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5. Other Types of Waves

So far we have concentrated on waves that are movements of a medium. In Physics 7B we discussed how an object perturbed from equilibrium underwent simple harmonic motion if the displacement was small (and the technical condition that the equilibrium was stable). At the beginning of this discussion we illustrated how a material wave arose: as pieces of the medium were displaced they pulled or pushed on the next piece of the medium.  Even though we only disturb one piece of the medium other pieces can be affected. The traveling disturbance was what we referred to as the wave. Because for small oscillations each part of the medium undergoes simple harmonic motion (as we discussed in Physics 7B) it does not seem all that surprising that all the different media we discussed can support waves.

Later on in the course we will also discuss light and quantum particles. In each of these cases we have “waves” that have no medium, and so are not propagated along by forces pulling and pushing a material. Why do we call them waves? The answer is that these light and quantum waves exhibit much of the same behavior as the material waves we are currently discussing. In the next chapter we will show how to combine material waves (Superposition) and talk about how material waves travel (diffraction and geometric optics). The light  and quantum waves have these properties too! We take the view that we will use material waves to build up our intuition, and then define a general wave as anything that has similar behavior to the material waves we have come to know.

A Wave Model of Sound

You have probably heard the term “sound waves”. That is because sound is a wave in the sense that we have already described. In this case the medium is the material the sounds travels through. Often this is the air, but sound can also travel through liquids and solids. As we discussed in Physics 7A the bonds between particles in the air are virtually non-existent, so it is difficult to think of the air molecules as having an equilibrium position. Instead,  it's easier to think of sound as a pressure wave. While this is slightly different in form from the material waves we have already discussed, we can apply almost all of the same techniques that we have already learned to sound waves. Fundamentally, however, sound waves are just material waves in a medium and so we may not be surprised that the same techniques work.

In seeking to model sound as a wave, we can choose to model sound in two different ways. The first model is that the individual molecules that make up the air are vibrating back in forth in the direction of propagation. That is, sound waves are longitudinal. We might try to describe an air molecule with the equation \[y(x,t) - y_0 = A \sin{\left( 2 \pi \dfrac{t}{T} \pm 2 \pi \dfrac{x}{\lambda} + \phi \right)}\] where (\(y(x, t)\) - y_0\)) is the particle's displacement from its equilibrium position.  However, we previously described gases as molecules zooming around one another, so it makes no sense for any one of them to have an "equilibrium position;" there is a model that better fits with reality.

Instead of focusing on the position of the air molecules (which is difficult to calculate or measure) we can instead consider the density or the pressure of the air. We learned in Physics 7A that air molecules move around rapidly in random motion, but there are well-defined averages for density and pressure. Sound waves in air are the oscillation of the average value of particle density (and therefore pressure) over distance scales much larger than the mean distance between particles. Thus we can model sound wave considering only the pressure or density of the air.

Choosing pressure, we can describe sound by

\[P(x,t)  -P_{atm}= A \sin{\left( 2 \pi \dfrac{t}{T} \pm 2 \pi \dfrac{x}{\lambda} + \phi \right)}\]

where \(P(x, t)\) is the absolute pressure of the air at a given position \(x\) along the tube, and at a time \(t\). \(P_{atm}\) is the atmospheric pressure (i.e. the equilibrium pressure), and \(A\) is the amplitude of the pressure fluctuation (gauge pressure) from equilibrium (recall from Wave Properties that the amplitude may not be constant).

Note the similarity between this equation and the equation above for \(y(x, t)\) (from Harmonic Waves). We can therefore ascribe familiar parameters to sound waves, like wavelength, period, and wave speed.  We can also use the same techniques  from Graphical Representations of Waves to plot the pressure against time at constant position, or pressure against position at constant time.