Lecture notes wave equation

Solving the Wave Equation in PDEs
Author Name
November 2, 2023


1 Introduction
The wave equation is a second-order linear partial differential equation which
describes the propagation of waves, be they sound waves, light waves, or the
oscillations of a string. In one dimension, the wave equation is typically written
as:
∂2u ∂2u
2
= c2 2
∂t ∂x
where c is the wave speed and u(x, t) describes the displacement of the wave at
position x and time t.


2 General Solution
For the wave equation to have unique solutions, boundary and initial conditions
are often required. Given an interval [0, L], common boundary conditions are:
• Dirichlet boundary conditions: u(0, t) = 0 and u(L, t) = 0 for all t.
• Neumann boundary conditions: ux (0, t) = 0 and ux (L, t) = 0 for all
t.
In addition, we often need initial conditions, which are typically of the form:

u(x, 0) = f (x) and ut (x, 0) = g(x)


3 Example
Consider a string of length L = π fixed at both ends. The wave speed on this
string is given as c = 1. We aim to solve the wave equation with the following
initial and boundary conditions:


u(0, t) = u(π, t) = 0
u(x, 0) = sin(x)
ut (x, 0) = 0


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