Green's law

In

parallel to each other (and the coast), it states:

${\displaystyle H_{1}\,\cdot \,{\sqrt[{4}]{h_{1}}}=H_{2}\,\cdot \,{\sqrt[{4}]{h_{2}}}}$   or   ${\displaystyle \left(H_{1}\right)^{4}\,\cdot \,h_{1}=\left(H_{2}\right)^{4}\,\cdot \,h_{2},}$

where ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the wave heights at two different locations – 1 and 2 respectively – where the wave passes, and ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ are the mean water depths at the same two locations.

Green's law is often used in

Description

According to this law, which is based on linearized shallow water equations, the spatial variations of the wave height ${\displaystyle H}$ (twice the amplitude ${\displaystyle a}$ for

in water of mean depth ${\displaystyle h}$ and width ${\displaystyle b}$ (in case of an open channel) satisfy^[4][5]

${\displaystyle H\,{\sqrt {b}}\,{\sqrt[{4}]{h}}={\text{constant}},}$

where ${\displaystyle {\sqrt[{4}]{h}}}$ is the

of ${\displaystyle h.}$ Consequently, when considering two cross sections of an open channel, labeled 1 and 2, the wave height in section 2 is:

${\displaystyle H_{2}={\sqrt {\frac {b_{1}}{b_{2}}}}\;{\sqrt[{4}]{\frac {h_{1}}{h_{2}}}}\;H_{1},}$

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 ${\displaystyle b}$ a constant, say 1 metre or yard.

For refracting long waves in the ocean or near the coast, the width ${\displaystyle b}$ can be interpreted as the distance between wave rays. The rays (and the changes in spacing between them) follow from the geometrical optics approximation to the linear wave propagation.^[6] In case of straight parallel depth contours this simplifies to the use of Snell's law.^[7]

Green published his results in 1838,

${\displaystyle b\,{\sqrt {gh}}\,{\tfrac {1}{8}}\rho gH^{2}={\text{constant}},}$

where ${\displaystyle {\sqrt {gh}}}$ is the

phase speed

in shallow water), ${\displaystyle {\tfrac {1}{8}}\rho gH^{2}={\tfrac {1}{2}}\rho ga^{2}}$ is the mean wave energy density integrated over depth and per unit of horizontal area, ${\displaystyle g}$ is the gravitational acceleration and ${\displaystyle \rho }$ is the water density.

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

) 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 ${\displaystyle h}$ and width ${\displaystyle b}$ along the path of wave propagation.[9]

Wave equation for an open channel

Starting point are the linearized

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

) ${\displaystyle \tau }$ 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

${\displaystyle c={\sqrt {gh}}.}$ 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

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

) 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

${\displaystyle T,}$ 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

. 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

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

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.

Notes

References

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