| The plasma acoustic source(PAS) is a new generation of intensive sound source. Taking advantage of its “high source level, adjustable waveform structure and frequency range”, the PAS has broad applications, such as in national defense, industry and medical domains etc. Based on different sound generation mechanisms, the PAS mainly has two kinds of discharge type, i.e., arc discharge and corona discharge. The former produces higher sound pressure than the latter one, but the phase of the sound pulse generated by arc discharge is difficult to control. Generally, the methods of amplifying an acoustical signal can be divided into three categories, i.e., phase array technology, acoustic lenses and via concave reflectors. Due to the uncertainty of the phase of an arc discharge PAS, the former two methods are hard to apply. Therefore, a concave metallic reflector is adopted to amplify the acoustic signals generated by the PAS currently. In this paper, the reflection and propagation of intensive acoustic pulse from a concave parabolic or ellipsoidal reflector is studied theoretically, numerically and experimentally.Firstly, assuming the wave at the focus of the concave reflector spreads spherically, the wavelength of the pulse is much smaller than the dimension of the reflector and the observation point is located on the axis of the reflector, based on Kirchhoff diffraction integral and linear system theory, a transient analytical solution for the reflected pressure wave is derived. To overcome the 1D limitation of analytical solution, the Discrete Representation(DR) method proposed by Piwakowski[J. Acoust. Soc. Am. 86(6), 2422(1989)] is introduced to calculate the convolutional Kirchhoff integral, and a 3D numerical solution is obtained consequently. Further, an augmented KZK equation is obtained by introducing the relaxation effect of seawater into the equation of state. The complicated reflector system is modeled as an equivalent circular piston source, and the calculation of the reflected pressure field is equal to the solution of the augmented KZK equation. The stability, precision and speed of the numerical method have been improved by adding new procedures into the original FDTD method used to solve the KZK equation. The cavitaion problem is studied and QX equation is re-derived and corrected.Secondly, C-MEX method is adopted in the computer code, and the evolution of the reflection sound field is simulated based on our own program. The on-axis and off-axis evolution of the pressure waveform is studied; the 2D distribution of the sound field is discussed; the effect of depth-to-focal-distance ratio of a parabolic reflector on the peak power density is analyzed. And the effect of nonlinearity on the location of the actual focus spot is analyzed.Thirdly, based on the theoretical model, parameters study is performed. The influence of reflector parameters and source waveform parameters on the reflection sound field is discussed. The reflector parameters include the change of its geometry, e.g., a small hole drilled at the bottom of the reflector, the lateral truncation of the reflector and the variation of eccentricity. The source waveform parameters includes the rise time, duration and amplitude. Due to the limited computational resources, the study of source waveform parameters is especially for the ellipsoidal focusing case.Finally, an experimental platform is built up, which consists of a water tank, PAS, ellipsoidal reflector, pressure measurement system and high-speed photography system. The directed pulse parameters are studied and the spherical wave property of the directed pulse has been proved. The evolution of the reflected pulse both on-axis and off-axis is studied and the negative pressure generation mechanism is explained. The focusing process and deviation of the actual focal spot are observed. In addition, the dynamics of a spherical cavitation bubble in the sound field is recorded by high-speed photography.The following conclusions are reached:1. Due to diffraction effect, the reflected waveform includes the “center wave”, “edge wave” and “wake”. The arrival time, amplitude and phase of the three waves depend on the geometry of the reflector and the position of the observation point. For a parabolic reflector, the phase of the center wave is positive and that of the edge wave and wake is negative, and the amplitude of center wave is much larger than that of the edge wave and wake. For an ellipsoidal reflector, the phase of the center wave is positive and that of the edge wave and wake is negative before the far focus, but the case is reversed beyond the far focus. In the far field of a parabolic reflector or at the far focus of an ellipsoidal reflector, the reflected pressure waveform is equivalent to the derivative of the source waveform. Besides sound frequency(df/dt), the amplitude of the reflected pressure also depends on the geometry of the reflector.2. Nonlinear effect leads to a deviation of the actual focal spot position with respect to the far focus of an ellipsoidal reflector, i.e., the peak positive pressure appears beyond the far focus and the peak negative pressure appears before the far focus. This kind of phenomenon can be explained by “self-diffraction” theory.3. Base on the result of the effect of depth-to-focal-distance ratio on peak power density, a concept of “diffraction-leading zone” for a parabolic reflector is proposed. Similar to the saturation phenomenon, inside the diffraction-leading zone, the peak power density will not be improved by simply enlarging the dimension of the reflector. However, if the aperture size if fixed, an optimum depth-to-focal-distance ratio exists, i.e., d/zF=3.92, to maximize the reflected pressure in the far field.4. A change of the geometry of the reflector has special effect on the resulting sound field.(1) A dip in the reflected pressure wave signals appears near the axis of a parabolic reflector if a small hole is drilled at the bottom of the reflector.(2) The lateral truncation of an ellipsoidal reflector leads to the reduction of the amplitude of the reflected pulse at the far focus and the-6d B focal width will increase in the lateral truncation direction.5. The rise time tr, duration T and amplitude p0 of the source waveform have different effect on the reflection sound field. The rise time has non-obvious effect on the reflection sound field. If the duration of the source waveform is increased, the amplitude of the reflected wave will decrease and the-6d B focal spot for negative moves toward the aperture of the reflector. If the amplitude of the source waveform is increased, the linear focusing gain will decrease.6. Under the experimental condition, the rise time of the directed pulse is about 5μs, the duration is about 25 μs and the amplitude is about 1MPa(L=30cm). The directed wave indeed spreads spherically, and the measurement focusing property of an ellipsoidal reflector agrees well with the theoretical calculation results. The cavitation bubble behaviors in the sound field demonstrated the waveform structure of the reflected wave indirectly. |
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