The wake of a duck

We all know that when a duck swims steadily on a lake, it leaves a wedge-shaped wake behind it. What’s less obvious is that the wake always has the same angle, about 39°, regardless of the size or speed of the duck, provided that the water is deep enough. The same is true of any other swimmer or ship. This phenomenon was studied by Lord Kelvin over a hundred years ago, and related problems have been of interest to shipmakers ever since.

The full explanation of where the 39° comes from needs quite a lot of physics, and is in fact given in the third year of the maths course at Cambridge! However, the basic idea comes down to a balance between how fast the duck is travelling and how fast surface waves travel. The duck splashes around the water around it and the disturbances travel outwards as waves. If the duck were sitting still, those disturbances would just go outwards as concentric circles. But if the duck is moving, then the centres of these circles move along with the duck:

Figure 1: The wake behind a duck is created by all of the waves generated by the duck at its previous positions interacting with each other.

The big black dot on the right is the current position of the duck. The centres of the circles are previous positions of the duck. For example, the red circle shows the position of waves that were generated when the duck was at the red point. The radius of the circles is equal to the wave speed, multiplied by the time since the duck was at that position.

However, water waves do not have a fixed wave speed, and in this way they are unlike electromagnetic waves in vacuo. Light travelling through a vacuum has a constant wave speed c related to the frequency f and wavelength λ through the equation c = . By convention, it is more typical to work with the angular wavenumber k = 2π/λ and angular frequency ω = 2πf, and rewrite the above formula as ω = ck. (We will now drop the adjective ‘angular’; when we say ‘frequency’ we shall mean ω, not f.) The speed of light is a constant and the frequency is proportional to the wavenumber.

This is not the case for water waves, and it can be shown that the frequency actually depends on the wavenumber according to the formula ω = (gk)1/2, where g is the gravitational acceleration (approximately 10 m s-2). We then define two different wave speeds. The phase velocity is defined as cp = ω/k = (g/k)1/2, and is the speed at which wave crests travel. On the other hand, the group velocity is defined as the derivative cg = dω/dk = (g/k)1/2/2, and is the speed at which wave packets travel. Note that for EM waves the phase and group velocities are the same, but for water waves the group velocity is equal to half the phase velocity.

Figure 2: Phase and group velocity.

This property of non-constant wave speeds is called dispersion and we say that water waves are dispersive, while EM waves in vacuo are non-dispersive. Famously, however, EM waves are dispersive in certain media such as glass, which is why a glass prism can split a beam of white light up into its components of different frequencies and wavenumbers (figure 3).

Figure 3: Light dispersion of a mercury-vapor lamp with a prism made of flint glass. (Credit: Wikipedia user D-Kuru)

I have been a little sloppy with language: the phase and group velocities are velocities and are vectors, not scalars. The direction of the vector is simply the direction of propagation of the wave: radially outwards from the point of generation.

Going back to the duck problem, in order to calculate the radii of the circles in figure 1 we need to identify the wave speed, and therefore the wavenumber, that we care about. The key idea is that the waves at the edge of the wake should move with the same phase velocity as the duck. Otherwise, they would not follow the duck and the wake would not be steady. The radius of the circle, on the other hand, is equal to the group velocity multipled by the elapsed time.

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