Math 257/316 Assignment 8
Due Mon. Mar. 23 in class
1. Consider the waveequation utt = c2 uxx ,  −∞ < x < ∞, with initial position
 x + 1 if −1 < x ≤ 0 
1 − x if 0 < x < 1
u(x, 0) = f (x) =
and with initial velocity ut (x, 0) = 0.


0
otherwise
Sketch the shape of the solution u(x, t) at t = 0, t = 1/(2c), t = 1/c, and t = 2/c.
2. spreadsheet project: Consider the following initial-boundary value problem:
utt = 16uxx ,
u(0, t) = 0
0 < x < π/2, t > 0
and
−50(x−π/4)2
u(x, 0) = e
u(π/2, t) = 0
,
ut (x, 0) = 0.
a) Use a spreadsheet and a finite difference approximation to solve the problem
numerically, taking ∆x = π/80 and ∆t = 0.004 (as usual, you can use the
examples on the website, or build your own from previous spreadsheets, but
remember to properly account for the initial conditions). Turn in plots of the
solution at times t = 0.1, t = 0.2, t = 0.3, and t = 0.4. Describe in a few words
the behaviour of the solution.
b) Change the boundary conditions to Neumann BCs: ux (0, t) = ux (π/2, t) = 0,
and repeat the computation from part (a), generating the same plots. Describe
in a few words how the behaviour differs from the case of Dirichlet BCs.
3. Solve the wave equation problem
utt = 4uxx ,
u(x, 0) =
0 < x < 1, t > 0,
0 0 ≤ x < 1/4, 3/4 < x ≤ 1
ut (x, 0) = 0
1
1/4 ≤ x ≤ 3/4
a) with Dirichlet BCs: u(0, t) = 0 = u(1, t)
b) with Neumann BCs: ux (0, t) = 0 = ux (1, t).
In each case, find the earliest positive time t at which the solution u(x, t) is exactly
the same as at time 0: u(x, t) = u(x, 0) for all x. Interpret your answer in terms of
reflecting waves.
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