A. Dispersive waves
So far in this chapter we have concentrated on sound waves. However, these are by no means the only type of wave that we might be interested in: vibration of structures like beams and plates can also be viewed in a “wavy” manner. Many of these other types of waves share a property that sound waves lack, and this property has some counter-intuitive consequences that it is interesting to be aware of. For sound waves, we have been talking about “the” speed of sound, but for most other types of waves there is no such thing: instead, waves of different wavelengths or frequencies travel at different speeds.
These are known as “dispersive waves”, for a simple reason. Suppose you make a short-lived disturbance; by clapping your hands to make a pulse of sound waves, or by tapping on a beam or plate (or indeed on the body of a guitar or violin). We know from the discussion of frequency analysis in section 2.2 that our disturbance can be expressed as a mixture of sine waves at different frequencies, and because the disturbance is short-lived we know that a wide range of frequencies must be involved. As the effects of our initial clap or tap spread out, away from the source, each of these frequency components will travel at its own speed. For the hand-clap, all the components travel at the same speed and thus remain in step with each other. If you listen at a distance you hear the sound of the clap as a short-lived pulse.
But for a tap on a beam or plate, the waves are dispersive (we will see an example shortly). The different sine-wave components travel at different speeds, so if you observe the response at a distance from the source you should not expect to see a tightly localised disturbance, more or less retaining its shape as it travels as was the case with the sound wave from your hand-clap. Instead, you will see a pattern that has been spread out (or “dispersed”).
The most familiar example of this behaviour, admittedly not one with obvious musical significance, is the behaviour of waves on the surface of water after you throw a stone in and make a splash. Look at the example in Fig. 1. After the initial splash, you see a pattern of ripples that move gradually outwards. This pattern does not consist of a single ring of disturbed water: it is spread over a range encompassing several ripples. If you look carefully you can see that the outermost ripples have a relatively longer wavelength, while the inner ones are shorter: perhaps this becomes easier to see later in the video sequence.
Figure 2 shows a computer-simulated example of wave dispersion for something of more obvious musical relevance. An impulse has been applied at the centre of a stretched string which has some bending stiffness, as all real strings do. You see a rounded step travelling outwards on both sides of the plucking point, but you also see a train of wiggles moving out ahead of that step. As the next link shows, for anything involving bending stiffness, shorter wavelengths (or high frequencies) tend to travel faster. The side link also gives the theory for water waves, and shows that the converse happens for that case: longer wavelengths travel faster.
B. Group velocity
But the story of dispersive waves has a more complicated twist. Think again about the water waves in Fig. 1. If a few seconds after throwing the stone you take a careful snapshot of the wave pattern, you can find an area of waves with a particular wavelength and measure how far they have travelled from where the stone was dropped. If you do so you will find that the waves have not travelled at the speed you would expect from the theory explained in the previous link, but at one-half this speed!
To get an inkling of what is going on, we can look at the simplest possible manifestation of the effect. If we were able to excite a single sinusoidal wave, whether in the water or on our stiff string, it would travel at a particular speed, called the phase velocity. The waves are dispersive, so the speed will depend on the frequency or wavelength — the two are connected together via the governing equation for the waves, as explained in the previous link.
Now suppose we generate, simultaneously, two sinusoidal waves with slightly different frequencies or wavelengths. The combination gives a phenomenon known as beating, leading to a pattern with a sinusoidal “carrier wave” having an amplitude that is modulated with a slower sinusoidal wave. Figure 3 shows an example, and if you run the video you will see what happens when the two sinusoidal waves have the same phase velocity, as they would for sound waves. The beating pattern simply travels to the right, without changing shape.
But now look at Fig. 4. The initial beating pattern is very similar to Fig. 3, but this time when you play the video something more interesting happens. This case corresponds to water waves, for which the component with the shorter wavelength has a lower speed. If you watch the video carefully, you will see that the little wiggles making up the carrier wave travel at one speed, but that the sinusoidal modulation pattern moves at a slower speed. The short-wavelength ripples move through the modulation pattern, overtaking it. The small ripples are travelling at the phase velocity appropriate to their wavelength, but the modulation pattern is travelling at a different speed known as the group velocity. If you watch the video in Fig. 1 carefully, you can see a similar effect occurring: if you follow a single wave-crest, it moves forwards through the region of disturbance, then fades away after it has popped out from the leading edge.
Figure 5 shows the opposite behaviour. This case corresponds to waves on a bending beam, so that the component with the shorter wavelength has a higher speed. This time the short-wavelength wiggles travel slower than the sinusoidal modulation pattern. So for this case the group velocity is higher than the phase velocity, whereas for the water waves in Fig. 4 it was lower.
Finally, Fig. 6 shows an unusual case in which the wavelength gets longer as the frequency gets higher: in all the cases so far, higher frequency has always given a shorter wavelength. This time, the video shows the carrier wave ripples moving in the opposite direction to the modulation pattern: the group velocity is negative relative to the phase velocity! The next link describes the theory behind all these examples, and gives a slightly more sophisticated analysis of group velocity to justify applying the concept to the case we saw in Fig. 1, which doesn’t look very much like the simple beating pattern of Figs. 3–6.
C. Kelvin’s wave pattern in the wake of a slowly-moving vessel
The interplay between phase velocity and group velocity in the case of water waves has a very striking consequence. It has nothing whatever to do with music, but I include a brief account here for interest. Figure 7 shows a familiar sight, the wake pattern behind a moving duck. Figure 8 shows a particularly clear image of the wake behind a slow-moving boat. Remarkably, these two wake patterns share an identical structure, first explained by William Thompson, Lord Kelvin, in 1887 [1].
As is explained in the next link, wakes like this are confined to a wedge with a fixed angle of $19.5^\circ$ on either side of the line of travel of the source. This “Kelvin wedge” does not depend on the speed of travel, the properties of water, the strength of gravity or anything else: it is determined entirely by the fact that the group velocity of water waves is half the phase velocity. Usually, the strongest waves are found on the edge of the wedge. The wave crests at the point where they fade away outside the wedge are always inclined at an angle of about $55^\circ$ to the line of travel. Again, this angle does not depend on anything, it is fixed by a geometric argument given in the link.
Kelvin’s argument led to a mathematical expression for the shape of the successive wave crests. These take the form of “curvilinear triangles”, as illustrated by the red lines in Fig. 9. The black dashed lines show the boundaries of the Kelvin wedge. This pattern can be seen very clearly in Fig. 8, and somewhat less clearly in Fig. 7.
[1] William Thomson “On ship waves,” Institution of Mechanical Engineers, Proceedings, 38, 409–34 (1887)