Interpretation of Ultrasonic Experiments on Finite-Amplitude Waves

Abstract

An analysis was made for the purpose of relating higher-order elastic constants to the growth of harmonics in an initially sinusoidal wave. It is supposed that the face of a semiinfinite nondissipative elastic medium—for which the equation of one-dimensional motion is ü = (∂2u/∂x2)g(∂u/∂x)—is subjected to the sinusoidal displacement u = u0 cos(ωt−φ). The function g(∂u/∂x) is given as g(ξ) = (1/ρ0)[M2+M3ξ+M4ξ2+⋯+Miξi−2], where Mi depends on elastic constants up to and including order i. A solution, including the effects due to terms as high as M5 in the function g(ξ), is expressed as u = u0 Σn{Dn sin[n(ωt− Kx)]+En cos[n(ωt− Kx)]}+F(x), where the summation over n goes from 1 to ∞. Dn and En are expanded in powers of a Mach number M and a dimensionless distance X (M = ωu0/c0, X = ωx/c0, c02 = M2/ρ0). It is found that as u0 → 0, the limiting value of the ratio of the second-harmonic amplitude, H2 = u0(D22+E22)12, to u0M is given by −M3X/8M2, independently of M4, M5, etc. Thus, with M2 already known, M3 can be calculated from an experimental determination of this ratio. Similarly, it is found that the limiting value of the ratio of the third harmonic amplitude to u0M2 depends on elastic constants only up through order four. [Tutorial material is included to make the paper reasonably self-contained and to relate the present analysis to corresponding work for fluids. Careful derivations show that purely longitudinal waves in elastic solids and nondissipative fluids are governed by an equation of motion of precisely the same form. Hence, with appropriate choice of the parameters, an analysis of the one case applies equally well to the other.]