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[[Image:Slope picture.svg|right|thumb|The slope of a line is defined as the rise over the run, ''m'' = Δ''y''/Δ''x''. When ''m'' << 1, the slope is said to be ''mild''.]]
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.
In [[mathematics]], the '''slope''' or '''gradient''' of a [[Line (mathematics)|line]] describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.


The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (''x''<sub>1</sub>,''y''<sub>1</sub>) and (''x''<sub>2</sub>,''y''<sub>2</sub>) on a line, the slope ''m'' of the line is
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]].
:<math>m=\frac{y_2-y_1}{x_2-x_1}.</math>


The '''mild-slope equation''' is when the slope is not very steep (m << 1). Although the equation is the same, the mild-slope equation can be comforting to people who are [[Vertigo|afraid of heights]], or who are not [[physical fitness|physically fit]].
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 13<sup>th</sup> International Conference on Coastal Engineering | location=Vancouver | pages=471–490 }}</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, PhD. Thesis }}</ref> There after, 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.


==See also==
In case of a constant depth, the mild-slope equation reduces to the [[Helmholtz equation]] for wave diffraction.
*[[Slope]]

*[[Mountain climbing]]
== 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 [[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 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 [[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, PhD. Thesis }}</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
}}</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
| 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
}}</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)
= 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>

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 bottom curvature, provides accurate results for the wave field over bottom 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
}}</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 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
| pages = 27–63
}}
*{{citation
| title= The applied dynamics of ocean surface waves
| first= Chiang C.
| last= Mei
| publisher= World Scientific
| year= 1994
| isbn= 997 15 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
| pages = 13–53
}}

{{physical oceanography}}

[[Category:Coastal geography]]
[[Category:Equations of fluid dynamics]]
[[Category:Water waves]]

Revision as of 00:55, 28 April 2010

The slope of a line is defined as the rise over the run, m = Δy/Δx. When m << 1, the slope is said to be mild.

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is

The mild-slope equation is when the slope is not very steep (m << 1). Although the equation is the same, the mild-slope equation can be comforting to people who are afraid of heights, or who are not physically fit.

See also

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