Airbus Operations SAS, Acoustics Department, Turbomachinery Acoustics, 31060 Toulouse, France.
J Acoust Soc Am. 2011 Feb;129(2):604-15. doi: 10.1121/1.3531799.
A model is developed for the propagation of finite amplitude acoustical waves and weak shocks in a straight duct of arbitrary cross section. It generalizes the linear modal solution, assuming mode amplitudes slowly vary along the guide axis under the influence of nonlinearities. Using orthogonality properties, the model finally reduces to a set of ordinary differential equations for each mode at each of the harmonics of the input frequency. The theory is then applied to a two-dimensional waveguide. Dispersion relations indicate that there can be two types of nonlinear interactions either called "resonant" or "non-resonant." Resonant interactions occur dominantly for modes propagating at a rather large angle with respect to the axis and involve mostly modes propagating with the same phase velocity. In this case, guided propagation is similar to nonlinear plane wave propagation, with the progressive steepening up to shock formation of the two waves that constitute the mode and reflect onto the guide walls. Non-resonant interactions can be observed as the input modes propagate at a small angle, in which case, nonlinear interactions involve many adjacent modes having close phase velocities. Grazing propagation can also lead to more complex phenomena such as wavefront curvature and irregular reflection.
建立了一个模型来研究有限振幅声波和弱激波在任意截面直管道中的传播。该模型假设在非线性的影响下，模态幅度在导波轴上缓慢变化，从而推广了线性模态解。利用正交性性质，模型最终在输入频率的各谐频下，为每个模态都简化为一组常微分方程。然后，将该理论应用于二维波导。色散关系表明，可能存在两种类型的非线性相互作用，分别称为“共振”或“非共振”。对于相对于轴以较大角度传播的模态，主要发生共振相互作用，并且主要涉及与相同相速度传播的模态。在这种情况下，导波传播类似于非线性平面波传播，构成模态的两个波逐渐变陡，直至形成激波，并反射到波导壁上。在输入模态以小角度传播的情况下，可以观察到非共振相互作用，在这种情况下，非线性相互作用涉及许多具有相近相速度的相邻模态。掠射传播也可能导致更复杂的现象，如波前曲率和不规则反射。
