(2) For a monochromatic wave, the wave phase function

Ω (*x*,*y*,*t*) ' (*k *cosθ % *k *sinθ & ω*t*)

(II-3-3)

can be used to define the wave number vector P by

k

P

(II-3-4)

(3) Since P is a vector, one can take the curl of P

k

k

P

L*k*'0

(II-3-5)

which is zero because P by definition is the gradient of a scaler and the curl of a gradient is zero.

k

(4) Substituting the components of P Equation II-3-5 yields

k,

M (*k *sin θ)

M (*k *cos θ)

(II-3-6)

'0

&

M*x*

M*y*

(5) Since the problem is defined to have straight and parallel contours, derivatives in the *y *direction are

zero and using the dispersion relation linking *k *and *C *(and noting that *k =2π/CT *and wave period is constant)

Equation II-3-6 simplifies to

(II-3-7)

'0

or

sin θ

' *constant*

(II-3-8)

(6) Let *C*0 be the deepwater celerity of the wave. In deep water, sin *(θ*0)/c0 is known if the angle of the

wave is known, so Equation II-3-8 yields

sin θ0

sin θ

(II-3-9)

'

along a ray. This identity is the equivalent of Snell's law in optics. The equation can be readily solved by

starting with a point on the wave crest in deep water and incrementally estimating the change in C because

of changes in depth. The direction s of wave travel is then estimated plotting the path traced by the ray. The

size of increment is selected to provide a smooth estimate of the ray.

(7) The wave-height variation along the ray can be estimated by considering two rays closely spaced

together (Figure II-3-5). In deep water, the energy flux (*EC*n), which is also *EC*g, across the wave crest

distance b0 can be estimated by *(ECn)*0b0. Considering a location a short distance along the ray, the energy

flux is *(ECn)*1b1. Since the rays are orthogonal to the wave crest, there should be no transfer of energy across

the rays and conservation principles give

Estimation of Nearshore Waves

II-3-9

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