Green's law

Wave equation for an open channel

Starting point are the linearized one-dimensional Saint-Venant equations for an open channel with a rectangular cross section (vertical side walls). These equations describe the evolution of a wave with free surface elevation ${\displaystyle \eta (x,t)}$ and horizontal flow velocity ${\displaystyle u(x,t),}$ with ${\displaystyle x}$ the horizontal coordinate along the channel axis and ${\displaystyle t}$ the time:

${\displaystyle {\begin{aligned}&b\,{\frac {\partial \eta }{\partial t}}+{\frac {\partial (b\,h\,u)}{\partial x}}=0,\\&{\frac {\partial u}{\partial t}}+g\,{\frac {\partial \eta }{\partial x}}=0,\end{aligned}}}$

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

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-{\frac {g}{b}}\,{\frac {\partial }{\partial x}}\left(b\,h\,{\frac {\partial \eta }{\partial x}}\right)=0,}$   and with the velocity following from  ${\displaystyle u=-g\int {\frac {\partial \eta }{\partial x}}\;\mathrm {d} t.}$

(1)

In the Liouville–Green method, the approach is to convert the above wave equation with non-homogeneous coefficients into a homogeneous one (neglecting some small remainders in terms of ${\displaystyle \mu }$).

Transformation to the wave phase as independent variable

The next step is to apply a coordinate transformation, introducing the travel time (or wave phase) ${\displaystyle \tau }$ given by

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \tau }}={\sqrt {g\,h}},}$   so   ${\displaystyle \tau =\int _{x_{0}}^{x}{\frac {1}{\sqrt {g\,h(\mu s)}}}\;\mathrm {d} s}$

and ${\displaystyle x}$ are related through the celerity ${\displaystyle c={\sqrt {gh}}.}$ Introducing the slow variable ${\displaystyle X=\mu x}$ and denoting derivatives of ${\displaystyle b(X)}$ and ${\displaystyle h(X)}$ with respect to ${\displaystyle X}$ with a prime, e.g. ${\displaystyle b'=\mathrm {d} b/\mathrm {d} X,}$ the ${\displaystyle x}$-derivatives in the wave equation, Eq. (1), become:

${\displaystyle {\begin{aligned}&{\frac {\partial \eta }{\partial x}}=\left({\frac {\mathrm {d} x}{\mathrm {d} \tau }}\right)^{-1}\,{\frac {\partial \eta }{\partial \tau }}={\frac {1}{\sqrt {g\,h}}}\,{\frac {\partial \eta }{\partial \tau }}\qquad {\text{and}}\\&{\frac {\partial }{\partial x}}\left(b\,h\,{\frac {\partial \eta }{\partial x}}\right)=b\,h\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}+\mu \left(b'\,h+b\,h'\right)\,{\frac {\partial \eta }{\partial x}}={\frac {b}{g}}\,{\frac {\partial ^{2}\eta }{\partial \tau ^{2}}}+\mu \,{\frac {b'\,h+{\tfrac {1}{2}}\,b\,h'}{\sqrt {gh}}}\,{\frac {\partial \eta }{\partial \tau }}.\end{aligned}}}$

Now the wave equation (1) transforms into:

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-{\frac {\partial ^{2}\eta }{\partial \tau ^{2}}}-\mu \,{\sqrt {g\,h}}\,\left({\frac {b'}{b}}+{\tfrac {1}{2}}\,{\frac {h'}{h}}\right)\,{\frac {\partial \eta }{\partial \tau }}=0.}$

(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 ${\displaystyle \mu ^{2}.}$

Further transformation towards homogeneity

The homogeneous wave equation (i.e. Eq. (2) when ${\displaystyle \mu }$ is zero) has solutions ${\displaystyle \eta =F(t\pm \tau )}$ for travelling waves of permanent form propagating in either the negative or positive ${\displaystyle x}$-direction. For the inhomogeneous case, considering waves propagating in the positive ${\displaystyle x}$-direction, Green proposes an approximate solution:

${\displaystyle \eta (\tau ,t)=\Theta (X)\,F(t-\tau ).}$

(3)

Then

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}\eta }{\partial t^{2}}}&=\Theta \,{\frac {\mathrm {d} ^{2}F}{\mathrm {d} \tau ^{2}}},\\{\frac {\partial \eta }{\partial \tau }}&=\Theta \,{\frac {\mathrm {d} F}{\mathrm {d} \tau }}+\mu \,{\sqrt {g\,h}}\,\Theta '\,F\qquad {\text{and}}\\{\frac {\partial ^{2}\eta }{\partial \tau ^{2}}}&=\Theta \,{\frac {\mathrm {d} ^{2}F}{\mathrm {d} \tau ^{2}}}+2\,\mu \,{\sqrt {g\,h}}\,\Theta '\,{\frac {\mathrm {d} F}{\mathrm {d} \tau }}+\mu ^{2}\,g\,h\,\Theta ''\,F.\end{aligned}}}$

Now the left-hand side of Eq. (2) becomes:

${\displaystyle \mu \,{\sqrt {g\,h}}\,\left(2\,{\frac {\Theta '}{\Theta }}+{\frac {b'}{b}}+{\tfrac {1}{2}}\,{\frac {h'}{h}}\right)\,\Theta \,{\frac {\mathrm {d} F}{\mathrm {d} \tau }}+\mu ^{2}\,g\,h\,\left({\frac {\Theta ''}{\Theta '}}+{\frac {b'}{b}}+{\tfrac {1}{2}}\,{\frac {h'}{h}}\right)\,\Theta '\,F.}$

So the proposed solution in Eq. (3) satisfies Eq. (2), and thus also Eq. (1) apart from the above two terms proportional to ${\displaystyle \mu }$ and ${\displaystyle \mu ^{2}}$, with ${\displaystyle \mu \ll 1.}$ The error in the solution can be made of order ${\displaystyle {\mathcal {O}}(\mu ^{2})}$ provided

${\displaystyle 2\,{\frac {\Theta '}{\Theta }}+{\frac {b'}{b}}+{\tfrac {1}{2}}\,{\frac {h'}{h}}=0.}$

This has the solution:

${\displaystyle \Theta (X)=\alpha \,b^{-{\tfrac {1}{2}}}\,h^{-{\tfrac {1}{4}}}.}$

Using Eq. (3) and the transformation from ${\displaystyle x}$ to ${\displaystyle \tau }$, the approximate solution for the surface elevation ${\displaystyle \eta (x,t)}$ is

${\displaystyle \eta (x,t)=b^{-{\tfrac {1}{2}}}(x)\;h^{-{\tfrac {1}{4}}}(x)\;F\left(t-\int _{x_{0}}^{x}{\frac {1}{\sqrt {g\,h(s)}}}\;\mathrm {d} s\right)+{\mathcal {O}}(\mu ^{2}),}$

(4)

where the constant ${\displaystyle \alpha }$ has been set to one, without loss of generality. Waves travelling in the negative ${\displaystyle x}$-direction have the minus sign in the argument of function ${\displaystyle F}$ 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 sinusoidal in time, with period ${\displaystyle T,}$ are considered. That is

${\displaystyle \eta (x,t)=a(x)\,\sin(\omega \,t-\phi (x))={\tfrac {1}{2}}\,H(x)\,\sin(\omega \,t-\phi (x)),}$

where ${\displaystyle a}$ is the amplitude, ${\displaystyle H=2a}$ is the wave height, ${\displaystyle \omega =2\pi /T}$ is the angular frequency and ${\displaystyle \phi (x)}$ is the wave phase. Consequently, also ${\displaystyle F}$ in Eq. (4) has to be a sine wave, e.g. ${\displaystyle F=\beta \sin(\omega t-\phi (x))}$ with ${\displaystyle \beta }$ a constant.

Applying these forms of ${\displaystyle \eta }$ and ${\displaystyle F}$ in Eq. (4) gives:

${\displaystyle H\,b^{\tfrac {1}{2}}\,h^{\tfrac {1}{4}}=\beta ,}$

which is Green's law.

Flow velocity

The horizontal flow velocity in the ${\displaystyle x}$-direction follows directly from substituting the solution for the surface elevation ${\displaystyle \eta (x,t)}$ from Eq. (4) into the expression for ${\displaystyle u(x,t)}$ in Eq. (1):[10]

${\displaystyle {\begin{aligned}u(x,t)&=g^{\tfrac {1}{2}}\;b^{-{\tfrac {1}{2}}}(x)\;h^{-{\tfrac {3}{4}}}(x)\;F\left(t-\int _{x_{0}}^{x}{\frac {1}{\sqrt {g\,h(s)}}}\;\mathrm {d} s\right)\\&+\mu \,{\tfrac {1}{2}}\,g\,b^{-{\tfrac {1}{2}}}(x)\;h^{-{\tfrac {1}{4}}}(x)\;{\frac {(b\,{\sqrt {h}})'}{b\,{\sqrt {h}}}}\,\Phi (x,t)+{\frac {Q}{b(x)\,h(x)}}+{\mathcal {O}}\left(\mu ^{2}\right),\\&\qquad {\text{with}}\qquad \Phi (x,t)=\int _{t_{0}}^{t}F\left(\sigma -\int _{x_{0}}^{x}{\frac {1}{\sqrt {g\,h(s)}}}\;\mathrm {d} s\right)\;\mathrm {d} \sigma \end{aligned}}}$

and ${\displaystyle Q}$ an additional constant discharge.

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

For sinusoidal waves with velocity amplitude ${\displaystyle V,}$ the flow velocities shoal to leading order as[8]

${\displaystyle V\,b^{\tfrac {1}{2}}\,h^{\tfrac {3}{4}}={\text{constant}}.}$

This could have been anticipated since for a horizontal bed ${\displaystyle V={\sqrt {g\,h}}\,(a/h)={\sqrt {g/h}}\,a}$ with ${\displaystyle a}$ the wave amplitude.