PROPERTIES OF SOUND WAVES 



23 



placement D and that the numerical value of this 

 actual displacement is the real part of the right-hand 

 side of equation (70). With this understanding, 

 equations (70) and (69) represent one and the same 

 physical process. 



The complex form for a vibration simplifies some 

 types of calculations and will be frequently used in 

 the remainder of this chapter. We notice that equa- 

 tion (70) can be rewritten in the form 



D = A^''^" (71) 



where A is the complex number ae~''^'^\ A is called 

 the compkx amplitude of the vibration described by 

 equation (69). It is apparent that 



A = a [cos 2irf€ — i sin 2ir/€]. 



Thus, the complex amplitude has a cos 27r/6 as its real 

 component and —a sin 2irf€ as its imaginary com- 

 ponent. 



As an example of the convenience afforded by the 

 complex representation of a vibration, we shall use it 

 to find the harmonic solution of the plane wave equa- 

 tion (30). We assume tentatively that 



pix,t) = Ae2"<^' + ™-) (72) 



and see if we can find a value of m which will make 

 equation (72) a solution of equation (30). Substi- 

 tuting equation (72) into equation (30), we have 



(2irt/)2p = c=(2xm)2p. 



In other words, a value of m equal to f/c or —f/c 

 makes the expression (72) a solution of equation (30) . 

 These two solutions are, explicitly, 



p = ^e'"^^'-'"'''^; A = ae-""'^'. (72a) 



These two solutions, interpreted according to the 

 convention of this section, are obviously identical 

 with the "real" solutions (34). 



Similarly, a point harmonic source in an infinite 

 homogeneous medium gives rise to spherical harmonic 

 waves according to the equation 



pir,t) = -e2«/C«-('/^)]; A = ae-^^'^\ (73) 



2.4.4 



Sound Sources 



The wave equation (27) governs the manner in 

 which disturbances will be propagated in the interior 

 of a fluid, but does not say anything about the initial 

 disturbances themselves. In this section we shall con- 

 sider the various types of initial disturbances which 

 can be produced by sound sources. First we shall dis- 

 cuss the quahty of the sound put out by various 



sources, where quality refers to the frequency char- 

 acteristics of the emitted sound. Next, since some 

 sources radiate equally in all directions while others 

 do not, we shall consider in a general way the 

 directivity properties of sources. 



Frequency Characteristics 



Strictly speaking, the concept of frequency can be 

 applied only to simple harmonic disturbances. A 

 simple harmonic disturbance of the pressure in a fluid 

 is described by an equation of the form 



p = a cos 2Trf(t — e) 



and gives rise to what is called a pure tone. Most of 

 the echo-ranging transducers used at present produce 

 sounds which are very nearly pure tones, but cannot 

 be heard by the ear because the frequencies are too 

 high. 



If two or more pure tones are put into the water at 

 the same time, the resultant is known as a compound 

 tone. Some transducers, used mainly in research work, 

 can produce compound tones. Any sound of this 

 nature can be expressed as the sum of a finite number 

 of harmonic vibrations. 



Many sources, however, produce in their immedi- 

 ate vicinity an irregular change in pressure which 

 cannot be represented as the sum of a finite number 

 of sinusoidal vibrations. Such sound outputs are 

 called noises. Ship sounds and torpedo sounds are 

 examples of noises, and the reader can doubtless 

 supply other examples. According to a mathematical 

 theorem called the Fourier theorem, it is often possi- 

 ble to represent such an irregular sound output as an 

 infinite sum of simple tones, whose intensities, fre- 

 quencies, and phase relationships are such that they 

 add up to the given noise. If most of the component 

 frequencies lie in a narrow frequency range, the sound 

 is called a narrow-band noise; otherwise it is called 

 a wide-band noise. 



Some types of echo-ranging gear put out a fre- 

 quency-modulated signal. In this type of output, the 

 pressure is at every instant a sinusoidal function of 

 time, but the frequency changes during the signal in 

 some designed way. In one type of frequency-modu- 

 lated signal, called a "chirp" signal, the frequency 

 increases linearly with time for the duration of the 

 pulse : 



p = a cos 27r[(/o + at)t']. 



In a typical chirp, 100 msec long, the frequency may 

 increase from 23.5 kc at the beginning of the pulse to 

 24.5 kc at the end of the pulse. 



