Please answer d ### Educational Content on Wave Equation Transformation**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 the 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:\[ Y(x, t) = \frac{1}{2}[f(x - at) + f(x + at)]. \]This problem involves transforming the classic wave equation using a change of variables and subsequently solving initial condition equations to arrive at the solution form typically used to describe wave propagation in one-dimensional infinite media.

Answered: d) Y(x, Solve the equations of part (c)…

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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Please answer d

### Educational Content on Wave Equation Transformation

**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 the 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:
\[ Y(x, t) = \frac{1}{2}[f(x - at) + f(x + at)]. \]

This problem involves transforming the classic wave equation using a change of variables and subsequently solving initial condition equations to arrive at the solution form typically used to describe wave propagation in one-dimensional infinite media.

Expert Solution

Step 1: Proof

$Y_{t t} = a^{2} Y_{x x} Y (x . 0) = f (x) Y_{t} (x, 0) = 0 ϕ (x) + ψ (x) = f (x) ϕ' (x) - ψ' (x) = 0$

$Y (x, t) = ϕ (x + a t) + ψ (x - a t) Y_{t} (x, t) = a ϕ' (x + a t) - a ψ' (x - a t) Y_{t} (x, 0) = 0 \Rightarrow ϕ' (x) - ψ' (x) = 0$

Integrating this equation with respect to $x$ to yield

$ϕ (x) - ψ (x) = \int_{x_{0}}^{x} 0 d x$

Where the constant of integration has been incorporated in the lower limit by introducing an arbitrary constant $x_{0}$ . Solving equations for $ϕ$ and $ψ$ , gives

$\Rightarrow ϕ (x) = ψ (x) ∴ ϕ (x) + ψ (x) = f (x) \Rightarrow 2 ϕ (x) = f (x) [∵ ϕ (x) = ψ (x)] \Rightarrow ϕ (x) = \frac{1}{2} f (x) ∴ ψ (x) = \frac{1}{2} f (x) ∴ Y (x, t) = ϕ (x + a t) + ψ (x - a t) = \frac{1}{2} f (x + a t) + \frac{1}{2} f (x - a t) = \frac{1}{2} [f (x + a t) + f (x - a t)]$