Distortion and harmonic generation in the nearfield of a finite amplitude sound beam are considered, assuming time-periodic but otherwise arbitrary on-source conditions. The basic equations of motion for a lossy fluid are simplified by utilizing the parabolic approximation, and the solution is derived by seeking a Fourier series expansion for the sound pressure. The harmonics are governed by an infinite set of coupled differential equations in the amplitudes, which are truncated and solved numerically. Amplitude and phase of the fundamental and the first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source. Two cases for the source are considered and compared: one with a uniformly excited circular piston, and one with a Gaussian distribution. Various source levels are used, and the calculations are carried out into the shock region. The on-axis results for the fundamental amplitude are compared with results derived using the linearized solution modified with various taper functions. Apart from a nonlinear tapering of the amplitude along and near the axis, the results are found to be very close to the linearized solution for the fundamental, and for the second harmonic close to what is obtained from a quasilinear theory. The wave profile is calculated at various ranges. An energy equation for each harmonic is obtained, and shown to be equivalent within our approximation to the three-dimensional version of Westervelt’s energy equation. Recent works on one-dimensional propagation are reviewed and compared.

A study of the propagation and interaction of two collinear finite amplitude sound beams, presented in a previous paper [Naze Tjo/tta et al., “Propagation and interaction of two collinear finite amplitude sound beams,” J. Acoust. Soc. Am. 88, 2859–2870 (1990)] is extended to include the effects of focusing. The validity of the parabolic equation when applied to strongly focused sound beams is discussed. Equations and algorithms based on a transformed parabolic equation are presented and used to compute the interaction between two collinear, focused sound beams, and between one plane wave and a focused sound beam. Numerical results are shown, with special emphasis on parametric generation and parametric reception of sound.

Nonlinear propagation of a periodic sound beam in a dissipative fluid is considered using Fourier series expansion and numerical methods to solve the governing equation of motion in the parabolic approximation. The nearfield was considered in a previous paper [Aanonsen et al., J.Acoust. Soc. Am. 75, 749-768 { 1984)]. The analysis is now extended to the farfield. Numerical and asymptotic results are derived and used to explain the development of the fundamental and harmonic components from the nearfield into the farfield. A discussion is also given of some earlier models for the farfield of directional waves. Emphasis is put on the importance of imposing the proper matching conditions between the nearfield solution and the spherical solution in the farfield in order to obtain a good approximation. Propagation and saturation curves are calculated, as well as beam patterns for various harmonic components. The results are compared with available experimental observations. Nonlinear effects, although generated in the nearfield, are found to be propagated to ranges many tens of Rayleigh distances if the absorption is weak. PACS numbers: 43.25.Jh LIST OF SYMBOLS r(n ) u o A2,o, A2,1 Co D f(g,r) A k P T (n) characteristic length transverse to the direcu tion of propagation/radius of the source see Eq. (16} u, u' isentropic speed of sound at ambient values of w pressure and density x, y, z diffusivity of sound farfield directivity function of the linearized x fundamental component a normal velocity distribution on the source, normalized to Uo; f(• )sin rfor an axisymmetric sinusoidal source Fourier transform off with respect to g e see Eqs. (3)and (26) r/ see Appendix 0, cO/Co, wavenumber {tike)-1, shock formation distance of a plane wave P pressure Po ambient pressure a, ao {P --Po)/poCoUo, acoustic pressure normalized to Po see Eq. (13) poCoUo, acoustic pressure peak amplitude on O'match the source see text below Eq. (16) (X2 q-y2 q_Z2)1/2 7' ka2/2, Rayleigh distance rp, time rs, see Eq. (11) co coefficient in the trivial expansion; see Eq. characteristic velocity peak amplitude on the source g/(1 q_ a) lul, lu'l longitudinal velocity normalized to Co dimensional coordinates, z along the direction of propagation co2D /2Co • , absorption coefficient parameter of nonlinearity O• 2/O•2, O• 2/0•'2 Uo/Co, Math number renormalized variable, see Eq. (20) polar angles of (x,y•) x/a Ixl/a r/% ambient density z/% a• • In a •ro • = •j Range at whifh the solutions of the Kutznetsov and the spherical Burgers equations are matched w(t --Z/Co), retarded time for a plane wave r -Ua, r -U(1 + a) o(t-UCo), wit -(r-%)/Co], retarded time for a spherical wave nonlinear retarded time, see Eq. (22) angular frequency 202

The nonlinear sound pressure field from finite amplitude, plane, rectangular apertures, is considered. Experimental measurements on a 2.25-MHz transducer are presented for various source geometries and excitation levels. These results are compared with simulations based on the transformed beam equation, using a new algorithm which generalizes the Bergen code to treat the case of nonaxisymmetric on-source boundary conditions.