Abstract
This paper deals with the transient acoustic scattering in the particular geometry of a flat objet (crack) inR 3. The boundary integral for the “crack opening displacement” is studied as a spatial pseudo-differential equation with the frequency variable as a parameter. Existence, Uniqueness and Continuous dependence of the solution with respect to the data are obtained in the framework of Sobolev spaces of causal functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bamberger and T. Ha-Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique. Math. Methods Appl. Sci.,8 (1986), 405–435.
A. Bamberger and T. Ha-Duong, Formulation variationnelle espace-temps …; Problème de Neumann, Math. Methods Appl. Sci.,8 (1986), 598–608.
C.L. Bennet and H. Mieras, Time domain integral equation for acoustic scattering from fluid targets. J. Acoust. Soc. Amer.,59 (1981), 1261–1265.
J. Charazain and A. Piriou, Introduction à la Théorie des Équations aux Dérivées Partielles. Gauthier-Villars, Paris, 1981.
T. Ha-Duong, Equations Intégrales pour la résolution numérique de problèmes de diffraction acoustique dansR 3. Thèse de doctorat d’Etat, Université Paris VI, 1987.
T. Ha-Duong, On the first kind boundary integral equation for flat cracks. Proc. of the IU-TAM Symposium on Elastic waves, Galway, Ireland, 1988, Rapport Interne n0 194, CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
M.A. Hamdi, Une formulation variationnelle par équations intégrales pour la résolution de l’équation de Helmholtz. C.R.Acad Sci. Paris, sér. II,292 (1981), 17–20.
D.S. Jones, A new method for calculating scattering with particular reference to the circular disc. Comm. Pure Appl. Math.,9 (1956), 713–746.
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, 1972.
K.M. Mitzner, Numerical solution of transient scattering from a hard surface of arbitary shape—Retarded potential technique. J. Acoust. Soc. Amer.,42 (1967), 391–397.
J.C. Nédélec, Integral equations with non integrable kernels. Integral Equations Operator Theory,5 (1982), 561–572.
W. Rudin, Real and Complex Analysis. MacGraw-Hill, New-York, 1974.
B.H. Sako, A Model for the Crack and Punch Problem in Elasticity. Thesis, UCLA, 1986.
L. Schwartz, Théorie des Distributions. Hermann, Paris, 1957.
R.P. Shaw and J. English, Transient acoustic scattering by a free (pressure release) sphere. J. Sound Vibration,20 (1972), 321–331.
P. Trèves, Basic Linear Partial Differential Equations. Academic Press, New-York, 1975.
Author information
Authors and Affiliations
About this article
Cite this article
Ha-Duong, T. On the transient acoustic scattering by a flat object. Japan J. Appl. Math. 7, 489–513 (1990). https://doi.org/10.1007/BF03167856
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167856