N. W. McLachlan and A. L. Meyers [Proc. Phys. Soc. (London) 47, 644 (1935)] and K. A. Naugol'nykh [Soviet Phys.—Acoustics 5, 79 (1959)] have obtained solutions valid to second order in the wave amplitude for spherically symmetric waves. We have generalized their results and obtained solutions for nonspherically symmetric waves. If the first order density wave satisfying the linear wave equation is taken to be ρ1 = (A/r)e−i(ωt−kr) × f(θ,φ), we find the second-order correction to the density to be ρ2 = −12 iρ0−1kA2r[ln(rr0)][2+c0−2ρ0(d2pdρ2)ρ=ρ0]×e−2t(ωt−kr)f2(θ,φ). This result is subject to the restrictions that kr≫1 and that |∇f| ≪k. This result corresponds to the solution found in the literature for the special case of the spherically symmetric wave, f(θ,φ) = 1. (This work was supported jointly by the U. S. Office of Naval Research and the Aeronautical Systems Division, Air Research and Development Command, U. S. Air Force.)
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