Problem 18 - Advanced Engineering Mathematics

Question

The vertical displacement \(u(x, t)\) of an infinitely long string is determined from the initial-value problem $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad-\infty0 \\ &u(x, 0)=f(x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=g(x) \end{aligned} $$ This problem can be solved without separating variables. (a) Show that the wave equation can be put into the form \(\partial^{2} u / \partial \eta \partial \xi=0\) by means of the substitutions \(\xi=x+a t\) and \(\eta=x-a t .\) (b) Integrate the partial differential equation in part (a), first with respect to \(\eta\) and then with respect to \(\xi\), to show that \(u(x, t)=F(x+a t)+G(x-a t)\), where \(F\) and \(G\) are arbitrary twice differentiable functions, is a solution of the wave equation. Use this solution and the given initial conditions to show that $$ \begin{aligned} &F(x)=\frac{1}{2} f(x)+\frac{1}{2 a} \int_{x_{0}}^{x} g(s) d s+c \\ &\text { and } \quad G(x)=\frac{1}{2} f(x)-\frac{1}{2 a} \int_{x_{0}}^{x} g(s) d s-c, \end{aligned} $$ where \(x_{0}\) is arbitrary and \(c\) is a constant of integration. (c) Use the results in part (b) to show that \(u(x, t)=\frac{1}{2}[f(x+a t)+f(x-a t)]+\frac{1}{2 a} \int_{x-a t}^{x+a t} g(s) d s .\) (14) Note that when the initial velocity \(g(x)=0\) we obtain $$ u(x, t)=\frac{1}{2}[f(x+a t)+f(x-a t)],-\infty

Step-by-Step Solution

Answer

The solution is d'Alembert's solution for wave equations: \( u(x, t) = \frac{1}{2}[f(x+at)+f(x-at)] + \frac{1}{2a} \int_{x-at}^{x+at} g(s) \, ds \). This represents two waves traveling in opposite directions.

Step 1: Substituting Variables

Consider the substitutions \( \xi = x + at \) and \( \eta = x - at \). Then, express the second derivatives \( \frac{\partial^2 u}{\partial x^2} \) and \( \frac{\partial^2 u}{\partial t^2} \) in terms of \( \xi \) and \( \eta \).

Step 2: Rewriting the Wave Equation

Using the chain rule, calculate \( \frac{\partial^2 u}{\partial x^2} \) and \( \frac{\partial^2 u}{\partial t^2} \) with respect to \( \xi \) and \( \eta \). Show that these substitutions transform \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \) \ into the form \( \frac{\partial^2 u}{\partial \eta \partial \xi} = 0 \).

Step 3: Integrating with Respect to \(\eta\)

Integrate the resulting equation \( \frac{\partial^2 u}{\partial \eta \partial \xi} = 0 \) first with respect to \( \eta \). Let the result be a function of \( \xi \).

Step 4: Integrating with Respect to \(\xi\)

Integrate the previous result with respect to \( \xi \) to obtain the general solution \( u(x,t) = F(x+at) + G(x-at) \), where \( F \) and \( G \) are arbitrary functions.

Step 5: Applying Initial Conditions

Use the initial conditions \( u(x,0) = f(x) \) and \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = g(x) \) to determine the forms of \( F \) and \( G \). Substitute \( t = 0 \) to find \( F(x) + G(x) = f(x) \) and differentiate to substitute into the velocity condition.

Step 6: Solving for \( F \) and \( G \)

Express \( g(x) \) in terms of \( F'(x) \) and \( G'(x) \) using the condition \( \frac{\partial}{\partial t}[F(x+at) + G(x-at)] \). Integrate to find expressions for \( F(x) \) and \( G(x) \).

Step 7: Verify d'Alembert's Solution

Substitute the expressions for \( F(x) \) and \( G(x) \) back into the solution \( u(x,t) \). Simplify to derive d'Alembert's solution, showing that \( u(x, t) = \frac{1}{2}[f(x+at)+f(x-at)] + \frac{1}{2a} \int_{x-at}^{x+at} g(s) \, ds \).

Key Concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are a type of mathematical equation involving multiple variables and their partial derivatives. These equations are crucial for modeling various phenomena in fields like physics, engineering, and finance. The wave equation is one of the most common PDEs encountered, often described mathematically as \( \frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2} \), where \( u(x, t) \) represents the vertical displacement of a wave.

This particular wave equation describes how waves propagate over time in a given medium. Unlike ordinary differential equations, PDEs can involve derivatives with respect to more than one variable, making their solutions more complex. Solutions to PDEs like the wave equation provide insights into the underlying physical processes and behaviors.

The wave equation specifically helps us understand vibrations, sound waves, and even electromagnetic waves by showing how displacement changes over time and space. Given initial conditions or constraints, solving a PDE can help determine the future state of the system.

Initial Value Problem

An initial value problem in the context of differential equations involves finding a solution to a differential equation that satisfies specific conditions at a starting point, known as initial conditions. It is a way to solve equations where the behavior at the beginning is known, and you want to understand how that behavior evolves.

For the given wave equation \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \), the initial conditions are defined as \( u(x,0) = f(x) \) and \( \left. \frac{\partial u}{\partial t} \right|_{t=0} = g(x) \). These conditions specify the wave's initial shape and velocity.

• \( u(x,0) = f(x) \): Defines the initial displacement of each point \( x \) on the string.
• \( \left. \frac{\partial u}{\partial t} \right|_{t=0} = g(x) \): Describes how fast the initial displacement is changing over time.

The goal is to find \( u(x, t) \) that satisfies the wave equation and matches these initial conditions. Through integration and application of the initial conditions, functions \( F(x) \) and \( G(x) \) are determined, contributing to the solution form \( u(x, t) = F(x+at) + G(x-at) \).

d'Alembert's Solution

d'Alembert's Solution is a method for solving the wave equation utilizing the linearity and characteristic properties of PDEs. Named after the French mathematician Jean le Rond d'Alembert, this solution expresses the wave equation solution as a superposition of two traveling waves.

The d'Alembert formula is represented by:\[ u(x, t) = \frac{1}{2}[f(x+at)+f(x-at)] + \frac{1}{2a} \int_{x-at}^{x+at} g(s) \, ds\]This formula decomposes the wave's behavior into two components:

• \( \frac{1}{2}[f(x+at)+f(x-at)] \): Describes two initial displacement-based traveling waves, one moving right and one moving left.
• \( \frac{1}{2a} \int_{x-at}^{x+at} g(s) \, ds \): Accounts for the initial velocity, showing how additional energy influences wave evolution.

When the initial velocity \( g(x) = 0 \), d'Alembert's solution simplifies to the expression: \( u(x, t) = \frac{1}{2}[f(x+at) + f(x-at)] \), which vividly represents a symmetric traveling wave. This symmetry clearly illustrates the wave's split and movement in two directions with speed \( a \), highlighting the elegance and power of d'Alembert's solution in wave mechanics.