Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 92

(iii) A third boundary condition arises from the dynamics in the form of the time
dependent linearised Bernoulli equation for irrotational flow (e.g. Kundu and
Cohen, 2008 ) which relates simultaneous changes in pressure, p, and the time
derivative

@ f/ @ t of the potential function:

p p 0

¼ @

f
@

t

gz :

Requiring that the pressure at the surface should equal the (constant) atmospheric
pressure ( p 0 ), the Bernoulli integral boils down to

1
g

@ f

@

¼

t

:
The latter two boundary conditions require that the waves are small (so called
infinitesimal waves). The solution of this set of equations for a harmonic progressive
wave travelling in the x direction with wavelength l and amplitude a takes the form:

z ¼ 0

gA 0
o

cosh ð k ð z þ h ÞÞ gA 0

o H a ð

f ¼ cos ð kx ot Þ¼ z Þ cos ð kx ot Þ ð 4 : 3 Þ

cosh kh

where o is the wave's angular frequency, and k ¼ 2p/l is the wave number. The

factor H a ð cosh kh controls the attenuation of wave motion with

depth. The phase speed c of the waves is given by:

z Þ¼ cosh ð k ð z þ h ÞÞ=

o 2
k 2 ¼

g
k tanh kh

c 2

¼ : ð 4 : 4 Þ

Velocity components for the water particles within the wave are obtained from ( 4.3 )
by differentiation:

¼ @

f gk

o A 0 cosh

ð k ð z þ h ÞÞ

u

x ¼

sin ð kx ot Þ

@ cosh kh

ð : Þ

4 5

¼ @

ð ð þ ÞÞ

f gk

o A 0 sinh

k z h

w

z ¼

cos ð kx ot Þ:

@ cosh kh

and the pressure variation p 0 induced by the wave at depth z

The surface elevation

are then:

z ¼ 0 ¼

g @

1 f

@

¼ A 0 sin ð kx ot Þ

t

ð 4 : 6 Þ

Þ¼ @

f
@

p 0 ð

z

t ¼

gA 0 H a ð z Þ sin ð kx ot Þ:

For long waves (l ≫ h), kh ≪ 1 so that cosh kh ! 1, tanh kh ! kh and H a (z) ¼ 1. The

g p . These results are pressure is then hydrostatic and the phase velocity becomes c ¼

thus consistent with our analysis of long waves in 3.6.1.