
# 12.2E: The Wave Equation (Exercises)



and, on the other hand,

$u(x,\tau)= \left\{\begin{array}{cl} x,&0\le x\le{L\over2}-a\tau,\\[5pt] {L\over2}-a\tau,&{L\over2}-a\tau\le x\le{L\over2}+a\tau,\\[5pt] L-x,&{L\over2}-a\tau\le x\le L. \end{array}\right. \tag{B}$

if $$0\le \tau\le L/2a$$. The first objective of this exercise is to show that (B) can be used to compute $$u(x,t)$$ for $$0\le x\le L$$ and all $$t>0$$.

1. Show that if $$t>0$$, there’s a nonnegative integer $$m$$ such that either ${\bf(i)}\quad t={mL\over a}+\tau\quad \text{or} \quad {\bf(ii)}\quad t={(m+1)L\over a}-\tau,\nonumber$ where $$0\le \tau\le L/2a$$.
2. Use (A) to show that $$u(x,t)=(-1)^mu(x,\tau)$$ if (i) holds, while $$u(x,t)=(-1)^{m+1}u(x,\tau)$$ if (ii) holds.
3. Perform the following experiment for specific values of $$L$$ and $$a$$ and various values of $$m$$ and $$k$$: Let $t_j={Lj\over 2ka},\quad j=0,1,\dots k;\nonumber$ thus, $$t_0$$, $$t_1$$, …, $$t_k$$ are equally spaced points in $$[0,L/2a]$$. For each $$j=0$$, $$1$$ , $$2$$,…, $$k$$, graph the $$m$$th partial sum of (A) and $$u(x,t_j)$$ computed from (B) on the same axis. Create an animation, as described in the remarks on using technology at the end of the section.

17. If a string vibrates with the end at $$x=0$$ free to move in a frictionless vertical track and the end at $$x=L$$ fixed, then the initial-boundary value problem for its displacement takes the form

$\begin{array}{c} u_{tt}=a^2u_{xx},\quad 0<x<L,\quad t>0,\\ u_x(0,t)=0,\quad u(L,t)=0,\quad t>0,\\ u(x,0)=f(x),\quad u_t(x,0)=g(x),\quad 0\le x\le L. \end{array} \tag{A}$

Justify defining the formal solution of (A) to be

$u(x,t)=\sum_{n=1}^\infty \left(\alpha_n\cos{(2n-1)\pi a t\over2L}+{2L\beta_n\over(2n-1)\pi a}\sin{(2n-1)\pi at\over2L}\right) \cos{(2n-1)\pi x\over2L},\nonumber$

where

$C_{M\!f}(x)=\sum_{n=1}^\infty\alpha_n\cos{(2n-1)\pi x\over2L} \quad \text{and} \quad C_{M\!g}(x)=\sum_{n=1}^\infty\beta_n\cos{(2n-1)\pi x\over2L}\nonumber$

are the mixed Fourier cosine series of $$f$$ and $$g$$ on $$[0,L]$$; that is,

$\alpha_n={2\over L}\int_0^Lf(x)\cos{(2n-1)\pi x\over2L}\,dx \quad \text{and} \quad \beta_n={2\over L}\int_0^Lg(x)\cos{(2n-1)\pi x\over2L}\,dx.\nonumber$

## Q12.2.3

In Exercises 12.2.18-12.2.31, use Exercise 12.2.17 to solve the initial-boundary value problem. In some of these exercises Theorem 11.3.5c or Exercise 11.3.42b will simplify the computation of the coefficients in the mixed Fourier cosine series.

18. $$u_{tt}=9u_{xx},\quad 0<x<2,\quad t>0$$,
$$u_x(0,t)=0,\quad u(2,t)=0,\quad t>0$$,
$$u(x,0)=4-x^2,\quad u_t(x,0)=0,\quad0\le x\le2$$

19. $$u_{tt}=4u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=x^2(1-x),\quad u_t(x,0)=0,\quad0\le x\le 1$$

20. $$u_{tt}=9u_{xx},\quad 0<x<2,\quad t>0$$,
$$u_x(0,t)=0,\quad u(2,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=4-x^2,\quad0\le x\le2$$

21. $$u_{tt}=4u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^2(1-x),\quad0\le x\le 1$$

22. $$u_{tt}=5u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=2x^3+3x^2-5,\quad u_t(x,0)=0,\quad0\le x\le1$$

23. $$u_{tt}=3u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u(\pi,t)=0,\quad t>0$$,
$$u(x,0)=\pi^3-x^3,\quad u_t(x,0)=0,\quad0\le x\le\pi$$

24. $$u_{tt}=5u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=2x^3+3x^2-5,\quad0\le x\le1$$

25. $$u_{tt}=3u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u(\pi,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=\pi^3-x^3,\quad0\le x\le\pi$$

26. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=x^4-2x^3+1,\quad u_t(x,0)=0,\quad0\le x\le1$$

27. $$u_{tt}=7u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=4x^3+3x^2-7,\quad u_t(x,0)=0,\quad0\le x\le1$$

28. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^4-2x^3+1,\quad0\le x\le1$$

29. $$u_{tt}=7u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=4x^3+3x^2-7,\quad0\le x\le1$$

30. $$u_{tt}=u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=x^4-4x^3+6x^2-3,\quad u_t(x,0)=0,\quad0\le x\le1$$

31. $$u_{tt}=u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^4-4x^3+6x^2-3,\quad0\le x\le1$$

## Q12.2.4

32. Adapt the proof of Theorem 12.2.2 to find d’Alembert’s solution of the initial-boundary value problem in Exercise 12.2.17.

33. Use the result of Exercise 12.2.32 to show that the formal solution of the initial-boundary value problem in Exercise 12.2.17 is an actual solution if $$g$$ is differentiable and $$f$$ is twice differentiable on $$[0,L]$$ and

$g'_+(0)=g(L)=f'_+(0)=f(L)=f''_-(L)=0.\nonumber$

HINT: See Exercise 11.3.57, and apply Theorem 12.2.3 with $$L$$ replaced by $$2L$$.

34. Justify defining the formal solution of the initial-boundary value problem

$\begin{array}{c} u_{tt}=a^2u_{xx},\quad 0<x<L,\quad t>0,\\ u(0,t)=0,\quad u_x(L,t)=0,\quad t>0,\\ u(x,0)=f(x),\quad u_t(x,0)=g(x),\quad 0\le x\le L \end{array}\nonumber$

to be

$u(x,t)=\sum_{n=1}^\infty \left(\alpha_n\cos{(2n-1)\pi a t\over2L}+{2L\beta_n\over(2n-1)\pi a}\sin{(2n-1)\pi at\over2L}\right) \sin{(2n-1)\pi x\over2L},\nonumber$

where

$S_{M\!f}(x)=\sum_{n=1}^\infty\alpha_n\sin{(2n-1)\pi x\over2L} \quad \text{and} \quad S_{M\!g}(x)=\sum_{n=1}^\infty\beta_n\sin{(2n-1)\pi x\over2L}\nonumber$

are the mixed Fourier sine series of $$f$$ and $$g$$ on $$[0,L]$$; that is,

$\alpha_n={2\over L}\int_0^Lf(x)\sin{(2n-1)\pi x\over2L}\,dx \quad \text{and} \quad \beta_n={2\over L}\int_0^Lg(x)\sin{(2n-1)\pi x\over2L}\,dx.\nonumber$

## Q12.2.5

In Exercises 12.2.35-12.2.46 use Exercise 12.2.34 to solve the initial-boundary value problem. In some of these exercises Theorem 11.3.5d or Exercise 11.3.50b will simplify the computation of the coefficients in the mixed Fourier sine series.

35. $$u_{tt}=64u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=x(2\pi-x),\quad u_t(x,0)=0,\quad0\le x\le \pi$$

36. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=x^2(3-2x),\quad u_t(x,0)=0,\quad0\le x\le 1$$

37. $$u_{tt}=64u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x(2\pi-x),\quad0\le x\le \pi$$

38. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^2(3-2x),\quad0\le x\le 1$$

39. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=(x-1)^3+1,\quad u_t(x,0)=0,\quad0\le x\le 1$$

40. $$u_{tt}=3u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=x(x^2-3\pi^2),\quad u_t(x,0)=0,\quad0\le x\le\pi$$

41. $$u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=(x-1)^3+1,\quad0\le x\le 1$$

42. $$u_{tt}=3u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x(x^2-3\pi^2),\quad0\le x\le\pi$$

43. $$u_{tt}=5u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=x^3(3x-4),\quad u_t(x,0)=0,\quad0\le x\le1$$

44. $$u_{tt}=16u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=x(x^3-2x^2+2),\quad u_t(x,0)=0,\quad0\le x\le1$$

45. $$u_{tt}=5u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^3(3x-4),\quad0\le x\le1$$

46. $$u_{tt}=16u_{xx},\quad 0<x<1,\quad t>0$$,
$$u(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x(x^3-2x^2+2),\quad0\le x\le1$$

## Q12.2.6

47. Adapt the proof of Theorem 12.2.2 to find d’Alembert’s solution of the initial-boundary value problem in Exercise 12.2.34.

48. Use the result of Exercise 12.2.47 to show that the formal solution of the initial-boundary value problem in Exercise 12.2.34 is an actual solution if $$g$$ is differentiable and $$f$$ is twice differentiable on $$[0,L]$$ and

$f(0)=f'_-(L)=g(0)=g_-'(L)=f''_+(0)=0.\nonumber$

HINT: See Exercise 11.3.58 and apply Theorem 12.2.3 with $$L$$ replaced by $$2L$$.

49. Justify defining the formal solution of the initial-boundary value problem

$\begin{array}{c} u_{tt}=a^2u_{xx},\quad 0<x<L,\quad t>0,\\ u_x(0,t)=0,\quad u_x(L,t)=0,\quad t>0,\\ u(x,0)=f(x),\quad u_t(x,0)=g(x),\quad 0\le x\le L. \end{array}\nonumber$

to be

$u(x,t)=\alpha_0+\beta_0t+\sum_{n=1}^\infty \left(\alpha_n\cos{n\pi at\over L}+{L\beta_n\over n\pi a}\sin{n\pi at\over L}\right) \cos{n\pi x\over L},\nonumber$

where

$C_f(x)=\alpha_0+\sum_{n=1}^\infty\alpha_n\cos{n\pi x\over L} \quad \text{and} \quad C_g(x)=\beta_0+\sum_{n=1}^\infty\beta_n\cos{n\pi x\over L}\nonumber$

are the Fourier cosine series of $$f$$ and $$g$$ on $$[0,L]$$; that is,

$\alpha_0={1\over L}\int_0^Lf(x)\,dx,\quad \beta_0={1\over L}\int_0^Lg(x)\,dx,\nonumber$

$\alpha_n={2\over L}\int_0^Lf(x)\cos{n\pi x\over L}\,dx, \quad \text{and} \quad \beta_n={2\over L}\int_0^Lg(x)\cos{n\pi x\over L}\,dx,\quad n=1,2,3,\dots.\nonumber$

## Q12.2.7

In Exercises 12.2.50-12.2.59 use Exercise 12.2.49 to solve the initial-boundary value problem. In some of these exercises Theorem 11.3.5a will simplify the computation of the coefficients in the Fourier cosine series.

50. $$u_{tt}=5u_{xx},\quad 0<x<2,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0$$,
$$u(x,0)=2x^2(3-x),\quad u_t(x,0)=0,\quad0\le x\le 2$$

51. $$u_{tt}=5u_{xx},\quad 0<x<2,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=2x^2(3-x),\quad0\le x\le 2$$

52. $$u_{tt}=4u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=x^3(3x-4\pi),\quad u_t(x,0)=0,\quad0\le x\le \pi$$

53. $$u_{tt}=7u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=3x^2(x^2-2),\quad u_t(x,0)=0,\quad0\le x\le 1$$

54. $$u_{tt}=4u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^3(3x-4\pi),\quad0\le x\le \pi$$

55. $$u_{tt}=7u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=3x^2(x^2-2),\quad0\le x\le 1$$

56. $$u_{tt}=16u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=x^2(x-\pi)^2,\quad u_t(x,0)=0,\quad0\le x\le \pi$$

57. $$u_{tt}=u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=x^2(3x^2-8x+6),\quad u_t(x,0)=0,\quad0\le x\le 1$$

58. $$u_{tt}=16u_{xx},\quad 0<x<\pi,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^2(x-\pi)^2,\quad0\le x\le \pi$$

59. $$u_{tt}=u_{xx},\quad 0<x<1,\quad t>0$$,
$$u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0$$,
$$u(x,0)=0,\quad u_t(x,0)=x^2(3x^2-8x+6),\quad0\le x\le 1$$

## Q12.2.8

60. Adapt the proof of Theorem 12.2.2 to find d’Alembert’s solution of the initial-boundary value problem in Exercise 12.2.49.

61. Use the result of Exercise 12.2.60 to show that the formal solution of the initial-boundary value problem in Exercise 12.2.49 is an actual solution if $$g$$ is differentiable and $$f$$ is twice differentiable on $$[0,L]$$ and

$f'_+(0)=f'_-(L)=g'_+(0)=g_-'(L)=0.\nonumber$

62. Suppose $$\lambda$$ and $$\mu$$ are constants and either $$p_n(x)= \cos n\lambda x$$ or $$p_n(x)=\sin n\lambda x$$, while either $$q_n(t)=\cos n\mu t$$ or $$q_n(t)=\sin n\mu t$$ for $$n=1$$, $$2$$, $$3$$, …. Let

$u(x,t)=\sum_{n=1}^\infty k_np_n(x)q_n(t), \tag{A}$

where $$\{k_n\}_{n=1}^\infty$$ are constants.

1. Show that if $$\sum_{n=1}^\infty |k_n|$$ converges then $$u(x,t)$$ converges for all $$(x,t)$$.
2. Use Theorem 12.1.2 to show that if $$\sum_{n=1}^\infty n|k_n|$$ converges then (A) can be differentiated term by term with respect to $$x$$ and $$t$$ for all $$(x,t)$$; that is, $u_x(x,t)= \sum_{n=1}^\infty k_np_n'(x)q_n(t)\nonumber$ and $u_t(x,t)= \sum_{n=1}^\infty k_np_n(x)q_n'(t).\nonumber$
3. Suppose $$\sum_{n=1}^\infty n^2|k_n|$$ converges. Show that $u_{xx}(x,y)= \sum_{n=1}^\infty k_np_n''(x)q_n(t)\nonumber$ and $u_{tt}(x,y)= \sum_{n=1}^\infty k_np_n(x)q_n''(t)\nonumber$
4. Suppose $$\sum_{n=1}^\infty n^2|\alpha_n|$$ and $$\sum_{n=1}^\infty n|\beta_n|$$ both converge. Show that the formal solution $u(x,t)=\sum_{n=1}^\infty\left(\alpha_n\cos{n\pi at\over L}+{\beta_nL\over n\pi a}\sin{n\pi at\over L}\right) \sin{n\pi x\over L}\nonumber$ of Equation 12.2.1 satisfies $$u_{tt}=a^2u_{xx}$$ for all $$(x,t)$$.
This conclusion also applies to the formal solutions defined in Exercises 12.2.17, 12.2.34,  and 12.2.49.

63. Suppose $$g$$ is differentiable and $$f$$ is twice differentiable on $$(-\infty,\infty)$$, and let

$u_0(x,t)={f(x+at)+f(x-at)\over2}\quad \text{and} \quad u_1(x,t)={1\over2a}\int_{x-at}^{x+at}g(u)\,du.\nonumber$

1. Show that ${\partial^2 u_0\over\partial t^2}=a^2{\partial^2u_0\over\partial x^2},\quad-\infty<x<\infty,\quad t>0,\nonumber$ and $u_0(x,0)=f(x),\quad {\partial u_0\over\partial t}(x,0)=0,\quad -\infty<x<\infty.\nonumber$
2. Show that ${\partial^2 u_1\over\partial t^2}=a^2{\partial^2u_1\over\partial x^2},\quad-\infty<x<\infty,\quad t>0,\nonumber$ and $u_1(x,0)=0,\quad {\partial u_1\over\partial t}(x,0)=g(x),\quad -\infty<x<\infty.\nonumber$
3. Solve $u_{tt}=a^2u_{xx},\quad-\infty<t<\infty,\quad t>0,\nonumber$ $u(x,0)=f(x),\quad u_t(x,0)=g(x),\quad-\infty<x<\infty.\nonumber$

## Q12.2.9

In Exercises 12.2.64-12.2.68 use the result of Exercise 12.2.63 to find a solution of $u_{tt}=a^{2}u_{xx},\quad -\infty <x<\infty \nonumber$ that satisfies the given initial conditions.

64. $$u(x,0)=x$$,$$u_t(x,0)=4ax$$,$$-\infty<x<\infty$$

65. $$u(x,0)=x^2$$,$$u_t(x,0)=1$$,$$-\infty<x<\infty$$

66. $$u(x,0)=\sin x$$,$$u_t(x,0)=a\cos x$$,$$-\infty<x<\infty$$

67. $$u(x,0)=x^3$$,$$u_t(x,0)=6x^2$$,$$-\infty<x<\infty$$

68. $$u(x,0)=x\sin x$$,$$u_t(x,0)=\sin x$$,$$-\infty<x<\infty$$