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 *K*R and *K*RKS

in terms of initial deepwater wave angle and *d/gT*2. Although the bathymetry of most coasts is more

complicated than this, these procedures provide a quick way of estimating approximate wave approach angles.

(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 M*k*

1 M*C*

(II-3-15)

'

'&

M*s*

and the ray path defined by

(II-3-16)

'*C*

' *C *cos θ

(II-3-17)

(II-3-18)

' *C *sin θ

(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

(II-3-19)

β

where β = *b/b*0 then

%*p*

(II-3-20)

% *q*β ' 0

2

with

cos θ M *C*

sin θ M *C*

(II-3-21)

&

Estimation of Nearshore Waves

II-3-11

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