THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

418 15. Underwater Acousticsthe inverse cube of the range. In the case of a sound channel of Figure 15.4b,time stretching that is proportional to the range causes the intensity to fall off asthe inverse square of the range instead of the inverse first power. In summary thespreading laws may be listed as follows:Propagation in Type of spreading Intensity varies as TL (dB)Tube None r 0 0Between parallel plates Cylindrical r −1 10 log rFree field Spherical r −2 20 log rHypersphere Free field with time stretching r −3 30 log rIt should be noted that the hyperspherical case applies in a hypothetical sense tosonar theory.Absorption varies with range in a manner different from the loss due to spreading.It occurs because acoustic energy converts into heat; and this conversionembodies a true loss of acoustic energy within the propagation medium. Considera plane wave passing through an absorbing medium. The fractional rate that theintensity of the wave decreases along distance x is proportional to the distancetraveled, i.e.dII=−kdx (15.1)where k denotes the proportionality constant and the minus sign signifies that dIdrops in the direction of increasing x. Integrating Equation (15.1) between rangesr 1 and r 2 , the intensity I 2 at r 2 is found fromRewriting (15.2) yieldsI 2 = I 1 e −k(r 2−r 1 ) . (15.2)10 log I 2 − 10 log I 1 =−10k(r 2 − r 1 ) log 10 e.Setting α = 10k log 10 e, the change of the intensity level in dB is now expressedas( )I210 log =−ω(r 2 − r 1 )I 1orα = 10 log( I 1/I 2 ).r 2 − r 1The quantity α is the logarithmic absorption coefficient and is usually expressed indecibels per kiloyard (dB/kyd) in the US or decibels per km (dB/km) in the metricsystem.