232 Chapter 3 The Wave Equation
9.Sketch the solution of the vibrating string problem, Eqs. (2)–(5), at times
ct=0, 0. 1 a,0. 3 a,0. 4 a,0. 5 a,0. 6 a,ifg(x)=0andf(x)=
0 , 0 <x< 0. 4 a,
10 h(x− 0. 4 a), 0. 4 a<x< 0. 5 a,
10 h( 0. 6 a−x), 0. 5 a<x< 0. 6 a,
0 , 0. 6 a<x<a.10.Ve r i f y d i r e c t l y t h a tu(x,t)as given by Eq. (1) is a solution of the wave
equation (2) ifφandψhave at least two derivatives.
11.Use the change of variables at the beginning of this section to transform
thewaveequation.Youneedtousethechainruleextensively—forin-
stance,
∂u
∂t=∂v
∂w∂w
∂t+∂v
∂z∂z
∂t=∂v
∂w·c+∂v
∂z·(−c).In this way, find expressions for the second derivatives ofu, and then sub-
stitute into the wave equation,∂^2 u
∂x^2 =1
c^2∂^2 u
∂t^2.12.The equation for the forced vibrations of a string is∂^2 u
∂x^2−^1
c^2∂^2 u
∂t^2=−^1
T
F(x,t) (∗)(see Section 1, Exercise 2). Changing variables tow=x+ct, z=x−ct, u(x,t)=v(w,z), f(w,z)=F(x,t),this equation becomes∂^2 v
∂w∂z=−^1
4 T
f(w,z).Show that the general solution of this equation isv(w,z)=−^1
4 T∫∫
f(w,z)dwdz+ψ(w)+φ(z).13.Find the general solution of Eq. (∗)inExercise12intermsofxandt,if
F(x,t)=Tcos(t).