The Application of the Normal Derivative From of Boundary Integral Equations to the Calculation of Transient Sound in a Room

为了在观众席上进行设计，有必要对房间的瞬时响应进行预测。以前已经提出了一些基于射线追踪法或其修改的近似方法。然而，用这种方法在低频范围内精确地计算声场是困难的。虽然我们提出了一种利用边界积分方程(BF)的基本形式的方法，但它在数值计算中存在稳定性问题。本文研究了用基尔霍夫公式的正态导数形式(NDF)求解同一问题的方法。阐明了用NDF得到精确的结果，需要用位势的二次插值法、关于时间导数的二阶差分公式和f(T)=t^2H(T)作为源波，还需要在计算机程序中使用双精度变量，如果使用NDF，数值解中网格尺寸与离散时间间隔之比对稳定性的影响不是很大；()SY.上填充三角形.()x/c()SY.上填充三角形.()/t的值可达20左右。此外，为了在稳态问题中获得唯一解而提出的BF-NDF组合系统，在暂态情况下应引入BF的时间导数。通过几次数值计算得出了以下结论：当数值较小的情况下。通过几次数值计算，得出如下结论：当上填充三角形之比S/c之比取小值时，数值解的稳定性有所提高；反之，比值大时稳定性降低。因此，单独使用NDF和选择适当的参数可以得到最准确的数值解。

For the purpose of designing at auditorium, it is necessary to predict transient responeses in a room. Some approximate methods based on the ray tracing method or its modefications have been presented previously. It is difficult, however, to calculate a sound field accurately by such methods in the low frewuency range. Though we proposed a method using the basic form of boundary integral equation (BF) for this purpose, it has a problem of stability in the numerical calculation. In this research a method using the normal derivative form (NDF) of Kirchhoff's formula for the same problem was investigated. The following was clarified : to obtain accurate results by using NDF, it is necessary to use the quadratic interpolation of potential, the second-order difference formula with respect to time derivative and f(t)=t^2H(t) as the source wave ; it is also necessary to use double-precision variables in computer programs ; the effect of the ratio mesh size to discrete time interval on stability in numerical solutions is not so critical if NDF is used ; the value of (〕SY.upper filled triangle.〔)x/c(〕SY.upper filled triangle.〔)/t can be up to about 20. Furthermore, the BF-NDF combined system, which has been proposed to obtain a unique solution in steady-state problems the time derivative of BF should be introduced in transient cases. Several numerical calculations derived the following conclusions : when a small value of cases. Several numerical calculations derived the following conclusions : when a small value of the ratio (〕SY.upper filled triangle.〔)s/c(〕SY.upper filled triangle.〔)t is taken, the stability of the numerical solution was improved to some extent ; on the contrary, the stability is reduced for large values of the ratio. Therefore the use of NDF alone and the choice of appropriate parameters yield the most accurate numerical solutins.