86
3 An Introduction to Surface Waves
The circular (or elliptical) orbits of water particles are only an approximate
representation of real particle motion. There is always a small net component
of forward motion, particularly in waves of large amplitude. The orbits are
not closed and the water particle, whilst in the crests, moves slightly further
forward then it moves backward while in the troughs. The small net forward
displacement of water in the direction of wave travel is known as wave drift
(see Fig. 3.5). Dots denote the position of water particles one wave period
apart.
3.4.3 Phase and Group Velocities
It is important to distinguish two different wave velocities associated with wave
motion, namely phase velocity and group velocity. The first, the phase
velocity, is more obvious and can be ascertained from the time taken for one
wavelength to pass a fixed point. In particular, for monochromatic wave train
with the period T we have:
L
c=-. T
(3.4)
So velocity, C, is a measure of the speed at which a particular wave phase (for
example the wave crest) is propagating on the sea surface. Combining Eqs.
(3.3), (3.4) and (3.1) we obtain:
L
w
C = - =-.
T
k
(3.5)
Basically, if we know any two of the variables in Eq. (3.5), we can calculate the
third. However, the combination of wavelengths, L, and wave period, T, can
not be arbitrary. The spatial wave characteristics, L or k, and temporal wave
characteristic, T or w, have to satisfy some relationship in order to represent
the surface wave. This relationship is called the dispersion relation. It
expresses the separation of waves by virtue of their differing rates of travel;
greater wavelengths correspond to longer periods and shorter wavelengths are
associated with shorter periods. We will develop the dispersion relation in
Sect. 4.2.2. Here we only mention that in deep water, the frequency, w, and
wave number, k, should satisfy the following dispersion relation:
G
(hl
w = ygk or T = Vg-g-'
(3.6)
in which 9 is gravitational acceleration, 9 = 9.81 m/s2. Combining Eq. (3.5)
and (3.6) we find that in deep water the phase velocity Cis:
(3.7)
3 An Introduction to Surface Waves
The circular (or elliptical) orbits of water particles are only an approximate
representation of real particle motion. There is always a small net component
of forward motion, particularly in waves of large amplitude. The orbits are
not closed and the water particle, whilst in the crests, moves slightly further
forward then it moves backward while in the troughs. The small net forward
displacement of water in the direction of wave travel is known as wave drift
(see Fig. 3.5). Dots denote the position of water particles one wave period
apart.
3.4.3 Phase and Group Velocities
It is important to distinguish two different wave velocities associated with wave
motion, namely phase velocity and group velocity. The first, the phase
velocity, is more obvious and can be ascertained from the time taken for one
wavelength to pass a fixed point. In particular, for monochromatic wave train
with the period T we have:
L
c=-. T
(3.4)
So velocity, C, is a measure of the speed at which a particular wave phase (for
example the wave crest) is propagating on the sea surface. Combining Eqs.
(3.3), (3.4) and (3.1) we obtain:
L
w
C = - =-.
T
k
(3.5)
Basically, if we know any two of the variables in Eq. (3.5), we can calculate the
third. However, the combination of wavelengths, L, and wave period, T, can
not be arbitrary. The spatial wave characteristics, L or k, and temporal wave
characteristic, T or w, have to satisfy some relationship in order to represent
the surface wave. This relationship is called the dispersion relation. It
expresses the separation of waves by virtue of their differing rates of travel;
greater wavelengths correspond to longer periods and shorter wavelengths are
associated with shorter periods. We will develop the dispersion relation in
Sect. 4.2.2. Here we only mention that in deep water, the frequency, w, and
wave number, k, should satisfy the following dispersion relation:
G
(hl
w = ygk or T = Vg-g-'
(3.6)
in which 9 is gravitational acceleration, 9 = 9.81 m/s2. Combining Eq. (3.5)
and (3.6) we find that in deep water the phase velocity Cis:
(3.7)