Far-field resonant third harmonic surface wave on a half-space of incompressible material of cubic nonlinearity

The analytical far-field solution for the cumulative third harmonic surface wave propagating on a half-space of isotropic incompressible cubically nonlinear material is obtained in a relative simple and systematic manner. Using the perturbation method, the governing equations and the boundary conditions for a weakly nonlinear material are separated into uncoupled equations at the zeroth and first-order. For a primary linear wave of frequency ω and amplitude A¯, the resonant third harmonic has frequency 3ω and amplitude AN which depends on A¯3 and a multiplying factor x, which is the distance of propagation. It is shown that, in the far field, the resonant third harmonic propagates with the classic Rayleigh wave velocity. We also consider the transmission of the resonant third harmonic across an interface at x=L into a linear material. The transmitted wave has the same general form as the incident third harmonic except that the multiplying factor x now is constant at L, t > L/c, x > L, and the amplitude also depends on the nonlinear constant G. Potential measurement of the transmitted wave can provide information on the location of the interface and the constant G of the material nonlinearity.