The Sturm–Liouville problem for the equation d2X/dx2+λ2X=0 on a closed interval with general unmixed boundary conditions leads to periodic, linear, and exponential eigenfunctions. However, examples of physical situations where different types of these eigenfunctions are employed together in a general solution are difficult to find. We present a physical problem of this type.

1.
The uniqueness theorems for second-order partial differential equations applied to general unmixed boundary conditions can be found in Ref. 7.
2.
Almost any text on mathematical physics covering the Sturm–Liouville problem and the method of separation of variables can be cited. See Ref. 8, Vol. I, pp. 287–289; Ref. 9, Chap. II, Paragraph 3 and Chap. III, Paragraph 2; Ref. 10, Chap. VIII, Paragraph 8 and Chap. IX, Paragraph 2; Ref. 11, pp. 581–582.
3.
See for example, the remark about the uniqueness theorem for zero boundary conditions in Ref. 12, p. 22.
4.
Values of the separation parameter λ2 for common initial-boundary value problems in physics tend to be positive so that linear and/or exponential solutions of Eq. (1) are absent. See Ref. 8, Vol. I, Chap. V; Ref. 13, Chap. II, Paragraph 3 and Chap. III, Paragraph 2.
5.
For the case of longitudinal oscillations of a rod with elastically attached ends, Hooke’s constant characterizes the degree of rigidity of the end attachment. See Ref. 13, p. 172.
6.
See Ref. 8, Vol. II, p. 183 on the substitution to eliminate first-order derivatives in partial differential equations of second order and Ref. 9, p. 206 for its application to Eq. (14).
7.
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, Berlin, 1985).
8.
R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1989).
9.
A. N. Tykhonov and A. A. Samarskii, Equations of Mathematical Physics (Pergamon, New York, 1963).
10.
E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications (Williams and Wilkins, Baltimore, 1976).
11.
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 1995), 4th ed., p. 544.
12.
J. R. Cannon, “The one-dimensional heat equation,” in Encyclopedia of Mathematics and its Applications, F. E. Browder, section editor (Addison–Wesley, Reading, MA, 1984), Vol. 23.
13.
B. M. Budak, A. A. Samarskii, and A. N. Tykhonov, A Collection of Problems in Mathematical Physics (Pergamon, New York, 1964; Dover, New York, 1988).
14.
Yu.
Sosov
and
C. E.
Theodosiou
, “
On the complete solution of the Sturm-Liouville problem d2X/dx2+λ2X=0 over a closed interval
,”
J. Math. Phys.
43
,
2831
–
2843
(
2002
).
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© 2004 American Association of Physics Teachers.
2004
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