Linear wave equation

Definition

The linear wave equation is a second-order partial differential equation that describes the propagation of linear waves, such as sound or light waves, in a medium. It is typically written as $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$, where $u$ represents the wave function and $c$ is the speed of the wave.

5 Must Know Facts For Your Next Test

1. The linear wave equation models how disturbances propagate through various media without changing shape.
2. $c$ represents the constant speed at which the wave travels through the medium.
3. Solutions to the linear wave equation can be expressed as sums of sinusoidal functions due to its linearity.
4. It applies to both longitudinal and transverse waves, depending on the context (e.g., sound vs. electromagnetic waves).
5. Boundary conditions and initial conditions are essential for solving specific problems involving the linear wave equation.

Review Questions

• What type of partial differential equation is the linear wave equation?
• In the linear wave equation, what does $c$ represent?
• How do initial and boundary conditions affect solutions to the linear wave equation?

Related terms

Partial Differential Equation: A type of differential equation that involves partial derivatives with respect to multiple variables.

Sinusoidal Functions: Functions that describe oscillations, often used in solutions to wave equations, including sine and cosine functions.

Boundary Conditions:

Constraints necessary for solving differential equations that specify values of the solution at particular locations or times.