EM 1110-2-1100 (Part II)
30 Apr 02
by noting that ray 2 is essentially ray 1 shifted downcoast. For straight and parallel contours, Figure II-3-6
is a solution nomogram. This is automated in the ACES program (Leenknecht, Szuwalski, and Sherlock
1992) and the program NMLONG (Kraus 1991). Figure II-3-6 provides the local wave angle KR and KRKS
in terms of initial deepwater wave angle and d/gT2. Although the bathymetry of most coasts is more
complicated than this, these procedures provide a quick way of estimating approximate wave approach angles.
c. Realistic bathymetry.
(1) The previous discussion was for the case of straight and parallel contours. If the topography has
variations in the y direction, then the full equation must be used. Dean and Dalrymple (1991) show the
derivation in detail for ray theory in this case. Basically, the (x,y) coordinate system is transformed to (s,n)
coordinates where s is a coordinate along a ray and n is a coordinate orthogonal to it. Algebraically, the
equation for wave angle can be derived in the ray-based coordinate system
Mθ
1 Mk
1 MC
(II-3-15)
'
'&
Ms
k Mn
C Mn
and the ray path defined by
ds
(II-3-16)
'C
dt
dx
' C cos θ
(II-3-17)
dt
dy
(II-3-18)
' C sin θ
dt
(2) Equation II-3-15 represents the discussion at the beginning of this section; the rate at which the wave
turns depends upon the local gradient in wave speed along the wave crest. Munk and Arthur's computation
for the refraction coefficient is more complicated: defining
1
1
2
Kr '
(II-3-19)
β
where β = b/b0 then
d 2β
dβ
%p
(II-3-20)
% qβ ' 0
ds
2
ds
with
cos θ M C
sin θ M C
p(s) ' &
(II-3-21)
&
C Mx
C My
Estimation of Nearshore Waves
II-3-11
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