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x = 0
x = L
T (x + A x, t)
V = T sin©
H= Tcos 0
FIGURE 10.B.1 (a) An elastic string under tension. (b) An element of the displaced string. (c) Resolution of the tension T into components.
which acts vertically downward, is assumed to be negligible, and has been neglected in Eq. (2).
If the vertical component of T is denoted by V , then Eq. (2) can be written as
V (x + Ax, t) — V (x, t) _
Passing to the limit as Ax ^ 0 gives
Vx(x, t) = putt(x, t). (3)
To express Eq. (3) entirely in terms of u we note that
V(x, t) = H(t) tan9 = H(t)ux(x, t).
Hence Eq. (3) becomes
(Hux )x = putt,
or, since H is independent of x,
Huxx = putt. (4)
For small motions of the string it is permissible to replace H = T cos 9 by T. Then Eq. (4) takes its customary form
Cl2uxx = Utt, (5)
C2 = T/p. (6)
We will assume further that a2 is a constant, although this is not required in our derivation, even for small motions. Equation (5) is called the wave equation for one space dimension. Since T has the dimension of force, and p that of mass/length, it follows that the constant a has the dimension of velocity. It is possible to identify a as the velocity with which a small disturbance (wave) moves along the string. According to Eq. (6) the wave velocity a varies directly with the tension in the string, but inversely with the density of the string material. These facts are in agreement with experience.
As in the case of the heat conduction equation, there are various generalizations of the wave equation (5). One important equation is known as the telegraph equation and has the form
utt + cut + ku = a2uxx + F (x, t), (7)
where c and k are nonnegative constants. The terms cut, ku, and F(x, t) arise from a viscous damping force, an elastic restoring force, and an external force, respectively. Note the similarity of Eq. (7), except for the term a2uxx, with the equation for the spring-mass system derived in Section 3.8; the additional term a2uxx arises from a consideration of internal elastic forces.
For a vibrating system with more than one significant space coordinate, it may be necessary to consider the wave equation in two dimensions,
a (uxx + uyy) = utt, (8)
or in three dimensions,
a2(uxx + uyy + uzz) = utt.
Chapter 10. Partial Differential Equations and Fourier Series
The following books contain additional information on Fourier series:
Buck, R. C., and Buck, E. F., Advanced Calculus (3rd ed.) (New York: McGraw-Hill, 1978). Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals (3rd ed.) (Cambridge: Cambridge University Press, 1930; New York: Dover, 1952).
Courant, R., and John, F., Introduction to Calculus and Analysis (New York: Wiley-Interscience, 1965; reprinted by Springer-Verlag, New York, 1989).
Kaplan, W., Advanced Calculus (4th ed.) (Reading, MA: Addison-Wesley, 1991).
A brief biography of Fourier and an annotated copy of his 1807 paper are contained in: Grattan-Guinness, I., Joseph Fourier 1768-1830 (Cambridge, MA: MIT Press, 1973).
Useful references on partial differential equations and the method of separation of variables include the following:
Churchill, R. V, and Brown, J. W., Fourier Series and Boundary Value Problems (5th ed.) (New York: McGraw-Hill, 1993).
Haberman, R., Elementary Applied Partial Differential Equations (3rd ed.) (Englewood Cliffs, NJ: Prentice Hall, 1998).
Pinsky, M. A., Partial Differential Equations and Boundary Value Problems with Applications (3rd ed.) (Boston: WCB/McGraw-Hill, 1998).
Powers, D. L., Boundary Value Problems (4th ed.) (San Diego: Academic Press, 1999).
Strauss, W. A., Partial Differential Equations, an Introduction (New York: Wiley, 1992). Weinberger, H. F., A First Course in Partial Differential Equations (New York: Wiley, 1965; New York: Dover, 1995).
Boundary Value Problems and Sturm-Liouville Theory
As a result of separating variables in a partial differential equation in Chapter 10 we repeatedly encountered the differential equation
X" + XX = 0, 0 < x < L
with the boundary conditions
X(0) = 0, X(L) = 0.
This boundary value problem is the prototype of a large class of problems that are important in applied mathematics. These problems are known as Sturm-Liouville boundary value problems. In this chapter we discuss the major properties of Sturm-Liouville problems and their solutions; in the process we are able to generalize somewhat the method of separation of variables for partial differential equations.
11.1 The Occurrence of Two-Point Boundary Value Problems
In Chapter 10 we described the method of separation of variables as a means of solving certain problems involving partial differential equations. The heat conduction problem consisting of the partial differential equation
a2uxx = ut, 0 < x < L, t > 0 (1)
Chapter 11. Boundary Value Problems and Sturm Liouville Theory
subject to the boundary conditions