Wavelength and period

Further, from Green's analysis, the wavelength ${\displaystyle \lambda }$ of the wave shortens during shoaling into shallow water, with^[4][8]

${\displaystyle {\frac {\lambda }{\sqrt {g\,h}}}={\text{constant}}}$

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

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 ${\textstyle V={\sqrt {g\,h}}\,(a/h)={\sqrt {g/h}}\,a}$ with ${\displaystyle a}$ the wave amplitude.