Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Y₁(x,0) = 0, -x 0. Using the form of the solution obtained in part (b), show that and must satisfy fo(x) + y(x) = f(x), o'(x) - '(x) = 0.
Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Y₁(x,0) = 0, -x 0. Using the form of the solution obtained in part (b), show that and must satisfy fo(x) + y(x) = f(x), o'(x) - '(x) = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Can you answer c and d please
![**a)** Show that the wave equation \( Y_{tt} = a^2 Y_{xx} \) where \( a \) is a constant, can be reduced to the form \( Y_{uv} = 0 \) by the change of variables \( u = x + at \) and \( v = x - at \).
---
**b)** Use the results in part (a) and show that \( Y(x, t) \) can be written as
\[
Y(x, t) = \phi(x + at) + \psi(x - at),
\]
where \( \phi \) and \( \psi \) are arbitrary functions.
---
**c)** Now, consider the wave equation in part (a) in an infinite one-dimensional medium subject to initial conditions
\[
Y(x, 0) = f(x), \quad Y_t(x, 0) = 0, \quad -\infty < x < \infty, \quad t > 0.
\]
Using the form of the solution obtained in part (b), show that \( \phi \) and \( \psi \) must satisfy
\[
\begin{cases}
\phi(x) + \psi(x) = f(x), \\
\phi'(x) - \psi'(x) = 0.
\end{cases}
\]
---
**d)** Solve the equations of part (c) for \( \phi \) and \( \psi \), and show that
\[
Y(x, t) = \frac{1}{2}[f(x - at) + f(x + at)].
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F56df4c5a-b856-4a3b-812b-89df46a85100%2Fua7mxpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**a)** Show that the wave equation \( Y_{tt} = a^2 Y_{xx} \) where \( a \) is a constant, can be reduced to the form \( Y_{uv} = 0 \) by the change of variables \( u = x + at \) and \( v = x - at \).
---
**b)** Use the results in part (a) and show that \( Y(x, t) \) can be written as
\[
Y(x, t) = \phi(x + at) + \psi(x - at),
\]
where \( \phi \) and \( \psi \) are arbitrary functions.
---
**c)** Now, consider the wave equation in part (a) in an infinite one-dimensional medium subject to initial conditions
\[
Y(x, 0) = f(x), \quad Y_t(x, 0) = 0, \quad -\infty < x < \infty, \quad t > 0.
\]
Using the form of the solution obtained in part (b), show that \( \phi \) and \( \psi \) must satisfy
\[
\begin{cases}
\phi(x) + \psi(x) = f(x), \\
\phi'(x) - \psi'(x) = 0.
\end{cases}
\]
---
**d)** Solve the equations of part (c) for \( \phi \) and \( \psi \), and show that
\[
Y(x, t) = \frac{1}{2}[f(x - at) + f(x + at)].
\]
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