SIMPLE TYPES OF SOUND WAVES 



17 



shallow depths, the pressure changes are relatively 

 unimportant, and salinity changes in the open ocean 

 are usually too small to matter much. Near the 

 mouths of large rivers, however, where fresh water is 

 continuously mixing with ocean water, the variations 

 in sound velocity may be largely controlled by varia- 

 tions in salinity. 



The quantitative dependence of soimd velocity on 

 temperature, pressure, and salinity is summarized in 

 Figures 4 and 5. In Figure 4, obtained from a report 

 by Woods Hole Oceanographic Institution [WHOI] ' 

 the value of the sound velocity at zero depth can be 

 read from the main charts for any given combination 

 of temperature and salinity. This velocity can theh be 

 corrected to the velocity at the actual depth by use 

 of the curve in the small box. Figure 5 gives the per- 

 centage changes in sound velocity caused by specified 

 absolute changes in the three determining variables. 

 It will be shown in Chapter 3 that it is the relative 

 changes in sound velocity which determine whether 

 sound transmission is expected to be good or bad 

 rather than the absolute changes. 



The direct measurement of sound velocity in the 

 ocean is very difficult. The intuitive method of di- 

 viding distance traveled by time is diflicult since 

 sufficiently accurate measurement of distances at sea 

 is usually not feasible. The U. S. Navy Electronics 

 Laboratory at San Diego, formerly the U. S. Navy 

 Radio and Sound Laboratory [USNRSL], developed 

 an acoustic interferometer for the determination of 

 the wavelength of sound at a point in the ocean ;^ 

 multiplication of this local wavelength by the known 

 frequency gives the local sound velocity. This instru- 

 ment was developed mainly for the purpose of check- 

 ing whether the temperature changes indicated by 

 the bathythermograph were correlated with the actual 

 changes of sound velocity in the ocean. Good general 

 agreement was observed between the velocity-depth 

 plots obtained with the interferometer or velocity 

 meter and those computed from bathythermograph 

 observations. However, since the bathythermograph 

 cannot follow rapid changes in water temperature with 

 the detail possible with the velocity meter, a velocity 

 microstructure was frequently recorded with the 

 meter which deviated as much as 0.1 per cent from 

 the velocity calculated from the simultaneous bathy- 

 thermograph reading. That these deviations were due 

 to the slow response of the bathythermograph rather 

 than to physical factors was verified by correlating 

 the velocity microstructure with the temperature 

 microstructure obtained by a thermocouple recorder. 



2.3.2 



Harmonic Waves 



Up to now we have allowed the initial disturbances 

 ■p{x,y,z) or p{t) to be arbitrary functions. However, 

 most initial disturbances which occur in practice are 

 of a very special type that originate in the elastic 

 vibration of some medium. Such disturbances are pro- 

 duced by small displacements of some parts of the 

 medium from their positions of equilibrium; these 

 displacements in turn produce restoring forces which 

 tend to restore the state of equilibrium. Such restor- 

 ing forces are, in first approximation, proportional to 

 the displacements. 



It is well known that under such conditions (re- 

 storing forces proportional to displacements) the 

 initial disturbance must be of the form of a harmonic 

 vibration; that is, it must be representable by trigo- 

 nometric functions of the time. In acoustics, such a 

 vibration produces a pure tone of a definite frequency. 

 Since echo-ranging pulses are very nearly pure tones, 

 the importance of a study of harmonic vibrations is 

 obvious. Also, harmonic vibrations are of crucial im- 

 portance because they are the most convenient build- 

 ing stones of the more general solutions of the wave 

 equation (see Section 2.7). 



Suppose, in the second example under plane waves, 

 that the initial disturbance of the plane a; = is a 

 harmonic vibration. That is, 



p(<) = a cos 2x/(< — e) 



for values of t between and t^. One solution of the 

 plane wave equation (30) under initial conditions 

 p(0,<) = p{t) is always 



,t) = v{t ± ^) 



p{x 



since pit + x/c) satisfies the wave equation and also 

 the imposed initial conditions. Thus, if we restrict our 

 attention to progressive waves traveling in a single 

 direction, the solution of the wave equation with 

 the given initial conditions is 



p = a cos 2wf{ t -\ « ) 



or 



p = a cos 27rfU e) (34) 



Clearly, the pressure changes represented by equa- 

 tion (34) are at most a; for that reason, a is called the 

 amplitude of the disturbance. Also, it is clear that at 

 a fixed point of space, p goes through / periods in one 

 second; and so / is called the frequency of the dis- 



