MATH 348 - Advanced Engineering Mathematics April 2, 2008
Homework 8, Spring 2008 Due: April 11, 2008
partial differential equations - the one-dimensional wave equation
Consider the one-dimensional wave equation,
∂2u
∂t2=c2∂2u
∂x2,(1)
x∈(0, L), t ∈(0,∞), c2=T
ρ.(2)
Equations (1)-(2) model the time-evolution of the displacement, u=u(x, t), of an elastic medium in one-dimension.
The object, of length L, is assumed to have a homogenous lateral tension T, and linear density ρ. That is, T, ρ ∈R+.
Define,
f(x) = x, 0< x ≤L
−x+ 2L, L < x < 2L(3)
1. Consider the one-dimensional wave equation, (1)-(2), with the boundary conditions1,
ux(0, t)=0, u(2L, t) = 0,(4)
and initial conditions
u(x, 0) = f(x), ut(x, 0) = g(x) (5)
(a) Assume that the solution to (1)-(2) is such that u(x, t) = F(x)G(t) and use separation of variables to find
the general solution to (1)-(2), which satisfies (4)-(5). 2
(b) Solve for the unknown constants assuming (3) and zero initial velocity for all points on the object.
2. Consider the one-dimensional wave equation, (1)-(2), with the boundary conditions3,
ux(0, t)=0, ux(2L, t)=0,(6)
and initial conditions
u(x, 0) = f(x), ut(x, 0) = g(x) (7)
(a) Assume that the solution to (1)-(2) is such that u(x, t) = F(x)G(t) and use separation of variables to find
the general solution to (1)-(2), which satisfies (6)-(7). 4 5
(b) Let L= 1 and solve for the unknown constants assuming (3) and zero initial velocity for all points on the
object.
3. Many applications consider traveling wave solutions, f(x, t) = f(x−ct), of the sinusoidal form, f(x, t) =
Acos(kx −ωt). Assume that u(x, t) = Aei(kx−ω t)is a solution to the following wave-like equation:6
utt −uxx +u= 0.(8)
Show that the phase velocity, cp=ω
k, of the traveling wave solutions to (8) is given by cp=±√1 + k−2.7
1These boundary conditions imply that the object must have zero curvature at the left endpoint and zero displacement at the right
endpoint.
2It is important to notice that the solution to the spatial portion of the problem is the same as the heat problem of hw5 prob1.
3These boundary conditions imply that the object must have zero curvature at each endpoint.
4It is important to notice that the solution to the spatial portion of the problem is the same as the heat problem of hw5 prob2.
5Remember that in this case we have nontrivial solutions for k0= 0. You should find that G0(t) = C1+C2t.
6Here we choose to work with complex exponential functions since calculation of derivatives is less clumsy than trigonometric functions.
Notice that the real-part of uis equal to f.
7This implies that different waves which solve (8) travel at different velocities.
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