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Green's law

## Summary

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, is a conservation law describing the evolution of non-breaking, surface gravity waves propagating in shallow water of gradually varying depth and width. In its simplest form, for wavefronts and depth contours 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 coastal engineering for the modelling of long shoaling waves on a beach, with "long" meaning wavelengths in excess of about twenty times the mean water depth.[1] Tsunamis 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, nonlinear effects become important and Green's law no longer applies.[2][3]

## Description

Convergence of wave rays (reduction of width ${\displaystyle b}$) at Mavericks, California, producing high surfing waves. The red lines are the wave rays; the blue lines are the wavefronts. 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 ${\displaystyle h}$).

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 sine waves, equal to the amplitude for a solitary wave) for travelling waves 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 fourth root 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,[8] 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:[4][5]

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

where ${\displaystyle {\sqrt {gh}}}$ is the group speed (equal to 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 ray. The oscillation 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 ${\displaystyle h}$ and width ${\displaystyle b}$ along the path of wave propagation.[9]

## Notes

1. ^ Dean & Dalrymple (1991, §3.4)
2. ^ Synolakis & Skjelbreia (1993)
3. ^ Synolakis (1991)
4. ^ a b c Lamb (1993, §185)
5. ^ a b Dean & Dalrymple (1991, §5.3)
6. ^ Satake (2002)
7. ^ Dean & Dalrymple (1991, §4.8.2)
8. ^ a b c Green (1838)
9. ^ The derivation presented below is according to the line of reasoning as used by Lamb (1993, §169 & §185).
10. ^ Didenkulova, Pelinovsky & Soomere (2009)

## References

### Green

• Green, G. (1838), "On the motion of waves in a variable canal of small depth and width", Transactions of the Cambridge Philosophical Society, 6: 457–462, Bibcode:1838TCaPS...6..457G

### Others

• Craik, A. D. D. (2004), "The origins of water wave theory", Annual Review of Fluid Mechanics, 36: 1–28, Bibcode:2004AnRFM..36....1C, doi:10.1146/annurev.fluid.36.050802.122118
• Dean, R. G.; Dalrymple, R. A. (1991), Water wave mechanics for engineers and scientists, Advanced Series on Ocean Engineering, 2, World Scientific, ISBN 978-981-02-0420-4
• Didenkulova, I.; Pelinovsky, E.; Soomere, T. (2009), "Long surface wave dynamics along a convex bottom", Journal of Geophysical Research, 114 (C7): C07006, 14 pp., arXiv:0804.4369, Bibcode:2009JGRC..114.7006D, doi:10.1029/2008JC005027
• Lamb, H. (1993), Hydrodynamics (6th ed.), Dover, ISBN 0-486-60256-7
• Satake, K. (2002), "28 – Tsunamis", in Lee, W. H. K.; Kanamori, H.; Jennings, P. C.; Kisslinger, C. (eds.), International Handbook of Earthquake and Engineering Seismology, International Geophysics, 81, Part A, Academic Press, pp. 437–451, ISBN 978-0-12-440652-0
• Synolakis, C. E. (1991), "Tsunami runup on steep slopes: How good linear theory really is", Natural Hazards, 4 (2): 221–234, doi:10.1007/BF00162789
• Synolakis, C. E.; Skjelbreia, J. E. (1993), "Evolution of maximum amplitude of solitary waves on plane beaches", Journal of Waterway, Port, Coastal, and Ocean Engineering, 119 (3): 323–342, doi:10.1061/(ASCE)0733-950X(1993)119:3(323)