~eoretical investigations of finite amplitude wave propagation from axisymmetric, circular piston sources has been considered by several researchers 1-3). Attention has been restricted to analyses of source conditions of either an unfocused piston or a weakly, spherically focused piston. However, little research has been reported on finite amplitude wave propagation from an array of discrete sources.~me domain simulations, based on the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, will be

more »

... ussed for an eight element discrete annular array immersed in fresh and sea water. TRODUC~ON men the peak pressure does not exceed approximately 150 MPa, the =K nonlinear parabolic wave equation yields numerical solutions in agreement with experiments. An augmented~K equation may be expressed in the following dimensionless form, where diffraction, thermoviscous absorption, and quadratic nonlinearity are described by the first three terms on the right-hand-side (RHS), respectively. The last term on the RHS is a summation over all relaxation processes that may be present in the fluid. The pressure is P; R = r/a and o = z/1 are unitless radial and z coordinates; 1, ZT = tioa2/2q, ZS = poc~/Puopo, and a are a reference length, Rayleigh distance, plane-wave sh~k fo~ation distance, and radius of the piston, respectively. For a focused source, 1 is typically taken to be the focal length. The ambient density and small-signal sound speed are PO and co while W. is a characteristic angular frequency for the finite pulse. Additionally, z. contains the peak positive pressure amplitude at the source, po, and the coefficient of nonlinearity, D. The thermoviscous attenuation coefficient is ao, and the vth relaxation process is described by a dimensionless relaxation time r. and unitless dispersion parameter m.. Finally,~denotes a dimensionless retarded time. D1SCUSS1ON Equation (1) is solved by a variety of numerical techniques including operator splitting, finite difference methods, trapezoidal quadrature, and an analytic solution for the quadratic nonlinearity term. Detailed comments concerning these techniques can be found in (1), but a brief summary is merited. Given an initial finite duration pulse at the source, Eq. (1) permits the forward propagation of the signal in the +0 direction. For a small step-size in a, each term on the RHS of Eq. (I) can be considered separately, and the order is unimportant. This is the essence of the operator splitting technique. To facilitate the computation of the diffraction term, the integral is first approximated by a trapezoidal quadrature. Both implicit backward and Crank-Nicolson finite difference algorithms are then used for the diffraction, thermoviscous absorption, and relaxation terms. Numerical accuracy is at least first order accurate, and stability depends upon proper choices of the discrete step sizes for a, R, and~. Finally, under the operator splitting, the quadratic nonlinearity term yields an analytic solution. This solution amounts to a finite amplitude distortion of the phase of the pulse at each a step, and then a resampling of the distorted pulse. The source condition is such that P(R, u = O, T) = g(R) exp{-at[(2r -L)/L]zm' } sin(~). Here, L is the dimensionless temporal length of the pulse and at and mt control the temporal envelop. g(R) provides amplitude shading. men the source is a focused piston, the initial phase of the signal at each discrete radial point R3 is determined by Tj = -z8R~/1. The negative sign indicates points further from the acoustical axis must radiate at earlier times to forma focus at 1. Figure I shows the initial dam plane a = O, and the continuous arc illustrates a wave front of constant phase corresponding to Tj. If the source condition is that of a discrete array, then g(R) and Tj become g(Rn) and~jn where Rn falls within the inner and outer radii of the nth annular element. The hash marks in Fig. 1 illustrate the relative phase and width for the eight elements of the array, For each element except the center, Rn is the average radius in the numerical simulation discussed below. 531