时间调制介质中波的调制傅里叶展开
The modulated Fourier expansion for waves propagating through time-modulated media
AI总结：
本文提出了一种用于时间调制声波方程的调制傅里叶展开方法，通过平滑变化的系数函数解决耦合演化偏微分方程组，适用于小振幅快速时间调制，并通过数值实验验证了其有效性。
摘要：
通过主动改变介质参数来控制波，可以开发新的声学和电气设备。调节材料会打破经典性质，如互易性和能量守恒，这使得数学分析复杂化。在没有限制幅度原理的情况下，时谐公式通常不适用。本文开发了一种用于时间调制声波方程的替代工具，即调制傅里叶展开（MFE）。解由多个平滑变化的系数函数特征化，这些函数解决了一个具有时间常数系数的耦合演化偏微分方程组。对于小振幅快速时间调制，该方程组被证明具有平滑变化的解，该解在小缺陷内精确描述解。导出的耦合系统的离散化产生了一种在较大时间步长下更稳定和准确的积分器，相比直接应用于时间调制声波方程的方案。数值实验展示了理论结果和该方法的使用。
英文摘要：
Controlling waves by actively changing the material parameters of a medium enables the development of new acoustic and electrical devices. Modulating the material breaks classical properties like reciprocity and the conservation of energy, which complicates the mathematical analysis. Without a limiting amplitude principle, time-harmonic formulations are generally inapplicable. The present manuscript develops an alternative tool for the time-modulated acoustic wave equation, that is based on a modulated Fourier expansion (MFE). The solution is characterized by multiple smoothly varying coefficient functions, which solve a coupled system of evolutionary partial differential equations with temporally constant coefficients. For small-amplitude fast-time modulations, this system of evolutionary partial differential equations is shown to possess a smoothly varying solution, which characterizes the exact solution up to a small defect. Discretization of the derived coupled system yields integrators that are stable and accurate when larger time steps are used, compared to those schemes that are applied to the time-modulated acoustic wave equation directly. Numerical experiments illustrate the theoretical results and the use of the approach.
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