onsider the D’Alembert IVP:
PDE: utt = uxx, 0 − ∞ < x < ∞, 0 < t < ∞
ICs:
(
u(x, 0) = f (x)
ut(x, 0) = 0 ∞ < x < ∞
where f (x) =
(
1 −1 < x < 1
0 otherwise
For the following values of t, graph the forward moving wave, the backward moving wave, and
the solution:
(a) t = 0.25 (b) t = 0.75
Forward moving wave
Backward moving wave
Solution
Forward moving wave
Backward moving wave
Solution
Orthogonality and D’Alembert, Page 4 of 4 Due: October 8, 2024
(c) t = 2
Forward moving wave
Backward moving wave
Solution

Here are the graphs of the forward-moving wave, backward-moving wave, and the solution at the given times:

1. **For \( t = 0.25 \)**:
   - The forward-moving wave is shifted right by 0.25.
   - The backward-moving wave is shifted left by 0.25.
   - The solution shows a non-zero overlap from \( -0.75 < x < 0.75 \).

2. **For \( t = 0.75 \)**:
   - The forward-moving wave is shifted right by 0.75.
   - The backward-moving wave is shifted left by 0.75.
   - The solution shows a non-zero overlap from \( -0.25 < x < 0.25 \).

3. **For \( t = 2 \)**:
   - The forward-moving wave is shifted right by 2.
   - The backward-moving wave is shifted left by 2.
   - There is no overlap between the waves, resulting in a zero solution everywhere.

Would you like more details or further clarification on any part of this process?

### Related Questions:
1. How would the solution change if the initial velocity \( u_t(x, 0) \) were not zero?
2. Can you explain the effect of different wave speeds on the solution of the wave equation?
3. How does D'Alembert's solution extend to multi-dimensional problems?
4. What is the physical significance of forward and backward-moving waves?
5. How would the solution change if \( f(x) \) were a different shape, such as a Gaussian?

**Tip:** In wave equations, the overlap of forward and backward-moving waves determines the shape of the solution at any given time.

Consider the D’Alembert IVP:
PDE: utt = uxx, -∞ < x < ∞, 0 < t < ∞
ICs:
u(x, 0) = f(x)
ut(x, 0) = 0, ∞ < x < ∞
where f(x) = 1 for -1 < x < 1 and 0 otherwise.
For the following values of t, graph the forward moving wave, the backward moving wave, and the solution:
(a) t = 0.25
(b) t = 0.75
(c) t = 2

Explore the solution to the D'Alembert IVP involving the wave equation utt = uxx with specific initial conditions. See graphs of forward and backward-moving waves at times t = 0.25, 0.75, and 2, illustrating how the solution evolves over time. Suitable for undergraduate students in mathematics or physics.

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