Transformation of Shallow-Water Waves The Certificate Program in Tsunami Science and Preparedness. Held at the University of Washington in Seattle, Washington.

Transformation of Shallow-Water Waves

TThe Certificate Program in Tsunami Science and Preparednesshe Certificate Program in Tsunami Science and Preparedness..

HeldHeld at the University of Washington in Seattle, Washington (USA) at the University of Washington in Seattle, Washington (USA) , , DDuring July 21- August 2, 2007.uring July 21- August 2, 2007.

The transformation of waves across the sea is a complex problem, including the processes of refraction, shoaling, breaking, energy dissipation by bottom friction, and reflection. As waves pass from deep water onto shallow water, the waves become highly nonlinear.



Wave energy is dissipated due to breaking, but energy is also transferred to both higher and lower frequencies in the wave spectrum, and the spectral shape becomes flat



Energy losses due to bottom friction are usually negligible in wave transformation across sandy beach profiles, but may be significant over shallow, rough reef flats where the bottom friction coefficient may be an order of magnitude larger than for a sandy bed.



Wave motion: Speed and energyWave motion: Speed and energyThere are mathematical relationships linking the There are mathematical relationships linking the characteristics of wavelength (L), wave period (T) and characteristics of wavelength (L), wave period (T) and wave height (H) to wave speed and to wave energy.wave height (H) to wave speed and to wave energy.Wave speed Wave speed speed = distance/time speed = distance/timeC = L/T (time one wavelength takes to pass a certain C = L/T (time one wavelength takes to pass a certain point)point)Wave speed can also be represented by the equation:Wave speed can also be represented by the equation:

Where:Where:g = acceleration due to gravityg = acceleration due to gravityL = WavelengthL = Wavelengthtanh = hyperbolic tangenttanh = hyperbolic tangenth = water depthh = water depth

)2

tanh(2 L

hgL

T

LC



Wave motion: Wave motion: Speed Speed ApproximationsApproximations

1)1) For h > L/2 (L < 2h) For h > L/2 (L < 2h) Deep-water waves Deep-water waves

If x is larger than If x is larger than , then tanh , then tanh x ~ 1 x ~ 1

tanh (2tanh (2h/L) = 1 (within 0.5%)h/L) = 1 (within 0.5%)

)2

tanh(2 L

hgL

T

LC

The The wavelengthwavelength is the is the onlyonly variable affecting the wave speed! variable affecting the wave speed!

2

gL

T

LC or L25.1 m/s

π

gT

T

LC

2 T56.1or m/s



Wave motion: Speed Wave motion: Speed ApproximationsApproximations

1)1) For h > L/2 For h > L/2 Deep-water waves Deep-water waves

Circular motion of particlesCircular motion of particlesExponential decrease of path with depth!Exponential decrease of path with depth!




2)2) For h < L/20 (L > 20h) For h < L/20 (L > 20h) Shallow-water Shallow-water waveswaves

If x is small, then tanh x ~ x If x is small, then tanh x ~ x

tanh (2tanh (2h/L) = (2h/L) = (2h/L) (within 3%)h/L) (within 3%))

2tanh(

2 L

hgL

T

LC

Or: C= L/T = (gh)Or: C= L/T = (gh)1/2 1/2 L = 3.13 (h) L = 3.13 (h)1/21/2.T.TThe water depth is the only variable affecting the wave speed!The water depth is the only variable affecting the wave speed!

or h1.3ghT

LC m/s




2)2) For h < L/20 For h < L/20 Shallow-water waves Shallow-water waves

Elliptical motion of Elliptical motion of particlesparticles

Exponential decrease of Exponential decrease of verticalvertical path with path with depthdepth

Not of horizontal path!Not of horizontal path!



A

B





Wave progression toward shore





Wave Diffraction- Propagation of a wave around an obstacleWave Reflection- a progressive wave striking a vertical barrier and being reflected in the direction from where they came



Transformation of Shallow-water Waves

• As waves enter shallow water

– Wavelength shortens

– Height increases

– Speed decreases



Transformation of Shallow-Water Waves The Certificate Program in Tsunami Science and Preparedness. Held at the University of Washington in Seattle, Washington.

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wave speed speed

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wave energy

wave transformation