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1.

Leander, Johan L.

Swedish National Defence College, Department of Military Studies, Military-Technology Division.

A note on transient underwater bubble sound1998In: Journal of the Acoustical Society of America, ISSN 0001-4966, E-ISSN 1520-8524, Vol. 103, no 2, p. 1205-1208Article in journal (Refereed)

Abstract [en]

This Letter considers scattered sound from transiently oscillating gas bubbles in liquids. The full transient problem including the finite duration of the excitation is analyzed. As a result, the wave front of the radiated sound pulse involving information about the excitation is also studied. The model presented is used to simulate sound pulses from sea-surface bubbles which have been generated by, for example, spilling breakers, capillary-gravity waves, and rain drops. Although very simple in relation to the actual physical process of excitation, this model enables us to predict some of the essential properties of scattered pulses observed experimentally. It is suggested that the time scale of duration of the initial driving that enters into the present analysis might be of some use in a further physical understanding of bubble generation and excitation.

The aim of this Comment is to suggest some possible improvements and developments of the investigation by Zhen Ye [J. Acoust. Sec. Am. 101, 3299-3305 (1997)]. Particular attention is given to the causality concept and the use of integral theorems.

This investigation concerns free linear gas bubble oscillations in liquids. Of prime interest is the eigenfrequency, and in particular its real part, here named as the transient frequency. The conceptual difference between the more frequently consulted resonance frequency and the transient frequency is first addressed by means of the classical mechanical oscillator. Next, bubble pulsations in liquids are discussed and an existing model is used for the gas-liquid interaction from which an approximate expression for the eigenfrequency is derived. A rationale for the approximate evaluation of the functions modeling the thermal processes is suggested which is independent of the frequency content of any possible pressure excitation, Moreover, compressibility effects are not approximated in the derivation presented here, The quantitative difference between the adiabatic resonance frequency and the derived estimate of the transient frequency is found to be of significance for small bubbles. Finally, the similarity between a standard mechanical oscillator and a bubble in a liquid for the case of liquid-compressibility effects only is discussed.

The relation between the wavefront speed and the group velocity concept is studied in this work. The relationship between the more well-known velocity concept named as the phase velocity and the speed of propagation of a front of an acoustic pulse is discussed. This is of interest since it concerns transient wave propagation and is, in general, not well known. The form and properties of a pulse can be obtained by means of a Fourier integral and estimates based on quantities derived for monochromatic waves, such as the phase velocity, can be severely misleading and confusing. The wavefront velocity is defined as the high-frequency Limit of the phase velocity. This quantity can be far less than the value of the phase velocity for finite frequencies which for example is the case for bubbly fluids. Then the group velocity concept is discussed, which was introduced in order to characterize the propagation of water waves of essentially the same wavelength. However, more confusion occurs in that it is sometimes believed that a wavefront is propagating with the group velocity (a limit process not mentioned) since it can be related to the propagation of energy. This interpretation of energy propagation is based on sinusoidal waves and involves time as well as space averages and is not applicable for pulses. However, by means of the expression for the group velocity given by Stokes it is shown that the speed of a wavefront can be found from the group velocity at a limiting high frequency. This result can be understood geometrically from the definition of the group velocity given by Lamb which is conservation of wavelength. A wavefront is a discontinuity and limiting short wavelengths will be found there.

In this paper it is demonstrated that a theoretical model for wave propagation may indeed correspond to a well-posed transient problem although the group velocity for finite frequencies becomes greater than the high frequency limit of the phase velocity, negative or even infinite. Sufficient conditions for causality dare derived and the particular cases of relaxing and bubbly fluids are considered so as to show-some of the properties of the group velocity concept.