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

在礼堂设计中，有必要对房间内的瞬态响应进行预测。在光线追踪法的基础上提出了一些近似方法或其修正。然而，用这种方法在低频范围内精确计算声场是困难的。虽然我们提出了一种基于边界积分方程基本形式的方法，但它在数值计算中存在稳定性问题。本文研究了用Kirchhoff公式的正导数形式（NDF）求解同一问题的方法。澄清了以下问题：为了得到准确的NDF结果，必须使用二次势插值、二阶时间导数差分公式和f(t)=t^2H(t)作为源波；在计算机程序中使用双精度变量也是必要的；采用NDF时，网格尺寸与离散时间间隔之比对数值解稳定性的影响不大；(] SY的值。上填充三角形。[)x/c(] SY。上部填充三角形。[)/t最多可达20左右。此外，为了获得稳态问题的唯一解，提出了BF- ndf组合系统，在瞬态情况下需要引入BF的时间导数。几次数值计算得出以下结论：当数值较小的情况下。若干数值计算得出以下结论：当比值（]SY）值很小时。上填充三角形。[)s/c(] SY。上部填充三角形。取[)t，一定程度上提高了数值解的稳定性；相反，当比值较大时，稳定性会降低。因此，仅使用NDF和选择适当的参数可得到最精确的数值解。

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.