Hydraulic Structures: Fourth Edition

and at the free surface, the vertical component of velocity vmust be such that

y

∂

∂



y



y



d

d



t

 (14.8)

whereis the surface elevation (Fig. 14.1). At the crest a, the ampli-
tude. Because of the small-amplitude assumption, equation (14.8) leads to

y

∂

∂



y



y 0



∂

∂



t

. (14.9)

Only the temporal derivative of is retained since convective terms are
negligible.
At the surface, y0,p0, and hence equation (14.6) becomes



∂

∂



t

g 0 (14.10)

The wave profile as shown in Fig. 14.1 is given by

asin 2 π (^) . (14.11a)
This wave moves along the positive x-direction with a celerity c; the fre-
quency of the wave, f, is 1/T. Denoting 2π/Tas the circular frequency, ,
and 2π/Las the wavenumber k, equation (14.11a) may be written as
asin (kx t). (14.11b)
For the wave profile of equation (14.11b), the solution for the velocity
potential satisfying equation (14.4) along with boundary conditions given
by equations (14.7) and (14.9) is
 cos(kx t). (14.12)
14.2.2 Wave celerity
Substituting for from equation (14.12) into equation (14.10), the follow-
ing expression is obtained for the celerity of the wave:
c^2 
k
g
tanh(kd)
g
2

L

π

tanh. (14.13)

2 πd



L

accosh[k(yd)]



sinh(kd)

t



T

x



L

WAVE MOTION 579