Homogeneous solutions in elastic wave propagation

Author:
John W. Miles

Journal:
Quart. Appl. Math. **18** (1960), 37-59

MSC:
Primary 73.00

DOI:
https://doi.org/10.1090/qam/111291

MathSciNet review:
111291

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Busemann's method of conical flows is formulated for two-dimensional elastic wave propagation. The equations of motion are reduced to either Laplace's equation in two dimensions or the wave equation in one dimension, and solutions then are obtained with the aid of complex variable or characteristics theory, respectively. Special attention is paid to that class of problems in which the hyperbolic domains (of the two-dimensional wave equation) are simple wave zones, in consequence of which the solutions may be continued into the elliptic domain (of Laplace's equation) without explicitly posing the boundary conditions on the boundary separating the two domains. The method is applied to the diffraction of - and -pulses by a perfectly weak half-plane.

**[1]**G. Green, Trans. Camb. Phil. Soc.**5**, 395 (1835)**[2]**H. Bateman,*The mathematical analysis of electrical and optical wave motion*, Cambridge University Press, 1915, Chap.**7**; Bateman gives extensive references to work prior to 1915**[3]**A. Busemann, Luftfahrt-Forsch.**12**, 210 (1935); Schr. Dtschen. Akad. Luftfahrt-Forsch.**7**B, 105 (1943)**[4]**S. A. Chaplygin, Sci. Annals Imp. Univ. of Moscow, Phys.-Math. Div. 21 (1904)**[5]**W. F. Donkin, Phil. Trans.**147**, 43 (1857); Bateman,*op. cit*., p. 114**[6]**G. N. Ward,*Linearized theory of steady high-speed flow*, Cambridge University Press, 1955, Chap. 7; Ward gives a comprehensive bibliography of papers on conical flows MR**0067649****[7]**A. A. Kharkevich, Zhur Tekh. Fiz.**19**, 828 (1949)**[8]**H. Davis, M. S. Thesis, University of California, Los Angeles, 1950**[9]**J. B. Keller and A. Blank, Communs. on Pure and Appl. Math.**4**, 75 (1951) MR**0043714****[10]**J. W. Miles, Proc. Roy. Soc. A**212**, 543-547; 547-551 (1952) MR**0053722****[11]**S. Sobolev, Publ. Inst. Seism. Acad. Sci. U. R. S. S., No. 18 (1932)**[12]**A. T. de Hoop,*Representation theorems for the displacement in an elastic solid and their application to elastodynamic diffraction theory*, Thesis, Technische Hogeschool te Delft, 1958**[13]**A. W. Maue, Z. angew. Math. Mech.**34**, 1-12 (1954) MR**0064620****[14]**M. Ewing, W. Jardetzky, and F. Press,*Elastic waves in layered media*, McGraw-Hill Book Co., New York, 1957, Chap. 2 MR**0094967****[15]**Ewing*et al., op. cit*., Secs. 1-5**[16]**R. Courant and K. O. Friedrichs,*Supersonic flow and shock waves*, Interscience Publishers, New York, 1948, Sec. 106 MR**0029615****[17]**Ewing*et al., op. cit*., p. 27**[18]***Ibid*., p. 28

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73.00

Retrieve articles in all journals with MSC: 73.00

Additional Information

DOI:
https://doi.org/10.1090/qam/111291

Article copyright:
© Copyright 1960
American Mathematical Society