In case of linear wave theory, the vertical integral in the Lagrangian density $L$ is split into a part from the bed $z = - h$ to the mean surface at $z = 0,$ and a second part from $z = 0$ to the free surface $z = ζ$ . Using a Taylor series expansion for the second integral around the mean free-surface elevation $z = 0,$ and only retaining quadratic terms in $Φ$ and $ζ,$ the Lagrangian density $L_{0}$ for linear wave motion becomes

L_{0} = - ρ {ζ {[\frac{\partial Φ}{\partial t}]}_{z = 0} + \int_{- h}^{0} \frac{1}{2} [{(\frac{\partial Φ}{\partial x})}^{2} + {(\frac{\partial Φ}{\partial y})}^{2} + {(\frac{\partial Φ}{\partial z})}^{2}] d z + \frac{1}{2} g ζ^{2}} .

The term $\partial Φ / \partial t$ in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the Euler–Lagrange equations, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to $h^{2}$ in the potential energy.

The waves propagate in the horizontal $(x, y)$ plane, while the structure of the potential $Φ$ is not wave-like in the vertical $z$ -direction. This suggests the use of the following assumption on the form of the potential $Φ :$

Φ (x, y, z, t) = f (z; x, y) φ (x, y, t)

with normalisation

f (0; x, y) = 1

at the mean free-surface elevation

z = 0 .

Here $φ (x, y, t)$ is the velocity potential at the mean free-surface level $z = 0 .$ Next, the mild-slope assumption is made, in that the vertical shape function $f$ changes slowly in the $(x, y)$ -plane, and horizontal derivatives of $f$ can be neglected in the flow velocity. So:

(\begin{matrix} \frac{\partial Φ}{\partial x} \\ \frac{\partial Φ}{\partial y} \\ \frac{\partial Φ}{\partial z} \end{matrix}) \approx (\begin{matrix} f \frac{\partial φ}{\partial x} \\ f \frac{\partial φ}{\partial y} \\ \frac{\partial f}{\partial z} φ \end{matrix}) .

As a result:

L_{0} = - ρ {ζ \frac{\partial φ}{\partial t} + \frac{1}{2} F [{(\frac{\partial φ}{\partial x})}^{2} + {(\frac{\partial φ}{\partial y})}^{2}] + \frac{1}{2} G φ^{2} + \frac{1}{2} g ζ^{2}},

with

F = \int_{- h}^{0} f^{2} d z

and

G = \int_{- h}^{0} {(\frac{d f}{d z})}^{2} d z .

The Euler–Lagrange equations for this Lagrangian density $L_{0}$ are, with $ξ (x, y, t)$ representing either $φ$ or $ζ :$

\frac{\partial L_{0}}{\partial ξ} - \frac{\partial}{\partial t} (\frac{\partial L_{0}}{\partial (\partial ξ / \partial t)}) - \frac{\partial}{\partial x} (\frac{\partial L_{0}}{\partial (\partial ξ / \partial x)}) - \frac{\partial}{\partial y} (\frac{\partial L_{0}}{\partial (\partial ξ / \partial y)}) = 0 .

Now $ξ$ is first taken equal to $φ$ and then to $ζ .$ As a result, the evolution equations for the wave motion become:^[4]

\begin{aligned} \frac{\partial ζ}{\partial t} & + \nabla \cdot (F \nabla φ) - G φ = 0 and \\ \frac{\partial φ}{\partial t} & + g ζ = 0, \end{aligned}

with ∇ the horizontal gradient operator: ∇ ≡(∂/∂x ∂/∂y)^T where T denotes the transpose.

The next step is to choose the shape function $f$ and to determine $F$ and $G .$

Vertical shape function from Airy wave theory

Since the objective is the description of waves over mildly sloping beds, the shape function $f (z)$ is chosen according to Airy wave theory. This is the linear theory of waves propagating in constant depth $h .$ The form of the shape function is:^[4]

f = \frac{\cosh (k (z + h))}{\cosh (k h)},

with $k (x, y)$ now in general not a constant, but chosen to vary with $x$ and $y$ according to the local depth $h (x, y)$ and the linear dispersion relation:^[4]

ω_{0}^{2} = g k \tanh (k h) .

Here $ω_{0}$ a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals $F$ and $G$ become:^[4]

\begin{aligned} F & = \int_{h}^{0} f^{2} d z = \frac{1}{g} c_{p} c_{g} and \\ G & = \int_{h}^{0} {(\frac{\partial f}{\partial z})}^{2} d z = \frac{1}{g} (ω_{0}^{2} - k^{2} c_{p} c_{g}) . \end{aligned}

Template:Hidden end The following time-dependent equations give the evolution of the free-surface elevation $ζ (x, y, t)$ and free-surface potential $ϕ (x, y, t) :$ ^[4]

\begin{aligned} g \frac{\partial ζ}{\partial t} & + \nabla \cdot (c_{p} c_{g} \nabla φ) + (k^{2} c_{p} c_{g} - ω_{0}^{2}) φ = 0, \\ \frac{\partial φ}{\partial t} & + g ζ = 0, with ω_{0}^{2} = g k \tanh (k h) . \end{aligned}

From the two evolution equations, one of the variables $φ$ or $ζ$ can be eliminated, to obtain the time-dependent form of the mild-slope equation:^[4]

- \frac{\partial^{2} ζ}{\partial t^{2}} + \nabla \cdot (c_{p} c_{g} \nabla ζ) + (k^{2} c_{p} c_{g} - ω_{0}^{2}) ζ = 0,

and the corresponding equation for the free-surface potential is identical, with $ζ$ replaced by $φ .$ The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around $ω_{0} .$

Monochromatic waves

Consider monochromatic waves with complex amplitude $η (x, y)$ and angular frequency $ω :$

ζ (x, y, t) = ℜ {η (x, y) e^{- i ω t}},

with $ω$ and $ω_{0}$ chosen equal to each other, $ω = ω_{0} .$ Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:^[4]

\nabla \cdot (c_{p} c_{g} \nabla η) + k^{2} c_{p} c_{g} η = 0 .

Applicability and validity of the mild-slope equation

The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3.^[11] However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.

Notes

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References

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, 2 Parts, 967 pages.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, 740 pages.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ ^4.00 ^4.01 ^4.02 ^4.03 ^4.04 ^4.05 ^4.06 ^4.07 ^4.08 ^4.09 See Dingemans (1997), pp. 248–256 & 378–379.
↑ See Dingemans (1997), p. 49.
↑ See Mei (1994), pp. 86–89.
↑ ^7.0 ^7.1 ^7.2 ^7.3 See Dingemans (1997), pp. 259–262.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[1] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[2] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[3] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[Dingemans_ms-4] 4.00 ^4.01 ^4.02 ^4.03 ^4.04 ^4.05 ^4.06 ^4.07 ^4.08 ^4.09 See Dingemans (1997), pp. 248–256 & 378–379.

[5] See Dingemans (1997), p. 49.

[6] See Mei (1994), pp. 86–89.

[Dingemans_259_263-7] 7.0 ^7.1 ^7.2 ^7.3 See Dingemans (1997), pp. 259–262.

[8] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[9] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[Miles1977-10] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[Booij1983-11] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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Fugue (hash function): Difference between revisions

Revision as of 23:14, 2 January 2014

Contents

Formulation for monochromatic wave motion

Transformation to an inhomogeneous Helmholtz equation

Propagating waves

Derivation of the mild-slope equation

Luke's variational principle

Linear wave theory

Vertical shape function from Airy wave theory

Monochromatic waves

Applicability and validity of the mild-slope equation

Notes

References

Navigation menu

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+[[File:CGWAVE Tedious Creek MD.jpg|thumb|right|Simulation of wave penetration—involving [[diffraction]] and [[refraction]]—into Tedious Creek, Maryland, using [[CGWAVE]] (which solves the mild-slope equation).]]
+In [[fluid dynamics]], the '''mild-slope equation''' describes the combined effects of [[diffraction]] and [[refraction]] for [[water wave]]s propagating over [[bathymetry]] and due to lateral boundaries—like [[Breakwater (structure)|breakwater]]s and [[coastline]]s. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in [[coastal engineering]] to compute the wave-field changes near [[harbour]]s and [[coast]]s.
+The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as [[cliff]]s, [[beach]]es, [[seawall]]s and breakwaters. As a result, it describes the variations in wave [[amplitude]], or equivalently [[wave height]]. From the wave amplitude, the amplitude of the [[flow velocity]] oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, [[sediment transport]] and resulting [[geomorphology]] changes of the sea bed and coastline, mean flow fields and [[mass transfer]] of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from [[numerical analysis]].
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+A first form of the mild-slope equation was developed by [[Carl Eckart|Eckart]] in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972.<ref>{{citation | first=C. | last=Eckart | authorlink=Carl Eckart | title=The propagation of gravity waves from deep to shallow water | year=1952 | publisher=National Bureau of Standards | journal=Circular 20 | pages=165–173 }}</ref><ref>{{citation | first=J. C. W. | last=Berkhoff | year=1972 | contribution=Computation of combined refraction–diffraction | title=Proceedings 13th International Conference on Coastal Engineering | location=Vancouver | pages=471–490 |url=http://journals.tdl.org/ICCE/article/view/2767 }}</ref><ref>{{citation | first=J. C. W. | last=Berkhoff | year=1976 | title=Mathematical models for simple harmonic linear water wave models; wave refraction and diffraction | publisher=Delft University of Technology |type=PhD. Thesis |url=http://repository.tudelft.nl/assets/uuid:381c691b-eea8-4f67-be8f-d471a7da1d58/261254.pdf }}</ref> Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: [[wave–current interaction]], wave [[nonlinearity]], steeper sea-bed slopes, [[drag (physics)|bed friction]] and [[wave breaking]]. Also [[Parabolic partial differential equation|parabolic]] approximations to the mild-slope equation are often used, in order to reduce the computational cost.
+In case of a constant depth, the mild-slope equation reduces to the [[Helmholtz equation]] for wave diffraction.
+== Formulation for monochromatic wave motion ==
+For [[monochromatic]] waves according to [[linear theory]]—with the [[free surface]] elevation given as <math>\zeta(x,y,t)=\Re\left\{\eta(x,y)\,\text{e}^{-i\omega t}\right\}</math> and the waves propagating on a fluid layer of [[average|mean]] water depth <math>h(x,y)</math>—the mild-slope equation is:<ref name=Dingemans_ms>See Dingemans (1997), pp. 248–256 & 378–379.</ref>
+:<math>\nabla\cdot\left( c_p\, c_g\, \nabla \eta \right)\, +\, k^2\, c_p\, c_g\, \eta\, =\, 0,</math>
+where:
+*<math>\eta(x,y)</math> is the [[complex analysis|complex-valued]] [[amplitude]] of the free-surface elevation <math>\zeta(x,y,t);</math>
+*<math>(x,y)</math> is the horizontal position;
+*<math>\omega</math> is the [[angular frequency]] of the monochromatic wave motion;
+*<math>i</math> is the [[imaginary unit]];
+*<math>\Re\{\cdot\}</math> means taking the [[real part]] of the quantity between braces;
+*<math>\nabla</math> is the horizontal [[gradient]] operator;
+*<math>\nabla\cdot</math> is the [[divergence]] operator;
+*<math>k</math> is the [[wavenumber]];
+*<math>c_p</math> is the [[phase speed]] of the waves and
+*<math>c_g</math> is the [[group speed]] of the waves.
+The phase and group speed depend on the [[dispersion (water waves)|dispersion relation]], and are derived from [[Airy wave theory]] as:<ref>See Dingemans (1997), p. 49.</ref>
+:<math>
+\begin{align}
+  \omega^2 &=\, g\, k\, \tanh\, (kh), \\
+  c_p &=\, \frac{\omega}{k} \quad \text{and} \\
+  c_g &=\, \frac12\, c_p\, \left[ 1\, +\, kh\, \frac{1 - \tanh^2 (kh)}{\tanh\, (kh)} \right]
+\end{align}
+</math>
+where
+*<math>g</math> is [[Earth's gravity]] and
+*<math>\tanh</math> is the [[hyperbolic tangent]].
+For a given angular frequency <math>\omega</math>, the wavenumber <math>k</math> has to be solved from the dispersion equation, which relates these two quantities to the water depth <math>h</math>.
+== Transformation to an inhomogeneous Helmholtz equation ==
+Through the transformation
+:<math>\psi\, =\, \eta\, \sqrt{c_p\, c_g},</math>
+the mild slope equation can be cast in the form of an [[inhomogeneous Helmholtz equation]]:<ref name=Dingemans_ms/><ref>See Mei (1994), pp. 86–89.</ref>
+:<math>
+  \Delta\psi\, +\, k_c^2\, \psi\, =\, 0
+  \qquad \text{with} \qquad k_c^2\, =\, k^2\, -\, \frac{\Delta\left(\sqrt{c_p\,c_g}\right)}{\sqrt{c_p\,c_g}},
+</math>
+where <math>\Delta</math> is the [[Laplace operator]].
+== Propagating waves ==
+In spatially [[Coherence (physics)|coherent]] fields of propagating waves, it is useful to split the complex amplitude <math>\eta(x,y)</math> in its amplitude and phase, both [[real number|real value]]d:<ref name=Dingemans_259_263>See Dingemans (1997), pp. 259–262.</ref>
+:<math>\eta(x,y)\, =\, a(x,y)\, \text{e}^{i\, \theta(x,y)},</math>
+where
+*<math>a=|\eta|\,</math> is the amplitude or [[absolute value]] of <math>\eta\,</math> and
+*<math>\theta=\arg\{\eta\}\,</math> is the wave phase, which is the [[Arg (mathematics)|argument]] of <math>\eta.\,</math>
+This transforms the mild-slope equation in the following set of equations (apart from locations for which <math>\nabla\theta</math> is singular):<ref name=Dingemans_259_263/>
+:<math>
+  \begin{align}
+  \frac{\partial\kappa_y}{\partial{x}}\, -\, \frac{\partial\kappa_x}{\partial{y}}\, =\, 0
+      \qquad &\text{ with } \kappa_x\, =\, \frac{\partial\theta}{\partial{x}} \text{ and } \kappa_y\, =\, \frac{\partial\theta}{\partial{y}},
+  \\
+  \kappa^2\, =\, k^2\, +\, \frac{\nabla\cdot\left( c_p\, c_g\, \nabla a \right)}{c_p\, c_g\, a}
+      \qquad &\text{ with } \kappa\, =\, \sqrt{\kappa_x^2 \, +\, \kappa_y^2} \quad \text{ and}
+  \\
+  \nabla \cdot \left( \boldsymbol{v}_g\, E \right)\, =\, 0
+      \qquad &\text{ with } E\, =\, \frac12\, \rho\, g\, a^2 \quad \text{and} \quad \boldsymbol{v}_g\, =\, c_g\, \frac{\boldsymbol{\kappa}}{k},
+  \end{align}
+</math>
+where
+* <math>E</math> is the [[average]] wave-energy density per unit horizontal area (the sum of the [[kinetic energy|kinetic]] and [[potential energy]] densities),
+* <math>\boldsymbol{\kappa}</math> is the effective wavenumber vector, with components <math>(\kappa_x,\kappa_y),\,</math>
+* <math>\boldsymbol{v}_g</math> is the effective [[group velocity]] vector,
+* <math>\rho</math> is the fluid [[density]], and
+* <math>g</math> is the acceleration by the [[Earth's gravity]].
+The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy <math>E</math> is transported in the <math>\boldsymbol{\kappa}</math>-direction normal to the wave [[crest (physics)|crest]]s (in this case of pure wave motion without mean currents).<ref name=Dingemans_259_263/> The effective group speed <math>|\boldsymbol{v}_g|</math> is different from the group speed <math>c_g.</math>
+The first equation states that the effective wavenumber <math>\boldsymbol{\kappa}</math> is [[irrotational]], a direct consequence of the fact it is the derivative of the wave phase <math>\theta</math>, a [[scalar field]]. The second equation is the [[eikonal equation]]. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with <math>\nabla\cdot(c_p\, c_g\, \nabla a)\ll k^2\, c_p\, c_g\, a,</math> the splitting into amplitude <math>a</math> and phase <math>\theta</math> leads to consistent-varying and meaningful fields of <math>a</math> and <math>\boldsymbol{\kappa}</math>. Otherwise, ''κ''<sup>2</sup> can even become negative. When the diffraction effects are totally neglected, the effective wavenumber ''κ'' is equal to <math>k</math>, and the [[geometric optics]] approximation for wave [[refraction]] can be used.<ref name=Dingemans_259_263/>
+{{hidden begin
+|toggle = left
+|bodystyle = font-size: 100%
+|title = Details of the derivation of the above equations
+}}
+When <math>\eta=a\mbox{ }\exp(i\theta)</math> is used in the mild-slope equation, the result is, apart from a factor <math>\exp(i\theta)</math>:
+:<math>
+  c_p\,c_g\, \left( \Delta a\, +\, 2i\, \nabla a \cdot \nabla\theta\, -\, a\, \nabla\theta \cdot \nabla\theta\, +\, i\, a\, \Delta\theta \right)\,
+  +\, \nabla \left( c_p\, c_g \right) \cdot \left( \nabla a\, +\, i\, a\, \nabla\theta \right)\,
+  +\, k^2\, c_p\, c_g\, a\,
+  =\, 0.
+</math>
+Now both the real part and the imaginary part of this equation have to be equal to zero:
+:<math>
+  \begin{align}
+  & c_p\,c_g\, \Delta a\, -\, c_p\, c_g\, a\, \nabla\theta \cdot \nabla\theta\,
+  +\, \nabla \left( c_p\, c_g \right) \cdot \nabla a\,
+  +\, k^2\, c_p\, c_g\, a\,
+  =\, 0
+  \quad \text{and} \\
+  & 2\, c_p\,c_g\, \nabla a \cdot \nabla\theta\, +\, c_p\, c_g\, a\, \Delta\theta\,
+  +\, \nabla \left( c_p\, c_g \right) \cdot \left( a\, \nabla\theta \right)\,
+  =\, 0.
+  \end{align}
+</math>
+The effective wavenumber vector <math>\boldsymbol{\kappa}</math> is ''defined'' as the gradient of the wave phase:
+:<math>\boldsymbol{\kappa}\, =\, \nabla\theta</math> &nbsp; and its [[vector length]] is &nbsp; <math>\kappa=|\boldsymbol{\kappa}|.</math>
+Note that <math>\boldsymbol{\kappa}</math> is an [[irrotational]] field, since the [[Vector calculus identities#Curl of the gradient|curl of the gradient]] is zero:
+:<math>\nabla\, \times\, \boldsymbol{\kappa}\, =\, 0.</math>
+Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by <math>a</math>:
+:<math>
+  \begin{align}
+    &\kappa^2\, =\, k^2\, +\, \frac{\nabla(c_p\, c_g)}{c_p\, c_g} \cdot \frac{\nabla a}{a}\, +\, \frac{\Delta a}{a}
+      \quad \text{and} \\
+    &c_p\, c_g\, \nabla\left(a^2\right) \cdot \boldsymbol{\kappa}\,
+    +\, c_p\, c_g\, \nabla\cdot\boldsymbol{\kappa}\,
+    +\, a^2\, \boldsymbol{\kappa} \cdot \nabla \left( c_p\, c_g \right)\,
+    =\, 0.
+  \end{align}
+</math>
+The first equation directly leads to the eikonal equation above for <math>\kappa\,</math>, while the second gives:
+:<math>\nabla \cdot \left( \boldsymbol{\kappa}\, c_p\, c_g\, a^2 \right)\, =\, 0,</math>
+which—by noting that <math>c_p=\omega/k</math> in which the angular frequency <math>\omega</math> is a constant for time-[[harmonic]] motion—leads to the wave-energy conservation equation.
+{{hidden end}}
+== Derivation of the mild-slope equation ==
+The mild-slope equation can be derived by the use of several methods. Here, we will use a [[Calculus of variations|variational]] approach.<ref name=Dingemans_ms/><ref>{{citation | last=Booij | first=N. | title= Gravity waves on water with non-uniform depth and current | year=1981 | publisher=Delft University of Technology |type=PhD. Thesis |url=http://repository.tudelft.nl/assets/uuid:05f9b2b1-b237-491f-927a-2a470e0808f3/Booij1981.pdf }}</ref> The fluid is assumed to be [[viscosity|inviscid]] and [[incompressible]], and the flow is assumed to be [[irrotational]]. These assumptions are valid ones for surface gravity waves, since the effects of [[vorticity]] and [[viscosity]] are only significant in the [[Stokes boundary layer]]s (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using [[potential flow]] theory.
+{{hidden begin
+|toggle = left
+|bodystyle = font-size: 100%
+|title = Details of the derivation of the mild-slope equation
+}}
+=== Luke's variational principle ===
+{{main|Luke's variational principle}}
+Luke's [[Lagrangian]] formulation gives a variational formulation for [[non-linear]] surface gravity waves.<ref>{{citation
+ | first = J. C.
+ | last = Luke
+ | year = 1967
+ | title = A variational principle for a fluid with a free surface
+ | journal = Journal of Fluid Mechanics
+ | volume = 27
+ | issue = 2
+ | pages = 395–397
+ | doi = 10.1017/S0022112067000412
+|bibcode = 1967JFM....27..395L }}</ref>
+For the case of a horizontally unbounded domain with a constant [[density]] <math>\rho</math>, a free fluid surface at <math>z=\zeta(x,y,t)</math> and a fixed sea bed at <math>z=-h(x,y),</math> Luke's variational principle <math>\delta\mathcal{L}=0</math> uses the [[Lagrangian]]
+:<math>
+  \mathcal{L} = \int_{t_0}^{t_1} \iint L\; \text{d}x\; \text{d}y\; \text{d}t,
+</math>
+where <math>L</math> is the horizontal [[Lagrangian density]], given by:
+:<math>
+  L = -\rho\, \left\{
+    \int_{-h(x,y)}^{\zeta(x,y,t)}
+    \left[
+      \frac{\partial\Phi}{\partial t}
+      +\, \frac{1}{2} \left(
+              \left( \frac{\partial\Phi}{\partial x} \right)^2
+            + \left( \frac{\partial\Phi}{\partial y} \right)^2
+            + \left( \frac{\partial\Phi}{\partial z} \right)^2
+          \right)
+    \right]\; \text{d}z\;
+    +\, \frac{1}{2}\, g\, (\zeta^2\, -\, h^2)
+  \right\},
+</math>
+where <math>\Phi(x,y,z,t)</math> is the [[velocity potential]], with the [[flow velocity]] components being <math>\partial\Phi/\partial{x},</math> <math>\partial\Phi/\partial{y}</math> and <math>\partial\Phi/\partial{z}</math> in the <math>x</math>, <math>y</math> and <math>z</math> directions, respectively.
+Luke's Lagrangian formulation can also be recast into a [[Hamiltonian mechanics|Hamiltonian formulation]] in terms of the surface elevation and velocity potential at the free surface.<ref name=Miles1977>{{citation
+ | first= J. W.
+ | last = Miles
+ | author-link = John W. Miles
+ | year = 1977
+ | title = On Hamilton's principle for surface waves
+ | journal = Journal of Fluid Mechanics
+ | volume = 83
+ | issue = 1
+ | pages = 153–158
+ | doi = 10.1017/S0022112077001104
+|bibcode = 1977JFM....83..153M }}</ref>
+Taking the variations of <math>\mathcal{L}(\Phi,\zeta)</math> with respect to the potential <math>\Phi(x,y,z,t)</math> and surface elevation <math>\zeta(x,y,t)</math> leads to the [[Laplace equation]] for <math>\Phi</math> in the fluid interior, as well as all the boundary conditions both on the free surface <math>z=\zeta(x,y,t)</math> as at the bed at <math>z=-h(x,y).</math>
+=== Linear wave theory ===
+In case of linear wave theory, the vertical integral in the Lagrangian density <math>L</math> is split into a part from the bed <math>z=-h</math> to the mean surface at <math>z=0,</math> and a second part from <math>z=0</math> to the free surface <math>z=\zeta</math>. Using a [[Taylor series]] expansion for the second integral around the mean free-surface elevation <math>z=0,</math> and only retaining quadratic terms in <math>\Phi</math> and <math>\zeta,</math> the Lagrangian density <math>L_0</math> for linear wave motion becomes
+:<math>
+  L_0 = -\rho\,
+    \left\{
+      \zeta\, \left[ \frac{\partial\Phi}{\partial t} \right]_{z=0}\,
+      +\, \int_{-h}^0 \frac12 \left[
+                  \left( \frac{\partial\Phi}{\partial x} \right)^2
+                + \left( \frac{\partial\Phi}{\partial y} \right)^2
+                + \left( \frac{\partial\Phi}{\partial z} \right)^2
+            \right]\; \text{d}z\;
+      +\, \frac{1}{2}\, g\, \zeta^2\,
+    \right\}.
+</math>
+The term <math>\partial\Phi/\partial{t}</math> in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the [[Euler–Lagrange equation]]s, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to <math>h^2</math> in the potential energy.
+The waves propagate in the horizontal <math>(x,y)</math> plane, while the structure of the potential <math>\Phi</math> is not wave-like in the vertical <math>z</math>-direction. This suggests the use of the following assumption on the form of the potential <math>\Phi:</math>
+:<math>\Phi(x,y,z,t) = f(z;x,y)\, \varphi(x,y,t)</math> &nbsp; with normalisation &nbsp; <math>f(0;x,y)=1</math> &nbsp; at the mean free-surface elevation <math>z=0.</math>
+Here <math>\varphi(x,y,t)</math> is the velocity potential at the mean free-surface level <math>z=0.</math> Next, the mild-slope assumption is made, in that the vertical shape function <math>f</math> changes slowly in the <math>(x,y)</math>-plane, and horizontal derivatives of <math>f</math> can be neglected in the flow velocity. So:
+:<math>
+  \begin{pmatrix}
+    \displaystyle \frac{\partial\Phi}{\partial{x}} \\[2ex]
+    \displaystyle \frac{\partial\Phi}{\partial{y}} \\[2ex]
+    \displaystyle \frac{\partial\Phi}{\partial{z}}
+  \end{pmatrix}\,
+  \approx\,
+  \begin{pmatrix}
+    \displaystyle f\, \frac{\partial\varphi}{\partial{x}} \\[2ex]
+    \displaystyle f\, \frac{\partial\varphi}{\partial{y}} \\[2ex]
+    \displaystyle \frac{\partial{f}}{\partial{z}}\, \varphi
+  \end{pmatrix}.
+</math>
+As a result:
+:<math>
+  L_0 = -\rho\, \left\{
+    \zeta\, \frac{\partial\varphi}{\partial t}\,
+    +\, \frac12\, F\, \left[
+                \left( \frac{\partial\varphi}{\partial{x}} \right)^2\,
+            +\, \left( \frac{\partial\varphi}{\partial{y}} \right)^2
+          \right]\,
+    +\, \frac12\, G\, \varphi^2\,
+    +\, \frac12\, g\, \zeta^2\,
+  \right\},
+</math> &nbsp; with &nbsp; <math>F\, =\, \int_{-h}^0 f^2\; \text{d}z</math> &nbsp; and &nbsp; <math>G\, =\, \int_{-h}^0 \left(\frac{\text{d}f}{\text{d}z}\right)^2\; \text{d}z.</math>
+The [[Euler–Lagrange equation]]s for this Lagrangian density <math>L_0</math> are, with <math>\xi(x,y,t)</math> representing either <math>\varphi</math> or <math>\zeta:</math>
+:<math>
+    \frac{\partial{L_0}}{\partial\xi}
+  - \frac{\partial}{\partial{t}}\left( \frac{\partial{L_0}}{\partial(\partial\xi/\partial{t})} \right)
+  - \frac{\partial}{\partial{x}}\left( \frac{\partial{L_0}}{\partial(\partial\xi/\partial{x})} \right)
+  - \frac{\partial}{\partial{y}}\left( \frac{\partial{L_0}}{\partial(\partial\xi/\partial{y})} \right)
+  = 0.
+</math>
+Now <math>\xi</math> is first taken equal to <math>\varphi</math> and then to <math>\zeta.</math>
+As a result, the evolution equations for the wave motion become:<ref name=Dingemans_ms/>
+:<math>\begin{align}
+  \frac{\partial\zeta}{\partial t}\, &+ \nabla \cdot \left( F\, \nabla\varphi \right)\, -\, G\, \varphi\, =\, 0
+  \quad \text{and} \\
+  \frac{\partial\varphi}{\partial t}\, &+\, g\, \zeta\, =\, 0,
+\end{align}</math>
+with ∇ the horizontal gradient operator: ∇&nbsp;≡(∂/∂''x'' ∂/∂''y'')<sup>T</sup> where T denotes the [[transpose]].
+The next step is to choose the shape function <math>f</math> and to determine <math>F</math> and <math>G.</math>
+=== Vertical shape function from Airy wave theory ===
+Since the objective is the description of waves over mildly sloping beds, the shape function <math>f(z)</math> is chosen according to [[Airy wave theory]]. This is the linear theory of waves propagating in constant depth <math>h.</math> The form of the shape function is:<ref name=Dingemans_ms/>
+:<math>f = \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\cosh\, (k h)},</math>
+with <math>k(x,y)</math> now in general not a constant, but chosen to vary with <math>x</math> and <math>y</math> according to the local depth <math>h(x,y)</math> and the linear dispersion relation:<ref name=Dingemans_ms/>
+:<math>\omega_0^2\, =\, g\, k\, \tanh\, (k h).</math>
+Here <math>\omega_0</math> a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals <math>F</math> and <math>G</math> become:<ref name=Dingemans_ms/>
+:<math>
+  \begin{align}
+    F &= \int_h^0 f^2\; \text{d}z = \frac{1}{g}\, c_p\, c_g \quad \text{and}
+    \\
+    G &= \int_h^0 \left( \frac{\partial{f}}{\partial{z}} \right)^2\; \text{d}z = \frac{1}{g} \left( \omega_0^2\, -\, k^2\, c_p\, c_g \right).
+  \end{align}
+</math>
+{{hidden end}}
+The following time-dependent equations give the evolution of the free-surface elevation <math>\zeta(x,y,t)</math> and free-surface potential <math>\phi(x,y,t):</math><ref name=Dingemans_ms/>
+:<math>
+  \begin{align}
+    g\, \frac{\partial\zeta}{\partial{t}}
+      &+ \nabla\cdot\left( c_p\, c_g\, \nabla \varphi \right)
+       + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \varphi
+       = 0,
+    \\
+    \frac{\partial\varphi}{\partial{t}} &+ g \zeta = 0,
+    \quad \text{with} \quad \omega_0^2\, =\, g\, k\, \tanh\, (kh).
+  \end{align}
+</math>
+From the two evolution equations, one of the variables <math>\varphi</math> or <math>\zeta</math> can be eliminated, to obtain the time-dependent form of the mild-slope equation:<ref name=Dingemans_ms/>
+:<math>
+  -\frac{\partial^2\zeta}{\partial{t^2}}
+       + \nabla\cdot\left( c_p\, c_g\, \nabla \zeta \right)
+       + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \zeta
+       = 0,
+</math>
+and the corresponding equation for the free-surface potential is identical, with <math>\zeta</math> replaced by <math>\varphi.</math> The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around <math>\omega_0.</math>
+=== Monochromatic waves ===
+Consider monochromatic waves with complex amplitude <math>\eta(x,y)</math> and angular frequency <math>\omega:</math>
+:<math>\zeta(x,y,t)\, =\, \Re\left\{ \eta(x,y)\; \text{e}^{-i\, \omega\, t} \right\},</math>
+with <math>\omega</math> and <math>\omega_0</math> chosen equal to each other, <math>\omega=\omega_0.</math> Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:<ref name=Dingemans_ms/>
+:<math>\nabla \cdot \left( c_p\, c_g\, \nabla \eta \right)\, +\, k^2\, c_p\, c_g\, \eta\, =\, 0.</math>
+== Applicability and validity of the mild-slope equation ==
+The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3.<ref name=Booij1983>{{citation
+ | first= N.
+ | last = Booij
+ | year = 1983
+ | title = A note on the accuracy of the mild-slope equation
+ | journal = Coastal Engineering
+ | volume = 7
+ | issue = 1
+ | pages = 191–203
+ | doi= 10.1016/0378-3839(83)90017-0
+}}</ref> However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.
+== Notes ==
+{{reflist}}
+== References ==
+*{{citation
+ | title=Water wave propagation over uneven bottoms
+ | first=M. W.
+ | last=Dingemans
+ | year=1997
+ | series=Advanced Series on Ocean Engineering
+ | volume=13
+ | publisher=World Scientific, Singapore
+ | isbn=981-02-0427-2
+ | oclc=36126836
+}}, 2 Parts, 967 pages.
+*{{citation
+ | last=Liu
+ | first= P. L.-F.
+ | contribution=Wave transformation
+ | year = 1990
+ | title = Ocean Engineering Science
+ | series = The Sea
+ | editor = B. Le Méhauté and D. M. Hanes
+ | publisher = Wiley Interscience
+ | volume = 9A
+ | isbn = 0-471-52856-0
+ | pages = 27–63
+}}
+*{{citation
+ | title= The applied dynamics of ocean surface waves
+ | first= Chiang C.
+ | last=  Mei
+ | author-link = Chiang C. Mei
+ | publisher= World Scientific
+ | year= 1994
+ | isbn= 9971-5-0789-7
+ | series= Advanced Series on Ocean Engineering
+ | volume= 1
+}}, 740 pages.
+*{{citation
+ | contribution = Linear wave scattering by two-dimensional topography
+ | first1 = D.
+ | last1 = Porter
+ | first2 = P. G.
+ | last2 = Chamberlain
+ | year = 1997
+ | title = Gravity waves in water of finite depth
+ | editor = J. N. Hunt
+ | publisher = Computational Mechanics Publications
+ | series = Advances in Fluid Mechanics
+ | volume = 10
+ | isbn = 1-85312-351-X
+ | pages = 13–53
+}}
+*{{citation
+| doi = 10.1017/S0022112003005846
+| volume = 494
+| pages = 51–63
+| last = Porter
+| first = D.
+| title = The mild-slope equations
+| journal = Journal of Fluid Mechanics
+| year = 2003
+|bibcode = 2003JFM...494...51P }}
+{{physical oceanography}}
+[[Category:Coastal geography]]
+[[Category:Equations of fluid dynamics]]
+[[Category:Water waves]]