Spherical waves in odd-dimensional space
Author:
J. G. Kingston
Journal:
Quart. Appl. Math. 46 (1988), 775-778
MSC:
Primary 35L05; Secondary 35Q05
DOI:
https://doi.org/10.1090/qam/973389
MathSciNet review:
973389
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Abstract: The general solution is given of the $\left ( {2N + 1} \right )$-dimensional wave equation with spherical symmetry, ${u_{tt}} - {u_{xx}} - \frac {{2N}}{x}{u_x} = 0$, in terms of two arbitrary functions and their first $N$ derivatives. Simple transformations then yield the general solutions to the Euler-Poisson-Darboux equation, ${u_{xy}} + \frac {N}{{\left ( {x + y} \right )}}\left ( {{u_x} + {u_y}} \right ) = 0$, for integer $N$, and also the one-dimensional wave equation, ${u_{tt}} - {c^2}{u_{xx}} = 0$, for certain variable wave speeds $c\left ( x \right )$.
- J. L. Synge, On the vibrations of a heterogeneous string, Quart. Appl. Math. 39 (1981/82), no. 2, 292–297. MR 625476, DOI https://doi.org/10.1090/S0033-569X-1981-0625476-6
- Brian Seymour and Eric Varley, Exact representations for acoustical waves when the sound speed varies in space and time, Stud. Appl. Math. 76 (1987), no. 1, 1–35. MR 881794, DOI https://doi.org/10.1002/sapm19877611
- George W. Bluman, On mapping linear partial differential equations to constant coefficient equations, SIAM J. Appl. Math. 43 (1983), no. 6, 1259–1273. MR 722940, DOI https://doi.org/10.1137/0143084
J. G. Kingston, Solutions of the generalized Stokes-Beltrami equations, Lett. Appl. Eng. Sci. 4, 395–400 (1976)
- E. T. Copson, Partial differential equations, Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR 0393746
J. L. Synge, On the vibrations of a heterogeneous string, Quart. Appl. Math. 39, 292–297 (1981)
B. R. Seymour and E. Varley, Exact representations for acoustical waves when the sound speed varies in space and time, Stud. Appl. Math. 76, 1–35 (1987)
G. W. Bluman, On mapping linear partial differential equations to constant coefficient equations, SIAM J. Appl. Math. 43, 1259–1273 (1983)
J. G. Kingston, Solutions of the generalized Stokes-Beltrami equations, Lett. Appl. Eng. Sci. 4, 395–400 (1976)
E. T. Copson, Partial Differential Equations, Cambridge University Press, Cambridge, 1975
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Article copyright:
© Copyright 1988
American Mathematical Society