Hydraulic Structures: Fourth Edition

From equations (14.13) and (14.14), respectively, for candc 0 ,

c^2 c^20 

L

L

0

tanh

2

L

πd

.

After substitution for L/L 0 from equation (14.29) and rearranging,



c

c

0

ln

1/2

. (14.30)

A wave approaching a shore of uniform slope is shown in Fig. 14.7. In deep
water, the wave crests form an angle
0 with the bed contours. The crests
are stretched and swung around so that they make an angle
as they
approach the shore, as shown in Fig. 14.7. The celerity cdepends on the
local depth and wavelength; it is possible to relate it to c 0 by applying
Snell’s law of refraction:

c/c 0 sin
/sin
 0. (14.31)

Consider a crest length b 0 between two adjacent rays in deep water. If the
normal distance between the chosen rays locally is b, then

b/b 0 cos
/cos
 0. (14.32)

For shoaling water, Snell’s law shows that

0 ; the rays tend to diverge as
the wave moves shorewards. (b 0 /b)1/2is called the refraction coefficient, Kr.
Consider the energy flux, R, normal to the wave crests:



1

8

^ gH^2 bCg

1

8

^ gH^20 b 0 Cg0. (14.33)

1 c/c 0



1 c/c 0

2 πd



L 0

WAVES APPROACHING A SHORE 587

Fig. 14.7 Waves approaching the shore