Suppose, for the sake of argument, that the wave train is of uniform transverse width . Consider a fixed line drawn downstream of the ship perpendicular to its path. The rate at which the length of the train is increasing ahead of this line is . Therefore, the rate at which the energy of the train is increasing ahead of the line is , where is the typical amplitude of the transverse waves in the train. As is well known (Fitzpatrick 2013), wave energy travels at the group velocity, rather than the phase velocity. Thus, the energy flux per unit width of a propagating gravity wave is simply . Wave energy consequently crosses our fixed line in the direction of the ship's motion at the rate . Finally, the ship does work against the drag force, which goes to increase the energy of the train in the region ahead of our line, at the rate . Energy conservation thus yields

(11.59) |

However, because , we obtain

where use has been made of Equation (11.44). Here, is determined implicitly in terms of the ship speed via expression (11.58). Note that this expression cannot be satisfied when the speed exceeds the critical value , because gravity waves cannot propagate at speeds in excess of this value. In this situation, no transverse wave train can keep up with the ship, and the drag associated with such waves consequently disappears. In fact, we can see, from the previous formulae, that when then , and so . However, the transverse wave amplitude, , generally increases significantly as the ship speed approaches the critical value. Hence, the drag due to transverse waves actually peaks strongly at speeds just below the critical speed, before effectively falling to zero as this speed is exceeded. Consequently, it usually requires a great deal of propulsion power to force a ship to travel at speeds faster than .

In the deep water limit , Equation (11.60) reduces to

(11.61) |

At fixed wave amplitude, this expression is independent of the wavelength of the wave train, and, hence, independent of the ship's speed. This result is actually rather misleading. In fact, (at fixed wave amplitude) the drag acting on a ship traveling through deep water varies significantly with the ship's speed. We can account for this variation by incorporating the finite length of the ship into our analysis. A real ship moving through water generates a

(11.62) |

Here, is the length of the ship. Moreover, the bow lies (instantaneously) at [hence, ], and the stern at [hence, ]. For the sake of simplicity, the upward water displacement due to the bow is assumed to equal the downward displacement due to the stern. At fixed bow wave displacement, the amplitude of transverse gravity waves of wave number (chosen such that the phase velocity of the waves matches the ship's speed, ) produced by the ship is

(11.63) |

that is, the amplitude is proportional to the Fourier coefficient of the ship's vertical displacement pattern evaluated for a wave number that matches that of the wave train (Fitzpatrick 2013). Hence, (at fixed bow wave displacement) the drag produced by the transverse waves is

where the dimensionless parameter

(11.65) |

is known as the Froude number. (See Section 1.16.)

Figure 11.2 illustrates the variation of the wave drag with Froude number predicted by Equation (11.64). As we can see, if the Froude number is much less than unity, which implies that the wavelength of the wave train is much smaller than the length of the ship, then the drag is comparatively small. This is the case because the ship is extremely inefficient at driving short wavelength gravity waves. It can also be seen that the drag increases as the Froude number increases, reaching a relatively sharp maximum when , and then falls rapidly. Now, corresponds to the case in which the length of the ship is equal to half the wavelength of the wave train. In this situation, the bow and stern waves interfere constructively, leading to a particularly large amplitude wave train, and, hence, to a particularly large wave drag. The smaller peaks visible in the figure correspond to other situations in which the bow and stern waves interfere constructively. (For instance, when the length of the ship corresponds to one and a half wavelengths of the wave train.) A heavy ship with large a displacement, and limited propulsion power, generally cannot overcome the peak in the wave drag that occurs when . Such a ship is, therefore, limited to Froude numbers in the range , which implies a maximum speed of

(11.66) |

This characteristic speed is sometimes called the