Question

Show that the wave equation $$ a^{2} u_{x x}=u_{t t} $$ can be reduced to the form \(u_{\xi \eta}=0\) by change of variables \(\xi=x-a t, \eta=x+a t .\) Show that \(u(x, t)\) can be written as $$ u(x, t)=\phi(x-a t)+\psi(x+a t) $$ where \(\phi\) and \(\psi\) are arbitrary functions.

Short answer

#tag_title#Short Answer#tag_content# The wave equation can be reduced to the form \(u_{\xi \eta} = 0\) using the change of variables \(\xi = x - at\) and \(\eta = x + at\). After solving the new equation, we obtain the solution \(u(x, t) = \phi(x - at) + \psi(x + at)\), where \(\phi\) and \(\psi\) are arbitrary functions.

Step-by-step solution

Change of variables

Let \(\xi = x - a t\) and \(\eta = x + a t\). We need to find the partial derivatives of \(u(x, t)\) with respect to \(\xi\) and \(\eta\) and substitute them in the wave equation. Start by finding the derivatives of \(x\) and \(t\) with respect to \(\xi\) and \(\eta\): $$ \frac{\partial x}{\partial \xi} = 1, \quad \frac{\partial x}{\partial \eta} = 1 \\ \frac{\partial t}{\partial \xi} = - \frac{1}{a}, \quad \frac{\partial t}{\partial \eta} = \frac{1}{a} $$

Compute the partial derivatives of \(u(x, t)\) with respect to \(\xi\) and \(\eta\)

Compute the first-order partial derivatives of \(u\): $$ u_\xi = \frac{\partial u}{\partial x} \frac{\partial x}{\partial \xi} + \frac{\partial u}{\partial t} \frac{\partial t}{\partial \xi} = u_x - a u_t \\ u_\eta = \frac{\partial u}{\partial x} \frac{\partial x}{\partial \eta} + \frac{\partial u}{\partial t} \frac{\partial t}{\partial \eta} = u_x + a u_t $$ Next, compute the second-order partial derivatives: $$ u_{\xi\xi} = \frac{\partial^2 u}{\partial x^2} - 2a \frac{\partial^2 u}{\partial x\partial t} + a^2 \frac{\partial^2 u}{\artial t^2} = u_{xx} - 2a u_{xt} + a^2 u_{tt} \\ u_{\eta\eta} = \frac{\partial^2 u}{\artial x^2} + 2a \frac{\partial^2 u}{\partial x\partial t} + a^2 \frac{\partial^2 u}{\partial t^2} = u_{xx} + 2a u_{xt} + a^2 u_{tt} $$

Substitution and simplification

Substitute variables into the wave equation \(a^2 u_{x x}=u_{t t}\): $$ a^2 (u_{\xi\xi} + u_{\eta\eta}) =2a^2u_{xx} $$ Now, let's add these two equations: $$ u_{\xi\xi} + u_{\eta\eta} = 0 \\ u_{\xi \eta} = 0 $$ We have successfully shown that the wave equation can be reduced to the form \(u_{\xi \eta} = 0\) using the change of variables.

Solve the new equation

We have the equation \(u_{\xi \eta} = 0\). Integrating it with respect to \(\xi\): $$ \int u_{\xi \eta} d\xi = \int 0 d\xi \\ u_\eta = F(\eta) $$ Now, we integrate with respect to \(\eta\): $$ \int u_\eta d\eta = \int F(\eta) d\eta \\ u(\xi, \eta) = \int F(\eta) d\eta + G(\xi) $$ Let \(\phi(\xi) = G(\xi)\) and \(\psi(\eta) = \int F(\eta) d\eta\) be arbitrary functions. Then, $$ u(\xi, \eta) = \phi(\xi) + \psi(\eta) $$

Reverse the change of variables

Now, we need to reverse the change of variables by substituting \(\xi = x - a t\) and \(\eta = x + a t\) back into the solution: $$ u(x, t) = \phi(x - a t) + \psi(x + a t) $$ Thus, we have shown that the solution to the wave equation can be written as \(u(x, t)=\phi(x-a t)+\psi(x+a t)\), where \(\phi\) and \(\psi\) are arbitrary functions.

Key concepts

Partial Derivatives

Understanding partial derivatives is critical for solving the wave equation. A partial derivative of a function with multiple variables is how the function changes with respect to one of those variables, while keeping the others constant. In the context of the wave equation, we denote the partial derivative of a function u with respect to time t as u_t and with respect to space x as u_x.

When solving the wave equation, we calculate partial derivatives to describe the change in the wave function in different dimensions, such as time and space. As seen in the step-by-step solution, the first and second-order partial derivatives of the function u are computed with respect to both x and t, which are important for transforming the equation under a change of variables. The nuances of this process include understanding how to manipulate these derivatives to simplify or alter the equation's form, a crucial skill in solving partial differential equations.

Change of Variables

The change of variables technique transforms a complex problem into a simpler one by substituting new variables, which make the problem easier to solve or interpret. In our case, the wave equation is transformed by introducing two new variables, ξ and η, that are combinations of the original variables x and t. This clever step simplifies the equation significantly. The process of finding new partial derivatives with respect to ξ and η, as opposed to x and t, allows us to 'decouple' the variables and examine the problem through a different lens.

As highlighted in the steps of the solution, once the change of variables is made, the original wave equation, involving second-order derivatives with respect to x and t, becomes a much more manageable equation where the mixed second-order derivative of u with respect to ξ and η equals zero. Being adept at transforming variables is thus vital to solving differential equations.

Arbitrary Functions

Arbitrary functions are the cornerstone of solutions to differential equations. After all transformations and calculations are done, it is often the case that the solution to a partial differential equation involves functions that are not specified—they can be any function that meet the conditions set by the problem. In the context of the wave equation, after reducing the equation and integrating, we are left with solutions that are expressed in terms of arbitrary functions φ(ξ) and ψ(η).

These arbitrary functions accommodate an infinite number of possible forms for the wave function u(x, t), representing the diverse range of physical wave behaviors that could satisfy the general conditions of the wave equation. This flexibility is a fundamental feature of partial differential equations and emphasizes the role of boundary conditions and initial conditions in finding specific solutions to physical problems. The concept of arbitrary functions teaches us that mathematics graciously adapts to the complexity of the natural world by allowing for a multiplicity of solutions under a single framework.