An Efficient Method for Calculation of Natural Frequencies for Wave Equation Inside a Trapezoid Region

In the classical problem of wave dynamics, we study the internal problem inside a two-dimensional trapezoid region. The specific property of wave problems in such domains is that any numerical algorithm loses its stability near irregular points of the boundary curve – near corners, sharp angles, slits, and so on. In the present work we propose a method which allows us to overcome this obstacle, to come to stable computations. The method is based on the fact that the instability of the algorithm is connected with the situation when both the point of observation and the point of integration approach simultaneously any corner, i.e. when the distance between them becomes small. Just in this case we apply an asymptotic expansion of the kernel for small argument, which is the argument of the Green’s function, being the Hankel function in the present problem. Then the integral on the interval adjacent to the corner can be calculated analytically in an explicit form that leads to a stable algorithm. The proposed algorithm is tested by an example taken from the Room acoustics, in calculation of low natural frequencies, below 200 Hz, in the case of a small studio of a trapezoid geometry.