Wave equation
Objective. Solve vibrating-string problems.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The wave equation uₜₜ = c²uₓₓ models vibrations such as a stretched string, with c the wave speed, and admits solutions that travel or stand still depending on boundary conditions. For a string fixed at both ends, separation of variables gives standing-wave modes uₙ(x,t) = sin(nπx/L)[Aₙcos(nπct/L) + Bₙsin(nπct/L)], each oscillating sinusoidally in time, unlike the heat equation's decay. Example: a string of length L plucked into shape sin(πx/L) with zero initial velocity oscillates as sin(πx/L)cos(πct/L), repeating periodically rather than smoothing out. General motion is a superposition of these modes, each with its own frequency nπc/L.
Worked example
A string satisfies the wave equation uₜₜ = 9uₓₓ on 0<x<π with fixed ends, initial shape u(x,0) = sin(3x), and zero initial velocity. Find u(x,t).
- 1Match initial shape to mode n=3 on [0,π]: spatial part sin(3x)
- 2Wave speed c is given by c² = 9, so c = 3
- 3Mode temporal frequency is nc = 3×3 = 9, giving cos(9t) since initial velocity is zero
- 4Combine: u(x,t) = sin(3x)cos(9t)
Answer: u(x,t) = sin(3x)cos(9t)
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students carry over the exponential-decay intuition from the heat equation, forgetting wave-equation modes oscillate forever in time as cosines/sines, with no energy loss in the idealised equation.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.