# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Problems 13 and 14 indicate the form of the general solution of the wave equation and the physical significance of the constant a.

13. Show that the wave equation

a2uxx = utt

can be reduced to the form ufn = 0 by change of variables f = x at, n = x + at. Show that u(x, t) can be written as

u(x, t) = ๔(x at) + ^(x + at),

where ๔ and ty are arbitrary functions.

14. Plot the value of ๔ (x at) for t = 0, 1/a, 2/a, and t0/a if ๔ (s) = sins. Note that for any t = 0 the graph of y = ๔(x at) is the same as that of y = ๔(x) when t = 0, but displaced a distance at in the positive x direction. Thus a represents the velocity at which a disturbance moves along the string. What is the interpretation of ๔ (x + at)?

15. A steel wire 5 ft in length is stretched by a tensile force of 50 lb. The wire has a weight per unit length of 0.026 lb/ft.

(a) Find the velocity of propagation of transverse waves in the wire.

u (x, t) = sin ๊ x cos ๊ at,

n

n

n

uu

ss TT

602

Chapter 10. Partial Differential Equations and Fourier Series

(b) Find the natural frequencies of vibration.

(c) If the tension in the wire is increased, how are the natural frequencies changed? Are the natural modes also changed?

16. A vibrating string moving in an elastic medium satisfies the equation

2 2 a uxx a u = utt,

where a2 is proportional to the coefficient of elasticity of the medium. Suppose that the string is fixed at the ends, and is released with no initial velocity from the initial position u(x, 0) = f (x), 0 < x < L. Find the displacement u(x, t).

17. Consider the wave equation

a2uxx = utt

in an infinite one-dimensional medium subject to the initial conditions

u(x, 0) = f (x), ut(x, 0) = 0, to < x < to.

(a) Using the form of the solution obtained in Problem 13, show that ๔ and f must satisfy

ิ(x) + f(x) = f (x),

๔'^) + f '(x) = 0.

(b) Solve the equations of part (a) for ๔ and f, and thereby show that

u(x, t) = 1[f (x at) + f (x + at)].

This form of the solution was obtained by DAlembert in 1746.

Hint: Note that the equation f'(x) = ๔'(x) is solved by choosing f(x) = ๔ (x) + c.

(c) Let

f( ) _ ฒ2, 1 < x < 1,

f(x ) = 0, otherwise.

Show that

2, 1 + at < x < 1 + at,

f(x at) =

0, otherwise.

Also determine f(x + at).

(d) Sketch the solution found in part (b)at t = 0, t = 1/2a, t = 1/a, and t = 2/a, obtaining the results shown in Figure 10.7.7. Observe that an initial displacement produces two waves moving in opposite directions away from the original location; each wave consists of one-half of the initial displacement.

18. Consider the wave equation

a2uxx = utt

in an infinite one-dimensional medium subject to the initial conditions

u(x, 0) = 0, ut(x, 0) = g(x), to < x < to.

(a) Using the form of the solution obtained in Problem 13, show that

๔(x) + f(x) = 0,

a^(x) + af' (x) = g(x).

(b) Use the first equation of part (a) to show that f'(x) = ๔'(x). Then use the second equation to show that 2a๔'(x) = g(x), and therefore that

1 fx

๔^) = ^ g(%) + ๔(x0),

Jx0

where x0 is arbitrary. Finally, determine f(x).

10.7 The Wave Equation: Vibrations of an Elastic String

603

FIGURE 10.7.7 Propagation of initial disturbance in an infinite one-dimensional medium. (c) Show that

1 px+at

u(x, t) = g(%) d

2a Jx-at

19. By combining the results of Problems 17 and 18 showthat the solution of the problem

a2uxx = utt,

u(x, 0) = f (x), ut(x, 0) = g(x), ๆ < x < ๆ

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