3.4 One-Dimensional Wave Equation: Generalities 233
3.4 One-Dimensional Wave Equation: Generalities
As for the one-dimensional heat equation, we can make some comments for
a generalized one-dimensional wave equation. For the sake of generality, we
assume that some nonuniform properties are present. For the sake of simplic-
ity, we assume that the equation is homogeneous and free ofu. Our initial
value–boundary value problem will be
∂
∂x(
s(x)∂∂ux)
=pc( 2 x)∂(^2) u
∂t^2 , l<x<r,^0 <t, (1)
α 1 u(l,t)−α 2 ∂u
∂x
(l,t)=c 1 , 0 <t, (2)
β 1 u(r,t)+β 2 ∂∂xu(r,t)=c 2 , 0 <t, (3)
u(x, 0 )=f(x), l<x<r, (4)
∂u
∂t(x,^0 )=g(x), l<x<r. (5)
We assume that the functionss(x)andp(x)are positive forl≤x≤r,be-
cause they represent physical properties, thats,s′,andpare all continuous,
and thatsandphave no dimensions. Also, suppose that none of the coeffi-
cientsα 1 ,α 2 ,β 1 ,β 2 are negative.
To obtain homogeneous boundary conditions we can write
u(x,t)=v(x)+w(x,t),
just as before. In the wave equation, however, neither of the names “steady-
state solution” nor “transient solution” is appropriate; for, as we shall see, there
is no steady state, or limiting case, nor is there a part of the solution that tends
to zero asttends to infinity. Nevertheless,vrepresents an equilibrium solu-
tion, and, more important, it is a useful mathematical device to consideruin
the form provided in the preceding equation.
The functionv(x)is required to satisfy the conditions
(sv′)′= 0 , l<x<r,
α 1 v(l)−α 2 v′(l)=c 1 ,
β 1 v(r)+β 2 v′(r)=c 2.
Thusv(x)is exactly equivalent to the “steady-state solution” discussed for the
heat equation.