EM 1110-2-1100 (Part II)
30 Apr 02
(2) For a monochromatic wave, the wave phase function
Ω (x,y,t) ' (k cosθ % k sinθ & ωt)
(II-3-3)
can be used to define the wave number vector P by
k
P
k'LΩ
(II-3-4)
(3) Since P is a vector, one can take the curl of P
k
k
P
Lk'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
&
Mx
My
(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
d sinθ
(II-3-7)
'0
dx C
or
sin θ
' constant
(II-3-8)
C
(6) Let C0 be the deepwater celerity of the wave. In deep water, sin (θ0)/c0 is known if the angle of the
wave is known, so Equation II-3-8 yields
sin θ0
sin θ
(II-3-9)
'
C0
C
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 (ECn), which is also ECg, 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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