Elements of Physical Oceanography 2.4 Surface Gravity Waves 23
Part A | 2.4
Intermediate
Shallow
Distance, x
Distance, x
a)
b)
0
–H
–H
0
Deep
Depth, z
0
Fig. 2.12a,b Kinematics of waves
propagating left to right in (top)
deep; (middle) intermediate depth;
and (bottom) shallow water –
(a) instantaneous snapshots of
the water parcel streamlines and
(b) trajectories of selected water
parcels over a full wave cycle
(after [2.8])
Crest
Trough
Particle orbits in
shallow water
Particle orbits in
deep water
Fig. 2.13 Deep water waves propagating toward the left into shallow water. Note how the waves feel the bottom and
water parcel orbits are squashed as the wave propagates into even shallower water
frequencies and wave numbers such that ! 2 > ! 1 and
k 2 > k 1 .
The amplitude of the resultant wave (shown in
Fig. 2.14) is modulated according to
Á D a cos
Ä k
2
.x c g t/
cos
k.x ct/
;
(2.4a)
where
k D
k 1 C k 2
2
; c D
w
k
and ! D
! 1 C ! 2
2
: (2.4b)
Note that on one hand, the cosŒ  term on the right
in (2.4) represents a wave with intermediate frequency
and wavelength and therefore is nearly indistinguishable from the parent waves. On the other hand, the cosŒ 
c p
c p
c g
Fig. 2.14 The wave envelope – resulting from the superposition of
a pair of surface gravity waves (with nearly identical phase speeds
c p ) – travels at the wave group velocity c g
term on the left represents a wave with much greater L
(smaller k) that is traveling at the speed c g given by
!
k
D
! 2 ! 1
k 2 k 1
D c g :