ウェブスターのホーン方程式における摂動論による高精度計算手法

Abstract

Applying mathematical and quantum mechanical techniques, this study develops numerical solutions to Webster's horn equation, which describes the sound inside brass instruments in acoustics. The wavenumbers and wave functions of the modes in the system are evaluated by perturbation theory, assuming a solvable system with relatively small perturbations. An obvious solvable example is a straight pipe, whose wavenumbers can be perturbed by varying the radius of the horn. Maintaining the second-order corrections, the method generated astonishingly accurate results for varying horn shapes. Moreover, in tests of various pipe shapes, the perturbation method required far fewer computational resources than the finite element method. Two analytically solvable shapes and two non-solvable models (one of them is a periodic shape described by a trigonometric function) are analyzed. The results imply the applicability of the method to highly non-solvable systems.