Ship Wakes

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.

In shallow water, all gravity waves propagate at the same phase velocity: that is,

(11.68) |

where is the water depth. Hence, Equation (11.67) yields

This equation can only be satisfied when

(11.70) |

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 (11.69). This implies the scenario illustrated in Figure 11.4. 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 (11.69), 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. Note, finally, that 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 .

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 (11.67) yields

It follows that in deep water any obliquely propagating gravity wave whose wave number exceeds the critical value

(11.72) |

can keep up with the ship, as long as its direction of propagation is such that Equation (11.71) is satisfied. In other words, the ship continuously excites gravity waves with a wide range of different wave numbers and propagation directions. The wake is essentially the interference pattern generated by these waves. As is well known (Fitzpatrick 2013), an interference maximum generated by the superposition of plane waves with a range of different wave numbers 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, .

Consider Figure 11.5. 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 . Of course, the distance is equal to . Simple trigonometry reveals that

(11.73) | ||

(11.74) |

Moreover,

(11.75) |

because is the tangent to the curve --that is, the curve --at point . It follows from Equation (11.71), and the fact that , that

(11.76) | ||

(11.77) |

where . The previous three equations can be combined to produce

(11.78) |

which reduces to

(11.79) |

This expression can be solved to give

(11.80) |

where is a constant. Hence, the locus of our interference maximum is determined parametrically by

(11.81) | ||

(11.82) |

Here, the angle ranges from to . The curve specified by the previous equations is plotted in Figure 11.6. 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 easily demonstrated, point , which corresponds to , lies at , . Moreover, the two cusps, and , which correspond to , lie at , .

The complete interference pattern that constitutes the wake is constructed out of many different wave maximum curves of the form shown in Figure 11.6, 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 11.7. This 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

(11.83) |

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.