longitudinal - Chapter 5 Longitudinal waves David Morin...

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Chapter 5 Longitudinal waves David Morin, [email protected] In Chapter 4 we discussed transverse waves, in particular transverse waves on a string. We’ll now move on to longitudinal waves. Each point in the medium (whatever it consists of) still oscillates back and forth around its equilibrium position, but now in the longitudinal instead of the transverse direction. Longitudinal waves are a bit harder to visualize than transverse waves, partly because everything is taking place along only one dimension, and partly because of the way the forces arise, as we’ll see. Most of this chapter will be spent on sound waves, which are the prime example of longitudinal waves. The outline of this chapter is as follows. As a warm up, in Section 5.1 we take another look at the longitudinal spring/mass system we originally studied in Section 2.4, where we considered at the continuum limit (the N → ∞ limit). In Section 5.2 we study actual sound waves. We derive the wave equation (which takes the same form as all the other wave equations we’ve seen so far), and then look at the properties of the waves. In Section 5.3 we apply our knowledge of sound waves to musical instruments. 5.1 Springs and masses revisited Recall that the wave equation for the continuous spring/mass system was given in Eq. (2.80) as 2 ψ ( x, t ) ∂t 2 = E μ 2 ψ ( x, t ) ∂x 2 , (1) where ψ is the longitudinal position relative to equilibrium, μ is the mass density, and E is the elastic modulus. This wave equation is very similar to the one for transverse waves on a string, which was given in Eq. (4.4) as 2 ψ ( x, t ) ∂t 2 = T μ 2 ψ ( x, t ) ∂x 2 , (2) where ψ is the transverse position relative to equilibrium, μ is the mass density, and T is the tension. These equations take exactly the same form, so all of the same results hold. However, the fact that ψ is a longitudinal position in the former case, whereas it is a transverse position in the latter, makes the former case a little harder to visualize. For example, if we plot ψ for a sinusoidal traveling wave (either transverse or longitudinal), we have the picture shown in Fig. 1. The interpretation of this picture depends on what kind of wave we’re talking about. x ψ A B C D E Figure 1 1
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2 CHAPTER 5. LONGITUDINAL WAVES For a transverse wave, ψ is the transverse displacement, so Fig. 1 is what the string actually looks like from the side. The wave is therefore very easy to visualize – you just need to look at the figure. It’s also fairly easy to see what the various points in Fig. 1 are doing as the wave travels to the right. (Imagine that these dots are painted on the string.) Points B and D are instantaneously at rest, points A and E are moving downward, and point C is moving upward. To verify these facts, just draw the wave at a slightly later time.
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