Page 35 - kursus eBook

2!D 1/2
(
for KD = >> 1, # = gK)
"
(10.4.34 a, b)
c = g / K) 1/2
(

and the wave frequency and phase speed becomes independent o f the water depth.
The wave amplitude for the free wave (p =0) is arbitrary in this linear theory and it
a
is convenient to choose the x axis to lie in the direction of the wave vector so that the y
wavenumber is zero. In that case we can write the solution as,

! = ! cos(kx " # t)
0 o
(10.4.35 a, b)
% # ( coshk(z + D)
$ = ' k ) ! sin(kx " # t) sinhkD
0
*
&
0
o
where (10.3. 35 b) is obtained from (10.4.10 a or b with p =0.
a
The velocities obtained from (10.4.35 b) are

!" coshk(z + D)
u = = # $ cos(kx % # t)
!x o o sinhkD o
(10.4.36 a, b)
!" sinhk(z + D)
w = = # $ sin(kx % # t)
!z 0 0 sinhkD o

Since we have chosen the propagation direction to be the x axis the motion is two

dimensional and so it is straight forward to construct the stream function for the motion
since it is incompressible,

"# "#
u = ! , w = (10.4.37 a, b)
"z "x

and it follows from (10.4.36 that,

Chapter 10 29