Symmetric flexural wave transmission and linear behaviour in a nonlinear system of two scatterers on a beam

The multiple scattering of flexural waves on a beam with two point scatterers attached to it, where one scatterer has a cubically nonlinear stiffness, is considered. The scattering behaviour is represented by first-order approximations of the reflection and transmission coefficients calculated with a perturbation approach, both at the fundamental frequency and at the generated third harmonic. Due to its nonlinearity and spatial asymmetry, such a system behaves in general nonreciprocally. By analyzing the fundamental-frequency and third-harmonic transmission coefficients, we derive conditions for particular scattering behaviours that result in transmission symmetry for the nonlinear system. We show that a sufficient condition for symmetric fundamental-frequency transmission is that the spacing between the scatterers is a multiple of half the incident wavelength, regardless of the scatterers' properties, although the values of the reflection and transmission coefficients will still depend on the incident-wave amplitude. Such a response is also possible if the real and imaginary parts of the admittance of the linear scatterer satisfy a certain mathematical relation, regardless of spacing. A particular choice of lossless linear scatterer combined with the separation distance being a multiple of half a wavelength leads to fully linear behaviour for the nonlinear multiple scattering system.