View source for Green's law

{{About-distinguish|the law in fluid dynamics|Green's theorem}}
{{short description|Equation describing evolution of waves in shallow water}}
[[File:Propagation du tsunami en profondeur variable.gif|thumb|right|Propagation of shoaling long waves, showing the variation of [[wavelength]] and [[wave height]] with decreasing water depth.]]

In [[fluid dynamics]], '''Green's law''', named for 19th-century British mathematician [[George Green (mathematician)|George Green]], is a [[conservation law]] describing the evolution of [[breaking wave|non-breaking]], [[surface gravity wave]]s [[Wave propagation|propagating]] in [[waves and shallow water|shallow water]] of gradually varying depth and width. In its simplest form, for [[wavefront]]s and [[depth contour]]s parallel to each other (and the coast), it states:

:<math>H_1\, \cdot\, \sqrt[4]{h_1} = H_2\, \cdot\, \sqrt[4]{h_2}</math> {{pad|1em}} or {{pad|1em}} <math>\left(H_1\right)^4\, \cdot\, h_1 = \left(H_2\right)^4\, \cdot\, h_2,</math>

where <math>H_1</math> and <math>H_2</math> are the [[wave height]]s at two different locations – 1 and 2 respectively – where the wave passes, and <math>h_1</math> and <math>h_2</math> are the [[mean]] water depths at the same two locations.

Green's law is often used in [[coastal engineering]] for the modelling of long [[wave shoaling|shoaling waves]] on a beach, with "long" meaning [[wavelength]]s in excess of about twenty times the mean water depth.<ref>{{harvtxt|Dean|Dalrymple|1991|loc=§3.4}}</ref> [[Tsunami]]s shoal (change their height) in accordance with this law, as they propagate – governed by [[refraction]] and [[diffraction]] – through the ocean and up the [[continental shelf]]. Very close to (and running up) the coast, [[nonlinearity|nonlinear effects]] become important and Green's law no longer applies.<ref>{{harvtxt|Synolakis|Skjelbreia|1993}}</ref><ref>{{harvtxt|Synolakis|1991}}</ref>

==Description==
[[File:Mavericks wave diagram.gif|thumb|right|Convergence of wave rays (reduction of width <math>b</math>) at [[Mavericks, California]], producing high [[surfing]] waves. The red lines are the wave rays; the blue lines are the [[wavefront]]s. The distances between neighboring wave rays vary towards the coast because of [[refraction]] by [[bathymetry]] (depth variations). The distance between wavefronts reduces towards the coast because of [[wave shoaling]] (decreasing depth <math>h</math>).]]
According to this law, which is based on [[linear system|linearized]] [[shallow water equations]], the spatial variations of the [[wave height]] <math>H</math> (twice the [[amplitude]] <math>a</math> for [[sine wave]]s, equal to the amplitude for a [[soliton|solitary wave]]) for [[travelling wave]]s in water of mean depth <math>h</math> and width <math>b</math> (in case of an [[open-channel flow|open channel]]) satisfy<ref name=Lamb>{{harvtxt|Lamb|1993|loc=§185}}</ref><ref name=Dean_Dalrymple>{{harvtxt|Dean|Dalrymple|1991|loc=§5.3}}</ref>

:<math>H\, \sqrt{ b }\, \sqrt[4]{h} = \text{constant},</math>

where <math>\sqrt[4]{h}</math> is the [[fourth root]] of <math>h.</math> Consequently, when considering two cross sections of an open channel, labeled 1 and 2, the wave height in section 2 is:

:<math>H_2 = \sqrt{\frac{b_1}{b_2}}\; \sqrt[4]{\frac{h_1}{h_2}}\; H_1,</math>

with the subscripts 1 and 2 denoting quantities in the associated cross section. So, when the depth has decreased by a factor sixteen, the waves become twice as high. And the wave height doubles after the channel width has gradually been reduced by a factor four. For wave propagation [[perpendicular]] towards a straight coast with depth contours parallel to the coastline, take <math>b</math> a constant, say 1 metre or yard.

For refracting long waves in the ocean or near the coast, the width <math>b</math> can be interpreted as the distance between wave [[ray (optics)|ray]]s. The rays (and the changes in spacing between them) follow from the [[geometrical optics]] approximation to the linear wave propagation.<ref>{{harvtxt|Satake|2002}}</ref> In case of straight parallel depth contours this simplifies to the use of [[Snell's law]].<ref>{{harvtxt|Dean|Dalrymple|1991|loc=§4.8.2}}</ref>

Green published his results in 1838,<ref name=Green>{{harvtxt|Green|1838}}</ref> based on a method – the [[Liouville–Green method]] – which would evolve into what is now known as the [[WKB approximation]]. Green's law also corresponds to constancy of the mean horizontal wave [[energy flux]] for long waves:<ref name=Lamb/><ref name=Dean_Dalrymple/>

:<math>b\, \sqrt{gh}\, \tfrac18 \rho g H^2 = \text{constant},</math>

where <math>\sqrt{gh}</math> is the [[group speed]] (equal to the [[phase speed]] in shallow water), <math>\tfrac18 \rho g H^2 = \tfrac12 \rho g a^2</math> is the mean wave [[energy density]] integrated over depth and per unit of horizontal area, <math>g</math> is the [[gravity of Earth|gravitational acceleration]] and <math>\rho</math> is the water [[density]].

===Wavelength and period===
Further, from Green's analysis, the [[wavelength]] <math>\lambda</math> of the wave shortens during shoaling into shallow water, with<ref name=Lamb/><ref name=Green/>

:<math>\frac{\lambda}{\sqrt{g\,h}} = \text{constant}</math>

along a wave [[ray (optics)|ray]]. The oscillation [[period (physics)|period]] (and therefore also the [[frequency]]) of shoaling waves does not change, according to Green's linear theory.

==Derivation==
Green derived his shoaling law for water waves by use of what is now known as the Liouville–Green method, applicable to gradual variations in depth <math>h</math> and width <math>b</math> along the path of wave propagation.<ref>The derivation presented below is according to the line of reasoning as used by {{harvtxt|Lamb|1993|loc=§169 & §185}}.</ref>

===Wave equation for an open channel===
Starting point are the linearized [[one-dimensional Saint-Venant equations]] for an [[open-channel flow|open channel]] with a rectangular cross section (vertical side walls). These equations describe the evolution of a wave with [[free surface]] elevation <math>\eta(x,t)</math> and horizontal flow velocity <math>u(x,t),</math> with <math>x</math> the horizontal coordinate along the channel axis and <math>t</math> the time:

:<math>
\begin{align}
  &b\, \frac{\partial \eta}{\partial t} + \frac{\partial (b\, h\, u)}{\partial x} = 0,
  \\ 
  &\frac{\partial u}{\partial t} + g\, \frac{\partial \eta}{\partial x} =0,
\end{align}
</math>

where <math>g</math> is the [[gravity of Earth]] (taken as a constant), <math>h</math> is the [[mean]] water depth, <math>b</math> is the channel width and <math>\partial/\partial t</math> and <math>\partial/\partial x</math> are denoting [[partial derivative]]s with respect to space and time. The slow variation of width <math>b</math> and depth <math>h</math> with distance <math>x</math> along the channel axis is brought into account by denoting them as <math>b(\mu x)</math> and <math>h(\mu x),</math> where <math>\mu</math> is a small parameter: <math>\mu \ll 1.</math> The above two equations can be combined into one [[wave equation]] for the surface elevation:

{{NumBlk|:|<math>
  \frac{\partial^2 \eta}{\partial t^2} 
  - \frac{g}{b}\, \frac{\partial}{\partial x} \left( b\, h\, \frac{\partial \eta}{\partial x} \right)
  = 0,
</math> {{pad|3em}} and with the velocity following from {{pad|2em}}<math>
  u = - g \int \frac{\partial \eta}{\partial x}\; \mathrm{d}t.
</math>|{{EquationRef|1}}}}

In the Liouville–Green method, the approach is to convert the above wave equation with [[homogeneity (physics)|non-homogeneous]] coefficients into a homogeneous one (neglecting some small remainders in terms of <math>\mu</math>).

===Transformation to the wave phase as independent variable===
The next step is to apply a [[coordinate transformation]], introducing the travel time (or [[wave phase]]) <math>\tau</math> given by

:<math>\frac{\mathrm{d} x}{\mathrm{d} \tau} = \sqrt{g\, h},</math> {{pad|3em}} so {{pad|2em}} <math>\tau = \int_{x_0}^x \frac{1}{\sqrt{g\,h(\mu s)}}\; \mathrm{d}s</math>

and <math>x</math> are related through the [[Phase speed|celerity]] <math>c=\sqrt{gh}.</math> Introducing the [[multiple-scale analysis|slow variable]] <math>X=\mu x</math> and denoting derivatives of <math>b(X)</math> and <math>h(X)</math> with respect to <math>X</math> with a prime, e.g. <math>b'=\mathrm{d}b/\mathrm{d}X,</math> the <math>x</math>-derivatives in the wave equation, Eq.&nbsp;({{EquationNote|1}}), become:

:<math>
\begin{align}
  & \frac{\partial\eta}{\partial x} 
    =\left (\frac{\mathrm{d}x}{\mathrm{d}\tau}\right)^{-1}\, \frac{\partial\eta}{\partial\tau}
    = \frac{1}{\sqrt{g\,h}}\, \frac{\partial\eta}{\partial\tau}
  \qquad \text{and}
  \\
  & \frac{\partial}{\partial x} \left( b\, h\, \frac{\partial\eta}{\partial x} \right)
    = b\, h\, \frac{\partial^2\eta}{\partial x^2} + \mu \left( b'\, h + b\, h' \right)\, \frac{\partial\eta}{\partial x} 
    = \frac{b}{g}\, \frac{\partial^2 \eta}{\partial \tau^2} 
    + \mu\, \frac{ b'\, h + \tfrac12\, b\, h' }{\sqrt{gh}}\, \frac{\partial \eta}{\partial \tau}.
\end{align}
</math>

Now the wave equation ({{EquationNote|1}}) transforms into:

{{NumBlk|:|<math>
  \frac{\partial^2 \eta}{\partial t^2} 
  - \frac{\partial^2 \eta}{\partial \tau^2}
  - \mu\, \sqrt{g\,h}\, \left( \frac{b'}{b} + \tfrac12\, \frac{h'}{h} \right)\, \frac{\partial \eta}{\partial \tau}
  = 0.
</math>|{{EquationRef|2}}}}

The next step is transform the equation in such a way that only deviations from homogeneity in the second [[order of approximation]] remain, i.e. proportional to <math>\mu^2.</math>

===Further transformation towards homogeneity===
The homogeneous wave equation (i.e. Eq.&nbsp;({{EquationNote|2}}) when <math>\mu</math> is zero) has solutions <math>\eta=F(t\pm\tau)</math> for [[travelling wave]]s of permanent form propagating in either the negative or positive <math>x</math>-direction. For the inhomogeneous case, considering waves propagating in the positive <math>x</math>-direction, Green proposes an approximate solution:

{{NumBlk|:|<math>\eta(\tau,t) = \Theta(X)\, F(t-\tau).</math>|{{EquationRef|3}}}}

Then

:<math>
\begin{align}
  \frac{\partial^2 \eta}{\partial t^2} 
  & = \Theta\, \frac{\mathrm{d}^2 F}{\mathrm{d} \tau^2},
  \\
  \frac{\partial \eta}{\partial \tau} 
  & = \Theta\, \frac{\mathrm{d} F}{\mathrm{d} \tau}
    + \mu\, \sqrt{g\,h}\, \Theta'\, F
  \qquad \text{and}
  \\
  \frac{\partial^2 \eta}{\partial \tau^2} 
  & = \Theta\, \frac{\mathrm{d}^2 F}{\mathrm{d} \tau^2}
    + 2\, \mu\, \sqrt{g\,h}\, \Theta'\, \frac{\mathrm{d} F}{\mathrm{d} \tau}
    + \mu^2\, g\, h\, \Theta''\, F.
\end{align}
</math>

Now the [[left-hand side]] of Eq.&nbsp;({{EquationNote|2}}) becomes:

:<math>
  \mu\, \sqrt{g\,h}\, \left( 2\, \frac{\Theta'}{\Theta} + \frac{b'}{b} + \tfrac12\, \frac{h'}{h} \right)\, 
    \Theta\, \frac{\mathrm{d} F}{\mathrm{d} \tau}
  +
  \mu^2\, g\, h\, \left( \frac{\Theta''}{\Theta'} + \frac{b'}{b} + \tfrac12\, \frac{h'}{h} \right)\, 
    \Theta'\, F.
</math>

So the proposed solution in Eq.&nbsp;({{EquationNote|3}}) satisfies Eq.&nbsp;({{EquationNote|2}}), and thus also Eq.&nbsp;({{EquationNote|1}}) apart from the above two terms proportional to <math>\mu</math> and <math>\mu^2</math>, with <math>\mu\ll 1.</math> The error in the solution can be made of order <math>\mathcal{O}(\mu^2)</math> provided

:<math>2\, \frac{\Theta'}{\Theta} + \frac{b'}{b} + \tfrac12\, \frac{h'}{h} = 0.</math>

This has the solution:

:<math>\Theta(X) = \alpha\, b^{-\tfrac12}\, h^{-\tfrac14}.</math>

Using Eq.&nbsp;({{EquationNote|3}}) and the transformation from <math>x</math> to <math>\tau</math>, the approximate solution for the surface elevation <math>\eta(x,t)</math> is

{{NumBlk|:|<math>
  \eta(x,t) = b^{-\tfrac12}(x)\; h^{-\tfrac14}(x)\; F\left( t - \int_{x_0}^x \frac{1}{\sqrt{g\,h(s)}}\; \mathrm{d}s \right) 
            + \mathcal{O}(\mu^2),
</math>|{{EquationRef|4}}}}

where the constant <math>\alpha</math> has been set to one, [[without loss of generality]]. Waves travelling in the negative <math>x</math>-direction have the minus sign in the argument of function <math>F</math> reversed to a plus sign. Since the theory is linear, solutions can be added because of the [[superposition principle]].

===Sinusoidal waves and Green's law===
Waves varying [[sine wave|sinusoidal]] in time, with [[period (physics)|period]] <math>T,</math> are considered. That is

:<math>\eta(x,t) = a(x)\, \sin( \omega\, t - \phi(x) ) = \tfrac12\, H(x)\, \sin( \omega\, t - \phi(x)),</math>

where <math>a</math> is the [[amplitude]], <math>H=2 a</math> is the [[wave height]], <math>\omega=2\pi/T</math> is the [[angular frequency]] and <math>\phi(x)</math> is the [[wave phase]]. Consequently, also <math>F</math> in Eq.&nbsp;({{EquationNote|4}}) has to be a sine wave, e.g. <math>F=\beta\sin(\omega t-\phi(x))</math> with <math>\beta</math> a constant.

Applying these forms of <math>\eta</math> and <math>F</math> in Eq.&nbsp;({{EquationNote|4}}) gives:

:<math>H\, b^{\tfrac12}\, h^{\tfrac14} = \beta,</math>

which is ''Green's law''.

===Flow velocity===
The horizontal flow velocity in the <math>x</math>-direction follows directly from substituting the solution for the surface elevation <math>\eta(x,t)</math> from Eq.&nbsp;({{EquationNote|4}}) into the expression for <math>u(x,t)</math> in Eq.&nbsp;({{EquationNote|1}}):<ref>{{harvtxt|Didenkulova|Pelinovsky|Soomere|2009}}</ref>

:<math>
\begin{align}
  u(x,t) &= g^{\tfrac12}\; b^{-\tfrac12}(x)\; h^{-\tfrac34}(x)\; 
            F\left( t - \int_{x_0}^x \frac{1}{\sqrt{g\,h(s)}}\; \mathrm{d}s \right)
  \\
         &+ \mu\, \tfrac12\, g\, b^{-\tfrac12}(x)\; h^{-\tfrac14}(x)\; \frac{(b\, \sqrt{h})'}{b\, \sqrt{h}}\, \Phi(x,t)
          + \frac{Q}{b(x)\, h(x)}
          + \mathcal{O}\left( \mu^2 \right),
  \\
  & \qquad \text{with} \qquad 
  \Phi(x,t) = \int_{t_0}^t F\left( \sigma - \int_{x_0}^x \frac{1}{\sqrt{g\,h(s)}}\; \mathrm{d}s \right)\; \mathrm{d}\sigma
\end{align}
</math>

and <math>Q</math> an additional constant [[discharge (hydrology)|discharge]].

Note that –  when the width <math>b</math> and depth <math>h</math> are not constants – the term proportional to <math>\Phi(x,t)</math> implies an <math>\mathcal{O}(\mu)</math> (small) phase difference between elevation <math>\eta</math> and velocity <math>u</math>.

For sinusoidal waves with velocity amplitude <math>V,</math> the flow velocities shoal to [[leading order]] as<ref name=Green/>

:<math>V\, b^{\tfrac12}\, h^{\tfrac34} = \text{constant}.</math>

This could have been anticipated since for a horizontal bed <math display="inline">V = \sqrt{g\,h}\,(a/h) = \sqrt{g/h}\,a</math> with <math>a</math> the wave amplitude.

==Notes==
{{reflist|30em}}

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{{refend}}

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{{refend}}

{{physical oceanography}}

[[Category:Fluid dynamics]]
[[Category:Water waves]]