Let us now make a detailed investigation of the wake pattern generated behind a ship as it travels over a body of water, taking
into account obliquely propagating gravity waves, in addition to transverse waves. For the sake of simplicity, the finite length of the ship
is neglected in the following analysis. In other words, the ship is treated as a point source of gravity waves. Consider Figure 9.6. This shows
a plane gravity wave generated on the surface of the water by a moving ship. The water surface corresponds to the - plane. The
ship is traveling along the -axis, in the negative -direction, at the constant speed . Suppose that the ship's bow is initially at point ,
and has moved to point after a time interval . The only type of gravity wave that is continuously excited by the passage of the ship
is one that maintains a constant phase relation with respect to its bow. In fact, as we have already mentioned, the bow should always correspond to a wave maximum.
An oblique wavefront associated with such a wave is shown in the figure. Here, the wavefront , which initially passes through the bow at point , has moved
to after a time interval , such that it again passes through the bow at point . The wavefront propagates at the
phase velocity, . It follows that, in the right-angled triangle , the sides and are of lengths and , respectively,
This, therefore, is the condition that must be satisfied in order for an obliquely propagating gravity wave to maintain a constant phase relation with respect to the ship.
An oblique plane wave generated on the surface of the water by a moving ship.
In shallow water, all gravity waves propagate at the same phase velocity. That is,
where is the water depth. Hence, Equation (9.308) yields
This equation can only be satisfied when
In other words, the ship must be traveling faster than the critical speed
Moreover, if this is the case then there is only one value of that satisfies Equation (9.310). This implies the scenario illustrated in
Figure 9.7. Here, the ship is instantaneously at , and the wave maxima that it previously generated—which all
propagate obliquely, subtending a fixed angle with the -axis—have interfered constructively
to produce a single strong wave maximum . In fact, the wave maxima generated when the ship was at have travelled to
and , the wave maxima generated when the ship was at have travelled to and , et cetera. We conclude
that a ship traveling over shallow water produces a V-shaped wake whose semi-angle, , is determined by the ship's speed.
Indeed, as is apparent from Equation (9.310), the faster the ship travels over the water, the smaller the angle becomes.
Shallow water wakes are especially dangerous to other vessels, and particularly destructive of the coastline, because all
of the wave energy produced by the ship is concentrated into a single large wave maximum. The wake
contains no transverse waves, because, as we have already mentioned, such waves cannot keep up with a ship
traveling faster than the critical speed
A shallow water wake.
Let us now discuss the wake generated by a ship traveling over deep water. In this case, the phase velocity of
gravity waves is
. Thus, Equation (9.308) yields
It follows that in deep water any obliquely propagating gravity wave whose wavenumber exceeds the critical value
can keep up with the ship, as long as its direction of propagation is such that Equation (9.312) is satisfied. In other words, the
ship continuously excites gravity waves with a wide range of different wavenumbers and propagation directions. The
wake is essentially the interference pattern generated by these waves. As described in Section 9.2, an interference maximum
generated by the superposition of plane waves with a range of different wavenumbers propagates at the group velocity, .
Furthermore, as we have already seen, the group velocity of deep water gravity waves is half their phase velocity; that is, .
Formation of an interference maximum in a deep water wake.
Consider Figure 9.8. The curve corresponds to a particular interference maximum in the wake. Here, is
the ship's instantaneous position. Consider a point on this curve. Let and be the coordinates of this point, relative to the
ship. The interference maximum at is part of the plane wavefront emitted some time earlier, when the ship was at point .
Let be the angle subtended between this wavefront and the -axis.
Because interference maxima propagate at the group velocity, the distance is equal to . The distance is
equal to . Simple trigonometry reveals that
because is the tangent to the curve —that is, the curve —at point . It follows from Equation (9.312), and the fact that , that
. The previous three equations can be combined to produce
which reduces to
This expression can be solved to give
where is a constant. Hence, the locus of our interference maximum is determined parametrically by
Here, the angle ranges
from to .
The curve specified by the previous equations is plotted in Figure 9.9. As usual, is the instantaneous
position of the ship. It can be seen that the interference maximum essentially consists of the transverse maximum , and the two radial maxima and . As is readily demonstrated, point , which corresponds to
, lies at , . Moreover, the
two cusps, and , which correspond to
, lie at
Locus of an interference maximum in a deep water wake.
The complete interference pattern that constitutes the wake is constructed out of many different wave maximum curves of the form shown in Figure 9.9, corresponding to
many different values of the parameter . However, these values must be chosen such that the wavelength of the pattern
along the -axis corresponds to the wavelength
of transverse (i.e.,
) gravity waves whose
phase velocity matches the speed of the ship. This implies that
, where is a positive integer. A complete deep water wake pattern is
shown in Figure 9.10. The pattern, which is made up of interlocking transverse and radial wave maxima, fills a wedge-shaped region—known
as a Kelvin wedge—whose semi-angle takes the value
. This angle is independent of the ship's speed. Finally, our initial assumption that the gravity waves that form
the wake are all deep water waves is valid provided
, which implies that
In other words, the ship must travel at a speed that is much less than the critical speed
. This explains why the
wake contains transverse wave maxima.
A deep water wake.