On the uniqueness of boundary integral equation for the exterior Helmholtz problem. (English) Zbl 0755.76078

Summary: From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated. It is proved that the Sommerfeld’s condition at the infinity is changed so that it is suitable not only for the radiative wave but also for the absorptive wave when we use the boundary integral equation to describe the exterior Helmholtz problem. Therefore, the total energy of the system is conservative. The mathematical dealings to guarantee the uniqueness are discussed based upon this explanation.


76Q05 Hydro- and aero-acoustics
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


[1] Sommerfeld, A.,Partial Differential Equation in Physics, Academic Press (1949). · Zbl 0034.35702
[2] Schenck, H. A., Improved integral formulation for acoustics radiation problems,J. Acoust. Soc. Am.,44 (1968), 41–58. · Zbl 0187.50302
[3] Burton, A.J. and G.F. Miller, The application of integral equation methods to the numerical solution of some exterior boundary value problems,Proc. R. Soc. London., Ser. A.,323 (1971), 201–210. · Zbl 0235.65080
[4] Waterman, P.C., New formulation of acoustic scattering,J. Acoust. Soc. Am.,45 (1969), 1417–1429. · Zbl 0187.24504
[5] Kleinman, R.E. and G.F. Roach, Boundary integral equations for the three-dimensional Helmholtz equation,SIAM, Review,16 (1974), 214–236. · Zbl 0282.35026
[6] Piaszczyk, C.M., Acoustic radiation from vibrating surface at characteristic frequencies,J. Acoust. Soc. Am.,75 (1984), 363–375. · Zbl 0529.73034
[7] Martin, P.A., On the null-field equations for the exterior problems of acoustics,Q.J. Mech. Appl. Math.,33 (1980), 385–396. · Zbl 0507.76081
[8] Brod, K., On the uniqueness of solution for all wave numbers in acoustic radiation,J. Acoust. Soc. Am.,76 (1984), 1238–243. · Zbl 0557.35020
[9] Filippi, P.J.T., Integral equations in acoustics,Theoretical Acoustics and Numerical Techniques, Ed. by P. Filippi, Springer-Verlag (1983), 1–49. · Zbl 0544.00039
[10] Meyer, W.L. W.A. Bell, B.T. Zinn and M.P. Stallybrass, Boundary integral solution of three dimensional acoustic radiation problems,J. Sound and Vib.,59, 2 (1978), 242–262. · Zbl 0391.76052
[11] Amini, S. and D.T. Wilton, An investigation of boundary element methods for the exterior acoustic problem,Compu. Meth. in Appl. Mech. and Eng.,54 (1986), 49–65. · Zbl 0563.73067
[12] Smirnov, V.I.,A Course of Higher Mathematics. Pergamon Press, Oxford (1964). · Zbl 0121.25904
[13] Courant, R. and D. Hilbert,Methods of Mathematical Physics,Vol II, Science Press, Beijing (1977). (Chinese version) · JFM 57.0245.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.