FLUCTUATION 



325 



phases with respect to each other, and the total 

 amplitude is large or small depending on the degree 

 of reinforcement or interference between these in- 

 coming individual amplitudes. 



An expression for the fluctuation of intensity 

 caused by the combination of a large number of 

 amplitudes of equal magnitude but random phase 

 was derived by Rayleigh.' The probability that the 

 resultant intensity will exceed the value / is given by 



p = e<^/i) 



(2) 



where 7 is the average intensity. A derivation of equa- 

 tion (2) is given in Chapter 7. The actual received 

 reverberation is of course a combination of ampli- 

 tudes of many different magnitudes, because the indi- 

 vidual scatterers are not all of equal strength, because 

 the projector is directional, and because the trans- 

 mission loss to different portions of the ocean may 

 not be the same. However, it can be shown that equa- 

 tion (2) remains valid, even if all the amplitudes are 

 not of equal magnitude, provided only that there are 

 a large number of amplitudes of each magnitude, and 

 that the number of amplitudes of each magnitude re- 

 mains essentially constant. Thus, formula (2) is im- 

 plied by the assumptions used in Chapter 12 to de- 

 rive the expression for the time variation of the 

 volume reverberation from one ping, with the proviso 

 that the transmission does not change from ping to 

 ping. 



The applicability of the Rayleigh distribution func- 

 tion (2) has been tested by the University of Cali- 

 fornia.^ They first chose sets of ten or more typical 

 reverberation records in deep water; all the records 

 in a given set were taken under similar conditions. 

 The ratio of the observed amplitude to the average 

 amplitude was computed for various times on each 

 set. All told, 420 values of this ratio were obtained 

 for the QB transducer, and 500 values of the ratio for 

 the QCH-3. The results are plotted in Figure 1. 

 There is apparently no major deviation of either ex- 

 perimental curve from the theoretical expression (2). 

 It appears from Figure 1 that the shape of the 

 fluctuation curve does not depend significantly on 

 such factors as the directivity pattern of the trans- 

 ducer, nor on the shape of the pulse sent out; both of 

 these factors are different for the two transducers. It 

 will be noted that equation (2) predicts this inde- 

 pendence. Since the times chosen on the records were 

 well distributed, and since no effort was made to 

 distinguish between surface and volume reverbera- 

 tion, it appears that the expression (2) applies fairly 



well to all portions of the deep-water reverberation 

 versus range curve. 



It is not .surprising that formula (2) fits the ob- 

 served fluctuation of surface reverberation levels 

 about as well as it fits the fluctuation of volume rever- 

 beration. If an assumption that the scattering power 

 of the surface remains essentially constant from ping 

 to ping is added to the other assumptions used in the 



AI 



4A 

 O 



80 



60 



40 



20 



^^ 



u O 0.5 1.0 1.5 2.0 2.5 3.0 3.5 



RATIO OF AMPLITUDE TO AVERAGE AMPLITUDE 



RAYLEIGH 



OBSERVED 



Figure 1. Agreement between Rayleigh distribution 

 and cumulative distribution of observed reverberation 

 amplitudes. 



derivation of equation (38) of Chapter 12, for the de- 

 cay of surface reverberation, the fluctuation formula 

 (2) would be predicted for surface reverberation also. 

 Separate tests of the appHcability of equation (2) to 

 surface and bottom reverberation have been re- 

 ported,^ which show that this equation is a reasonably 

 good fit to the fluctuation of both surface and bottom 

 reverberation levels. These results therefore support 

 the assumption that the surface scattering power can 



