Question

A stretched string of length \(L\) has the end \(x=0\) fastened down, and the end \(x=L\) moves in a prescribed manner: $$ y(L, t)=\cos \omega_{0} t \quad t>0 $$ Initially, the string is stretched in a straight line with unit displacement at the end \(x=L\) and zero initial velocities everywhere. Find the displacement \(y(x, t)\) as a function of \(x\) and \(t\) after \(t=0\).

Short answer

The displacement is given by \( y(x, t) = A \sin(\frac{n \pi x}{L}) \cos(\omega_{0} t) \).

Step-by-step solution

- Understanding the Boundary Conditions

Identify the boundary conditions given in the problem: \( y(0, t) = 0 \) and \( y(L, t) = \cos(\omega_{0} t) \).

- Initial Conditions

Determine the initial conditions: The string is initially straight, so \( y(x, 0) = 0 \). Also, the initial velocity is zero, so \( \frac{\partial y}{\partial t}(x, 0) = 0 \).

- Form of Solution

Propose a solution that satisfies the wave equation: Use the method of separation of variables where \( y(x, t) = X(x)T(t) \).

- Apply Boundary and Initial Conditions

Apply the boundary conditions to the general solution to find specific forms of \( X(x) \) and \( T(t) \): 1. \( T(t) = \cos(\omega_{0} t) \) considering the boundary condition at \( x = L \).2. Apply the initial condition at \( t=0 \), leading to \( X(x) = \sin(\frac{n \pi x}{L}) \) for the spatial part (assuming a sinusoidal form).

- Summarize the Displacement Function

Combine the spatial and temporal components to find the final expression: \( y(x, t) = A \sin(\frac{n \pi x}{L}) \cos(\omega_{0} t) \).

Key concepts

Boundary Conditions

Boundary conditions are essential when solving wave equations because they define how the ends of the vibrating string behave. In our problem:

• The end at \( x=0 \) is fixed, so the string displacement there is always zero: \( y(0, t) = 0 \).
• The end at \( x=L \) moves in a prescribed manner, following a cosine function: \( y(L, t) = \cos(\omega_0 t) \).

These boundary conditions help us determine the form of the solution.

Initial Conditions

Initial conditions describe the state of the string at the beginning of the observation. For this problem, the string starts off straight, meaning that:

• The initial displacement of the string is zero everywhere except at \( x=L \), leading to \( y(x, 0) = 0 \).
• The initial velocity of the string is zero everywhere, so \( \frac{\partial y}{\partial t}(x, 0) = 0 \).

These initial conditions help in finding the specific solution to the wave equation.

Separation of Variables

The method of separation of variables is used to simplify the wave equation by assuming the solution can be written as the product of a function of space and a function of time: \( y(x, t) = X(x)T(t) \).
This assumption transforms the wave equation into two separate, simpler, ordinary differential equations, one for \( X(x) \) and one for \( T(t) \).
Solving these equations individually is much easier and then they are combined to form the complete solution.

Displacement Function

The displacement function \( y(x, t) \) gives the position of the string at any point \( x \) along its length and at any time \( t \). Using the given boundary and initial conditions, the final form of the displacement function is: \( y(x, t) = A \sin\left(\frac{n \pi x}{L}\right) \cos(\omega_0 t) \).
This equation shows how the string's shape and position change over time.

• The sinusoidal part \( \sin\left(\frac{n \pi x}{L}\right) \) describes the wave pattern along the length of the string.
• The cosine part \( \cos(\omega_0 t) \) describes the oscillation over time.

This combined form satisfies both boundary and initial conditions.

String Vibration

String vibration refers to the way the string moves when disturbed. In this problem, the vibration is driven by the boundary condition at \( x=L \), making the string oscillate according to \( \cos(\omega_0 t) \).
The frequency of vibration is determined by the angular frequency \( \omega_0 \) and affects how fast the string vibrates.
To sum up:

• The pattern and mode of vibration are influenced by the boundary conditions.
• The initial conditions define the starting configuration.
• Separation of variables simplifies finding the exact vibratory motion.

Understanding these concepts helps us grasp the dynamics of string vibrations.