The Doppler Effect (1951)

Given a series of regularly spaced objects, each moving with the same constant speed in a straight line toward a fixed point, how many objects arrive per unit of time? Or what is the frequency of arrival? Suppose each object moves with the speed U. Then in one unit of time, each object moves a distance U. If the spacing between objects is lambda, the number of objects included within the distance U is U divided by lambda. Thus, the frequency of arrival is U over lambda. If for any reason the speed becomes greater, the spacing remaining the same, the frequency of arrival will be greater. On the other hand, if the spacing becomes smaller, the speed remaining the same, the frequency of arrival will again be greater. An important application of these ideas is provided by a phenomenon known as the Doppler effect. A vibrating source, vibrating with a single frequency, sends out longitudinal waves in the surrounding medium. The pitch of the sound heard by a listener depends on the frequency with which the waves strike the ear. Suppose the source and the listener are at rest with each other. The waves travel with speed U, and the source vibrates with frequency F0. The spacing of these waves is the wavelength lambda zero, where lambda zero equals U over F0. Or F0 equals U over lambda zero. As we have seen, the number of waves arriving at the ear per unit of time is the speed divided by the spacing, which is the frequency F0 of the source itself. In this case, therefore, there is no change of pitch, since the rate at which waves are sent out by the source is the same as the rate at which they arrive at the ear. This is true only when the source and the listener are at rest with respect to each other. The frequency of arrival of waves at the ear, which determines pitch, will differ from the frequency of the vibrating source if the source or the listener should move. This change of pitch is called the Doppler effect. If the source moves toward the listener, the waves become spaced closer together. When the source moves away from the listener, the waves are farther apart. The speed of the waves is independent of the motion of the source. Once the waves have left the source, they don't know what the source is doing. The waves still travel with speed U, but their spacing depends on the speed of the source and the direction of motion of the source. When the source moves toward the listener with the speed VS, the relative speed of the waves with respect to the source is the speed of the waves U minus the speed of the source VS. Thus, the spacing of the waves lambda is U minus VS divided by the frequency of the source F0. Since the frequency of arrival at the ear is U over lambda, by introducing this value of lambda, we get for the apparent frequency F this expression. When the source moves away from the listener with the speed VS, the speed of the waves with respect to the source is U plus VS. The spacing of the waves lambda is then U plus VS divided by the frequency of the source F0. Substituting this expression for lambda, we get this value for the apparent frequency. Thus, the pitch is higher when the source moves toward the listener, and lower when the source moves away from the listener. Now, if the listener moves toward or away from the source, the spacing of the waves lambda zero is the same as in the normal case. The motion of the listener has no effect on the wavelength, but the frequency of arrival of the waves at the ear is changed. If the listener advances toward the source, the relative speed of the waves with respect to the listener is the speed of the waves U plus the speed of the listener VL, and the frequency of arrival at the ear F equals plus VL over lambda zero. Since lambda zero equals U over F0, we get F equals F0 times U plus VL over U. If the listener moves away from the source with a speed VL, the speed of the waves with respect to the listener is U minus VL, and F then equals F0 times U minus VL over U. A familiar example of the Doppler effect is the drop in pitch that accompanies the passing of a rapidly moving train. Another example of the Doppler effect may be experienced by the engineer on the train as he passes a road signal.