## 8.5 The Doppler Effect

The discussion up to this point has implicitly assumed that the source and detector of radiation are stationary relative to one another. When the source and detector are in relative motion, an added complication, the Doppler effect, is introduced. This phenomenon has been observed for all types of radiation, though the theory is simplest for the acoustic situation. Since acoustic waves are longitudinal (compression) rather than transverse as are electromagnetic waves, the mathematical form of the effect is slightly different. Additionally, the acoustic Doppler effect has significant evolutionary implications.

Empirically, the Doppler effect is manifested in the difference between the pitch (frequency) of the sound heard by a listener in motion relative to a sound source and the pitch heard when the listener and source are relatively stationary. A familiar example is the sudden drop in pitch one hears from an automobile horn as one meets and passes a car going in the opposite direction. Alternatively, one might think of the changing pitch heard by a listener standing beside a track as a train whistle approaches, passes, and recedes.

The derivation presented below is for the Doppler effect as it applies to the special case when the motion of the source and listener lies along the line joining the two. The biological example considered later is of precisely this type.

Let $$V_L$$ and $$V_S$$ denote the velocities of the listener and source respectively. The positive direction for the velocities is taken to be from the position of the observer to the position of the source. Figure 8.5: The source $$S$$, moving with velocity $$V_S$$ is emiting sound at a constant frequency, $$f_S$$. The wave fronts of this sound, which travel radially outward at velocity $$C_S$$, are compressed together (wavelength equals $$\lambda_F$$) in front of the source and spread out (wavelength equals $$\lambda_B$$) behind the source. By the time the wavefront emitted by the source at $$x_0$$ at time t = 0 has reached a radius of $$C_ST$$, the source has moved to a position $$x_T$$. The listner $$L$$ moving with velocity $$V_L$$ will hear the sound at a different frequency than as actually emitted by the source because it encounters wave fronts which are $$\lambda_B$$ apart. An exact derivation is given in the text. [Adapted from Young 2013 p. 534.]

An illustration of the situation when the listener $$L$$ and the source $$S$$ are moving away from each other is presented in Fig. 8.5. At time $$t = 0$$, the source is at point $$X_T$$. The outermost circle is the representation of the position of the wave front at time $$t = T$$, caused by a disturbance at the source at time $$t = 0$$. The speed of propagation $$C_S$$ of a wave in a nondispersive medium such as air is dependent only upon the characteristics of the supporting medium and is independent of the motion of the source relative to the medium. Thus the outermost circle in Fig. 8.5 represents a sphere in 3 dimensions with center at $$x_0$$ and radius $$C_ST$$.

The source has moved a distance $$V_ST$$ in the time $$T$$ so that $x_0-x_T=V_ST$ and the following equalities are true: $a-x_T=(C_S+V_S)T$ $x_T-b=(C_S-V_S)T$ where $$a$$ and $$b$$ are respectively the positions at the rear and front of the outermost wave surface. In the time interval between $$t = 0$$ and $$t = T$$, the source has emitted a certain number of waves, $$f_ST$$, where $$f_S$$ is the frequency of the sound emitted at the source. The waves are spread out into the distance $$(x_T - b)$$ in front of the source. $\lambda_f=\frac{x_T-b}{f_ST}=\frac{(C_S-V_S)T}{f_ST}=(C_S-V_S)/f_S$ Similarly behind the source the wavelength $$\lambda_B$$ is $\lambda_B=\frac{a-x_T}{f_ST}=\frac{(C_S+V_S)T}{f_ST}=(C_S+V_S)/f_S$ The waves approaching the listener have a relative speed of $$C_S + V_L$$, so the detected frequency is $f_L=(C_S+V_L)/\lambda_B=\frac{C_S+V_L}{(C_S+V_S)/f_S}$ or $\begin{equation} f_L=\frac{C_S+V_L}{C_S+V_S}f_S \tag{8.14} \end{equation}$ When the source and listener are moving toward each other a frequency $f_L=\frac{C_S+V_L}{\lambda_f}=\frac{C_S+V_L}{(C_S-V_S)/f_S}$ or $\begin{equation} f_L=\frac{C_S+V_S}{C_S-V_S}f_S \tag{8.15} \end{equation}$ is detected.

In order to help remember the direction convention established in this derivation, it is useful to notice that the frequency heard by the listener will be less than that emitted by the source if they are moving away from each other, and will be greater than that emitted by the source if they are moving toward each other.

Example 5

The velocity of sound in still dry air is $$C_S = 350 ms^{-1}$$. For a source emitting sound of frequency $$f_S = 700 Hz$$ the wavelength of the sound emitted is $\lambda=C_S/\lambda_S=0.5m$ a) What are the wavelengths of the sound in front of and behind this source moving at $$V_S = 50 ms^{-1}$$?
b) If a listener is at rest and this source is moving away at $$V_S = 50 ms^{-1}$$, what is the frequency of the sound heard?

Solution:
\begin{align*} \mbox{a) } \lambda_F&=(C_S-V_S)/f_S=0.429m \\ \lambda_B&=(C_S+V_S)/f_S=0.571m \\ \\ \mbox{b) } f_L&=\frac{C_S}{C_S+V_S}=612Hz \end{align*}

Now that a quantitative formulation of the Doppler effect has been derived, it can be applied in the explanation of echolocation in bats, a sensory system which has only begun to be understood in the past several decades.

The bats, members of the order Chiroptera within the class Mammalia, have developed a system of echolocation which has been of immense evolutionary significance. It has enabled them to exploit a resource for which there are very few competitors, small night flying insects (see, e.g., Fenton 1974). The physical processes important in echolocation have both enhanced and constrained the evolution of the Chiroptera. Here it will suffice to show a few examples of the effect of these physical processes upon bat evolution and natural history. For a detailed and personable account of the actual experimentation that led to the elucidation of the physical principles employed in echolocation, see Griffin’s “Listening in the Dark” (Griffin 1958).

Almost all bats appear to be able to navigate in total darkness a closely spaced grid of wires, where the spacings of the wires are commensurate with the dimensions of the wingspread of the bats. Furthermore, insectivorous bats can pursue and capture flying insects on the wing in total darkness.

The ability to carry out these activities is significant in two respects. First, unlike the birds which are able to perform similar activities only in the presence of light, bats cannot be dependent upon a highly developed sense of sight for navigation. Second, no other animal can depend upon catching flying insects in the dark and so there are few competitors for this food resource. Prior to the evolution of bats, this resource had not been exploited.

The experiments of Griffin demonstrate that bats share another ability that sets them apart from most other animals. This is the ability to produce and detect sound of frequencies between 20-100 kHz, It is this attribute which is fundamental to the system of echolocation, as in the next example.

Example 6

A flying bat emits a pulse of sound. What is the minimum frequency and maximum length of a single frequency pulse, if the bat wants to use the echo of the pulse to avoid an obstacle 1 meter ahead?

Solution: One first solves for the maximum pulse length. Clearly the chief consideration is that the bat must not be producing the pulse when the echo returns. Otherwise the fainter echo might be masked by the bat’s own cry. The time of the echo’s return $$t$$ is dependent only on the speed of sound $$C_S$$ and the distance of the object $$d$$. Specifically: $t=d/C_S$ which works out to be: $t=2m/350m\:s^{-1}=5.71\times10^{-3}s=5.71ms$ Shorter pulses will allow even closer objects to be avoided, although the pulse length must have a lower limit of several wavelengths in order to avoid problems in interpretation. If a reasonable number of waves, in the pulse is 100, the predicted minimum frequency would be: $f=100/t=100/4\times10^3s^{-1}=25kHz$ In actual field measurements Griffin (1953 p.191) has observed pulse lengths from 1-15 ms, and pulse frequencies between 30-75 kHz with about 50 kHz being most common. Therefore, bats should be several times better at avoiding obstacles than one would expect from Example 6. This is not too surprising if the system is good enough to catch insects on the wing, which the bats must approach much closer than 1 meter.

Interestingly, bats appear to be rather economical in their use of pulses, using relatively long pulses (15 ms) at long repetition intervals when they are cruising far above the ground and relatively short pulses (1 ms) at short repetition intervals only when they are pursuing insect prey.

Many other characteristics of the bat echolocation system can be explained by an extension of the kinds of arguments given here. For example, theoretically one expects that the intensity of the returning echo from an object at a certain distance, should decrease with the size of the object. The echolocation system provides information about the size of objects in the environment from the intensity of the returning echo, and information about the distance of objects from the time delay between the emission of the pulse and the reception of the echo. However, the character of the echo should theoretically be relatively insensitive to the detailed nature of the object causing the echo, which perhaps explains the interest exhibited by bats in pebbles tossed in the air by small boys and naturalists.

In certain bats, for example those of the genus Rhinolophus, the Doppler effect appears to be the physical phenomena which is the key to an understanding of their system of echolocation (Schuller, et al. 1974). Several of the important considerations involved in the application of the Doppler effect to bats are developed in the following example.

Echo Example

A bat is flying along at $$5 ms^{-1}$$. It emits a 50 kHz sound pulse which is reflected by an insect which is 5 meters distant and is moving away at $$1 ms^{-1}$$. What is the frequency of the echo, and how long after the pulse begins does the echo begin to return?

\begin{align*} f_L&=51.74 kHz \\ t&=28.6 ms \end{align*}