THEORY OF SURFACE WAVES 
Waves of Infinitely Small Amplitude 
Equations required for subsequent develop- 
ments will be summarized here. Derivations and 
discussions of these equations are not repeated 
since they are readily available (Lamb, 1932, 
Sverdrup et al., 1942). In water of constant 
depth the wave velocity can be represented by 
means of the equation of classical hydrodynamics: 
C= © tanh ah (1) 
where h is the depth to the bottom and L is the 
wave length. 
Defining deep water waves and shallow water 
waves as: 
Deep water waves: 
ile 
h>5h 
Shallow water waves: (2) 
1 
h<5ph 
equation (1) becomes with sufficient accuracy for 
these special cases: 
Deep water waves: 
oat = (3a) 
Shallow water waves: 
C?=gh (3b) 
This paper deals only with deep water waves’ 
for which length, period, and velocity are inter- 
related as follows: 
a WE a 
=a Lt-£r 
2a 
a iar (4) 
Ps T1=0 
The water particles move in circles, the radii of 
which decrease exponentially with depth according 
to the relation: 
777333 O - 48 - 2 
Radius of particle orbit 
= 5 Het» (5a) 
where z is taken positive upwards; consequently 
the particle velocity is uniform and has the 
scalar value 
Particle velocity 
— "PF ,2n2/L 
= qe (5b) 
The mean energy per unit area of a wave equals 
(Lamb, 1932, p. 370) 
= Sent ; 
= pga” (6) 
The rate at which this energy is changed by 
dissipation equals (Lamb, 19382, p. 624) 
R,=—2pkh?e?C? (7) 
With every wave there is associated a flow of 
energy in the direction of propagation of the wave. 
Let V be the velocity at which energy is trans- 
mitted. Then VX, the rate at which energy 
flows across a vertical plane of unit width, equals 
0 
VE= {pu dz 
where p is pressure and wu is the horizontal com- 
ponent of the particle velocity. For deep water 
waves (Lamb, 1932, p. 383) 
VE=sega?C sin? k(x— Ct) (8a) 
with a mean value 
1 EC 
Va goga' C= oy (8b) 
Equation (8b) can be interpreted to mean that 
the entire energy is propagated at half the wave 
velocity or half the energy at full wave velocity. 
The significance of these interpretations will be 
discussed later (p. 6). 
