In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

中文翻译：

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Westervelt方程的周期解和多调和展开

在本文中，我们根据Westervelt方程考虑了非线性时间周期声音的传播，该方程是非线性声学的经典模型，并且具有潜在简并性的二阶拟线性强阻尼波方程。我们在具有吸收边界条件的有界域上证明了具有时间周期强迫和时间周期初端条件的解的存在性，唯一性和正则性。为了数学上恢复高次谐波的物理现象，我们将解扩展为在基本激励频率的倍数处的频率上的贡献叠加。事实证明，这种多谐展开可以在适当的功能空间中收敛到时域的周期解。