Maxima and Minima in the Spectrum of an Underwater Charge

The position of the maxima in the spectrum generated by an underwater explosive charge has been considered to occur at integral multiples of the inverse of the bubble-pulse period. The spectrum of the shock wave, represented by a decaying exponential, includes a frequency-dependent phase shift, while the bubble pulse, represented by a double exponential, has a phase-invariant spectrum. When this is considered, the positions of the maxima and minima are given, to a first approximation, by tanωnτ=ωnT, where τ is the bubble pulse period and T the shock-wave decay constant. The shift from integral values increases with the order and is asymptotic to 1/(4τ). In the actual case, where the shock wave has a finite rise time, the asymptotic limit is removed and the shift continues to increase with increasing order. For low-order maxima, the over-all effect is small for shallow detonations, but becomes appreciable as the depth increases. Theoretical predictions are compared with experimental data. [Work supported by the U. S. Office of Naval Research and Advanced Research Projects Administration.]