Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 76

Incident
progressive wave

Standing wave =
Incident wave + Reflected wave

λ 2

N N N

Reflected wave from barrier

Figure 3.11 A progressive sine wave propagating from left to right is reflected at the barrier
to produce a second wave of equal amplitude moving in the opposite direction. The two
waves combine to produce a standing wave pattern with nodes (N) at intervals of l/2. The
nodes are points of zero surface displacement but maximum current amplitude. Notice that
the first node is located at a distance l/4 from the barrier.

Since does not depend on z, it follows that the horizontal velocity u must be the

same at all depths. Differentiating the first of these equations with respect to x and
the second with respect to t, we can eliminate u to give the long wave equation:

2 2

@

c 2 @

1

c 2

x 2 ¼ t 2 ;

¼ gh : ð 3 : 58 Þ

@ @

g p which is independent
of the wave frequency and wavelength. While the general solution of Equation (3.58)
is any function varying as (x

¼

This equation describes waves travelling at a speed of c

ct), we are mainly concerned with harmonic solutions

such as:

¼ A 0 sin ð kx ot Þ ð 3 : 59 Þ

where the wave number k ¼ 2p/l, the angular frequency o ¼ 2p/T p , phase velocity

c signs in (3.59)

correspond to a sinusoidal wave travelling in the positive and negative x directions
respectively. The particle velocity u is found by substituting for

¼ o/k ¼ l/T p and A 0 is the wave amplitude. The - and þ

in either of the

Equations (3.57) and integrating to give:

gk
o A 0 sin

g
c A 0 sin

u f ¼ ð kx ot Þ¼ ð kx ot Þ

ð 3 : 60 Þ

g
c A 0 sin

u b ¼ ð kx þ ot Þ

for the forward and backward travelling waves respectively. The general solution for
waves travelling in the x direction along a channel of uniform depth is a combination
of a forward and a backwards travelling wave with amplitudes A f and A b respectively:

:

g
c

¼

u u f þ u b ¼ A f sin ð kx ot Þ A b sin ð kx þ ot Þ ð 3 : 61 Þ

As an example which will prove useful in our discussion of tidal waves in shelf seas,
consider the case of a wave approaching a barrier as shown in Fig. 3.11 . If the