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

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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.