Ben-Benjamin Jonathan S, Cohen Leon, Loughlin Patrick J
City University of New York, 695 Park Avenue, New York, New York 10065, USA.
University of Pittsburgh, 302 Benedum Hall, Pittsburgh, Pennsylvania 15261, USA.
J Acoust Soc Am. 2015 Aug;138(2):1122-31. doi: 10.1121/1.4926562.
A phase space approximation method for linear dispersive wave propagation with arbitrary initial conditions is developed. The results expand on a previous approximation in terms of the Wigner distribution of a single mode. In contrast to this previously considered single-mode case, the approximation presented here is for the full wave and is obtained by a different approach. This solution requires one to obtain (i) the initial modal functions from the given initial wave, and (ii) the initial cross-Wigner distribution between different modal functions. The full wave is the sum of modal functions. The approximation is obtained for general linear wave equations by transforming the equations to phase space, and then solving in the new domain. It is shown that each modal function of the wave satisfies a Schrödinger-type equation where the equivalent "Hamiltonian" operator is the dispersion relation corresponding to the mode and where the wavenumber is replaced by the wavenumber operator. Application to the beam equation is considered to illustrate the approach.
开发了一种用于具有任意初始条件的线性色散波传播的相空间近似方法。结果扩展了先前关于单模维格纳分布的近似。与先前考虑的单模情况不同，这里提出的近似是针对全波的，并且是通过不同的方法获得的。该解要求人们（i）从给定的初始波中获得初始模态函数，以及（ii）不同模态函数之间的初始交叉维格纳分布。全波是模态函数的总和。通过将方程变换到相空间，然后在新的域中求解，得到了一般线性波动方程的近似。结果表明，波的每个模态函数都满足一个薛定谔型方程，其中等效的“哈密顿”算子是与该模式相对应的色散关系，并且波数被波数算子所取代。考虑将其应用于梁方程以说明该方法。
