Problem 8 - Advanced Engineering Mathematics

Question

A string of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the displacement \(u(x, t)\). The ends are secured to the \(x\) -axis. Initially the string is undisplaced but has the initial velocity \(\sin (\pi x / L)\).

Step-by-Step Solution

Answer

The boundary-value problem involves the wave equation with specified boundary and initial conditions for a string oscillating along an x-axis from 0 to L.

Step 1: Identify the physical model

This problem involves a string that is free to oscillate. Please note, the ends of the string are secured along the x-axis, meaning that they will not move. Thus the basis of the model would be the wave equation \( u_{tt} = c^2 u_{xx} \), where \( u(x,t) \) represents the displacement at a point \( x \) along the string at time \( t \), and \( c \) is the wave speed.

Step 2: Define the boundary conditions

The ends of the string are secured to the x-axis. Therefore, the displacement at the endpoints must be zero at all times. Mathematically, this subject to the boundary conditions: \( u(0, t) = 0 \) and \( u(L, t) = 0 \) for all times \( t \).

Step 3: Incorporate initial conditions

The initial conditions describe the state of the system at time \( t = 0 \). The initial displacement is zero, thus \( u(x,0) = 0 \) for \( 0 \leq x \leq L \). Additionally, the initial velocity is given as \( u_t(x,0) = \sin(\pi x / L) \).

Step 4: Compile the boundary-value problem

Put together all components to form the boundary-value problem based on the wave equation and given conditions. The full setup is: 1. **Wave Equation**: \( u_{tt} = c^2 u_{xx} \) for \( 0 < x < L, \, t > 0 \)2. **Boundary Conditions**: \( u(0, t) = 0 \), \( u(L, t) = 0 \) for \( t > 0 \)3. **Initial Conditions**: \( u(x, 0) = 0 \) and \( u_t(x,0) = \sin(\pi x / L) \) for \( 0 \leq x \leq L \).

Key Concepts

Wave Equation

The wave equation is a fundamental concept in physics and mathematics. It describes how waves, like sound or light, move through space and time. In this context, we are looking at a one-dimensional wave equation that applies to the motion of a string fixed at both ends. More formally, the wave equation is expressed as:

• \( u_{tt} = c^2 u_{xx} \)

where \( u(x,t) \) is the displacement of the string at position \( x \) and time \( t \), \( u_{tt} \) is the second time derivative (representing acceleration), and \( u_{xx} \) is the second spatial derivative (representing curvature).
The term \( c \) refers to the wave speed, which depends on the tension and density of the string. The equation signifies how curvatures within the string cause accelerations in displacement over time. Understanding this equation helps in explaining vibrations and sound patterns.

Initial Conditions

Initial conditions play a crucial role in defining the starting state of any problem involving differential equations. Here, the initial conditions inform us about the shape and velocity of the string at time \( t = 0 \).
For this exercise:

• The initial displacement is zero: \( u(x, 0) = 0 \) for all points between \( 0 \) and \( L \). This means that, initially, the string rests along the \( x \)-axis without any elevation or depression.
• The initial velocity is given as \( u_t(x,0) = \sin(\pi x / L) \). This suggests that although the string starts flat, it begins with a speed that varies sinusoidally along its length.

These conditions set the problem's starting point for analyzing how the wave will evolve. They are crucial as they influence the future motion of the string described by the wave equation.

Boundary Conditions

Boundary conditions specify what happens at the boundaries, or ends, of the string. They are essential in defining how the system interacts with its surroundings. In our scenario, the string is fixed at both ends at \( x = 0 \) and \( x = L \).
This results in the boundary conditions:

• \( u(0, t) = 0 \)
• \( u(L, t) = 0 \)

These conditions mean that the displacement at the endpoints must always remain zero, irrespective of time. Physically, it implies that the string cannot move up or down at these points since they are secured to the \( x \)-axis. This constraint is fundamental in setting up the problem as it affects the form and behavior of the wave along the string.