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LETTERS TO THE EDITOR 297 5. F. H. FENLON 1972 Journal of the Acoustical Society of America S&284-289.An extension of the Bessel-Fubini series for a...

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5. F. H. FENLON 1972 Journal of the Acoustical Society of America S&284-289.An extension of the Bessel-Fubini series for a multiple-frequency cw acoustic source of finite amplitude. 6. M. B. MOFFE’I-I’, W. L. KONRAD, J. L. NELSON and J. C. LOCKWOOD 1979 Journal of the Acoustical Society of America 66, 1842-1847. A saturated parametric acoustic receiver. 7. P. J. WESTERVELT 1963 Journal of the Acoustical Society of America 35,535-531. Parametric acoustic array. AUTHOR’S



Drs Moffet and Mellen [l] have considered the question of whether a significant increase in linear attentuation coefficient at the second harmonic frequency of a finite-amplitude wave will reduce non-linear attenuation of the fundamental. The results they obtained by using the methods of Fubini and Fenlon agreed qualitatively with my own [2]: i.e., such saturation suppresssion will occur, provided the increase in second harmonic attenuation is large enough (their method does not provide quantitative results). While the Moffett-Mellen argument is intuitively appealing, and therefore is a very worthwhile contribution to arriving at an understanding of what Moffett and Mellen describe (correctly) as a rather “paradoxical situation”, the conclusions they have reached may be premature, in the light of some recent work by Trivett and Van Buren [3] and some further reflection on the subject by myself. 2.



In reference [3], Trivett and Van Buren have modified the generalized lossless Burgers’ equation to include frequency dependent absorption, as follows [3], equation (4)]: dV/dr+(a/r)V-bV(dV/&)=



where V is the particle velocity, r is the spatial co-ordinate, r is the retarded time a = 4 (cylindrical (t-r/Co), b = p/C,, /3 = (1 +B/2A) is the non-linearity parameter, waves), 1 (spherical waves), or 0 (plane waves). Also (Y(W)is the attenuation coefficient at frequency o. They have then solved equation (1) numerically for the various harmonic components. the results of their computations are quantitatively but not qualitatively different from my own. By retaining 40 terms in the harmonic series, Trivett and Van Buren were presumably better able to characterize the harmonic amplitudes in the region beyond the classical “discontinuity distance”. New calculations presented in reference [3] include results for a 1 atm 10 kHz primary wave in both fresh water and sea water (where relaxation affects linear attenuation coefficient). What was found was that the amount of increase in second harmonic attenuation required virtually to eliminate saturation losses was much for sea water than for fresh water: to reduce the excess non-linear attenuation of the fundamental component from 5.5 to 0.4 dB required a lo4 increase for a2 in fresh water, while in sea water a SO fold increase of (Yereduced the excess loss from 5.2 to 1.3 dB.




For high Goldburg number r, r=aL,


where L is the discontinuity distance (3)



and (Yis the primary wave attenuation coefficient, the interaction region of a parametric endfire array is confined to a region of length -L in front of the projector. Doubling the shock distance, which Moffett and Mellen have conceded might be possible by using the resonant absorption technique I suggested, would narrow the endfire difference frequency beam roughly by a factor of J2 which probably is significant for high resolution underwater imaging, or reverberation-limited sonar applications. The possible savings of several dB in transmitter power through reduced saturation loss (as implied by Trivett and Van Buren’s results) also makes the concept worthy of further study, even though a method of practical implementation might be hard to envision. For the case of resonant bubbles, which I suggested might merit consideration as a means of achieving the attenuation vs. frequency characteristic required, no consideration has yet been given to the effect dispersion of such a bubble-water mixture might have on harmonic generation. Some Soviet work on parametric receivers seems to suggest that saturation suppression might be achieved via dispersion alone [4]. These and other practical considerations (such as whether wake-bubbles might have the proper characteristics for this application) must be addressed before the practicality of achieving saturation suppression in a real sonar device can be assessed. ACKNOWLEDGMENTS

The author would like to thank D. Trivett and A. Van Buren for a preprint of reference

[31. Sanders Associates, Inc., 95 Canal Street, Nashua, New Hampshire 03061, U.S.A.

H. C.


(Received 27 January 1981) REFERENCES 1. M. B. MOFFETT and R. H. MELLEN 1981 Journal of Sound and Vibration 76,295297. On absorption as a means of saturation suppression. 2. H. C. WOODSUM 1980 Journal of sound and Vibration 69,27-33. Enhancement of parametric

efficiency by saturation


3. D. H. TRIVE-IT and A. L. VAN BUREN 1981 (to appear) Journal of the Acoustical Society of America. Propagation of plane, cylindrical, and spherical finite amplitude waves. 4. E. A. ZABOLOT-SKAYA and S. I. SOLUYAN 1967 Soviet Physics-Acoustics 13,254-256. A

possible approach to the amplification of sound waves.