Question

By combining the results of Problems 17 and 18 show that the solution of the problem $$ \begin{aligned} a^{2} u_{x x} &=u_{t t} \\ u(x, 0)=f(x), & u_{t}(x, 0)=g(x), &-\infty

Short answer

Question: Verify that the given solution satisfies the initial conditions of the wave equation and write the final solution. Answer: The given solution satisfies the initial conditions as follows: 1. For $u(x, 0) = f(x)$: $u(x, 0) = f(x)$ as shown in the step-by-step solution. 2. For $u_t(x, 0) = g(x)$: $u_t(x, 0) = g(x)$ as derived in the step-by-step solution. Thus, the final solution of the wave equation is: $$ u(x, t) = \frac{1}{2}[f(x - at) + f(x + at)] + \frac{1}{2a} \int_{x - at}^{x + at} g(\xi) d \xi $$

Step-by-step solution

Add the solutions of Problems 17 and 18

We are given that the solutions to Problems 17 and 18 are: $$ u_1(x, t) = \frac{1}{2} (f(x + at) + f(x - at)) \\ u_2(x, t) = \frac{1}{2a} \int_{x - at}^{x + at} g(\xi) d \xi $$ Adding the solutions together, we get: $$ u(x, t) = u_1(x, t) + u_2(x, t) $$ #2. Find the combined solution of wave equation#

Combine the solutions

Put the results of Problems 17 and 18 into the equation for \(u(x, t)\): $$ u(x, t) = \frac{1}{2} [f(x - at) + f(x + at)] + \frac{1}{2a} \int_{x - at}^{x + at} g(\xi) d \xi $$ #3. Verify initial conditions#

Verify the initial conditions

We need to verify that this solution satisfies the initial conditions \(u(x, 0) = f(x)\) and \(u_t(x, 0) = g(x)\): a) For \(u(x, 0) = f(x)\): $$ u(x, 0) = \frac{1}{2}[f(x) + f(x)] + \frac{1}{2a} \int_{x}^{x} g(\xi) d \xi = f(x) $$ b) For \(u_t(x, 0) = g(x)\): We have to find the time derivative of \(u(x, t)\) and then evaluate it at \(t=0\). $$ \frac{\partial u(x, t)}{\partial t} = -\frac{a}{2}[f'(x - at) - f'(x + at)] + \frac{g(x - at) - g(x + at)}{2 } $$ Now, evaluate the time derivative at \(t=0\): $$ u_t(x, 0) = -\frac{a}{2}[f'(x) - f'(x)] + \frac{g(x) - g(x)}{2 } = g(x) $$ Since both initial conditions are satisfied, the solution of the given wave equation is: $$ u(x, t) = \frac{1}{2}[f(x - at) + f(x + at)] + \frac{1}{2a} \int_{x - at}^{x + at} g(\xi) d \xi $$

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) represent a class of equations used to describe a wide range of physical phenomena, such as sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity. Unlike ordinary differential equations (ODEs), which involve derivatives with respect to a single variable, PDEs involve derivatives with respect to multiple variables.

In our exercise, we're analyzing a specific type of PDE known as the wave equation. It is typically written as:
\(a^{2} u_{xx} = u_{tt}\).

This equation is essential in physics as it describes the propagation of waves, whether they be sound, light, or water waves, through a given medium. The terms involving second derivatives of the function
\(u(x,t)\) with respect to space (
\(x\)) and time (
\(t\)) reflect the change of the wave shape in space and its evolution in time, respectively.

The constants and functions in a PDE provide specific information about the wave's behavior and properties of the medium through which it is traveling. In the given problem,
\(a\) represents the wave speed. The solution to a PDE like the one in our exercise often requires the application of various mathematical techniques, one of which is the method of characteristics, and sometimes involves a superposition of more basic solutions.

Boundary Value Problems

Boundary value problems (BVPs) are a type of problem in which a differential equation is accompanied by a set of constraints called boundary conditions. These conditions specify the behavior of the solution at the boundaries of the domain over which the equation is defined. While initial value problems specify the state of the system at the start time, boundary conditions may define the behavior at specific locations or times.

In the context of the wave equation, the domain is often the entire space, or
\(-\text{infinity} < x < \text{infinity}\), as in our exercise, signifying that the wave may theoretically travel an infinite distance in both directions. However, not all boundary value problems are this expansive; they often involve finite intervals with specific behaviors dictated at the endpoints.

Identifying and correctly applying the boundary conditions is crucial for finding a unique and physically meaningful solution to a boundary value problem. The exercise deals with an infinite domain, thus not having traditional boundary conditions, but still requires understanding the implications of such a setup on the form and behavior of the solution.

Initial Conditions

Initial conditions are a fundamental concept when solving both ordinary and partial differential equations. They specify the state of the system at the outset, essentially providing a starting point for the solution. The initial conditions in PDEs often take the form of the value of the function and some of its derivatives evaluated at the initial time.

In our problem, the initial conditions given are:
\(u(x, 0) = f(x)\) and
\(u_t(x, 0) = g(x)\).

These conditions provide the initial shape of the wave
\(f(x)\) and its initial velocity
\(g(x)\). When solving the wave equation, these initial conditions will determine how the wave evolves over time. It’s essential to confirm that any proposed solution satisfies the given initial conditions, as this ensures the solution is applicable to the physical scenario depicted by the problem.

Verifying the initial conditions as performed in steps 2 and 3 of the solution shows the rigor needed to establish a valid solution to a wave equation. It also demonstrates the importance of the initial conditions in defining a unique wave motion for given initial disturbances in the medium.