Wave motion

Derivation of One-Dimensional Wave Equation

Assumptions:

1. The wave is one-dimensional, propagating along the x-axis.
2. The string is flexible and has negligible mass.
3. Tension in the string is constant.
4. Small-angle approximation (\( \sin\theta \approx \theta \)) for small angles of displacement.

Setup:

Let \( y(x, t) \) be the displacement of a particle at position \( x \) along the string at time \( t \).

The net force acting on a small element \( \Delta x \) of the string is given by:

\[ F = T \cdot \frac{\partial y}{\partial x} \]

where \( T \) is the tension in the string.

Newton's Second Law:

\[ F = \frac{\partial^2 y}{\partial t^2} \cdot \Delta x \cdot \mu \]

where \( \mu \) is the linear density (mass per unit length) of the string.

Equating the Forces:

\[ T \cdot \frac{\partial y}{\partial x} = \frac{\partial^2 y}{\partial t^2} \cdot \Delta x \cdot \mu \]

Divide both sides by \( \Delta x \) and take the limit as \( \Delta x \) approaches zero:

\[ T \cdot \frac{\partial^2 y}{\partial x^2} = \mu \cdot \frac{\partial^2 y}{\partial t^2} \]

This is the one-dimensional wave equation!

Solution of the Wave Equation:

The general solution to the one-dimensional wave equation is a sum of two functions, each representing a wave traveling in the positive and negative x-directions:

\[ y(x, t) = f(x - vt) + g(x + vt) \]

where:

• \( f \) is a function representing a wave traveling to the right.
• \( g \) is a function representing a wave traveling to the left.
• \( v \) is the speed of the wave.

The functions \( f \) and \( g \) are determined by the initial conditions of the wave (e.g., the shape of the string at \( t = 0 \)).

This solution represents a superposition of two waves traveling in opposite directions, and it satisfies the one-dimensional wave equation.