Many introductory physics texts1–5 accompany the discussion of the Doppler effect and the formation of shock waves with diagrams illustrating the effect of a source moving through an elastic medium. Typically these diagrams consist of a series of equally spaced dots, representing the location of the source at different times. These are surrounded by a series of successively smaller circles representing wave fronts (see Fig. 1). While such a diagram provides a clear illustration of the shock wave produced by a source moving at a speed greater than the wave speed, and also the resultant pattern when the source speed is less than the wave speed (the Doppler effect), the texts do not often show the details of the construction. As a result, the key connection between the relative distance traveled by the source and the distance traveled by the wave is not explicitly made. In this paper we describe an approach emphasizing this connection that we have found to be a useful classroom supplement to the usual text presentation. As shown in Fig. 2 and Fig. 3, the Doppler effect and the shock wave can be illustrated by diagrams generated by the construction that follows.

1.
Paul A. Tipler, Physics for Scientist and Engineers, 4th ed. (W. H. Freeman, New York, 1999), pp. 463–468.
2.
Eugene Hecht, Physics: Algebra/Trig., 2nd ed. (Brooks/Cole, Pacific Grove, CA, 1998), pp. 467–473.
3.
Jerry D. Wilson and Anthony J. Buffa, College Physics, 4th ed. (Prentice Hall, Upper Saddle River, NJ, 1997), pp. 460–473.
4.
Raymond A. Serway and Jerry S. Faughn, College Physics, 5th ed. (Saunders, New York, 1999), pp. 452–457.
5.
Susan M. Lea and John Robert Burke, Physics: The Nature of Things, 4th ed. (Brooks/Cole, Pacific Grove, CA, 1998), pp. 542–547.
6.
Since the distance between the vertical lines represents the wavelength shift due to the motion of the source,
where λS and λD are the stationary and the Doppler-shifted wavelengths of the source, respectively, Vw is the wave velocity, and VS the velocity of the source. The ratio of the source frequency, fS, to the Doppler-shifted frequency, fD, is then
Thus,
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