Wave Equation
Vibrating strings and sound.
Key idea
Vibrating strings and sound.
In depth
u_tt = c²u_xx describes vibrating strings and sound waves. d’Alembert’s solution: u = f(x − ct) + g(x + ct), two waves travelling in opposite directions at speed c. Unlike heat, it preserves shape and does not smooth.
Worked example
A plucked string splits into two half-height pulses moving left and right.
Formula
Real-world uses
- Modelling guitar strings, sound in air and seismic waves through the ground
Common mistake
Forgetting that the solution depends on both initial position and velocity.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Wave Equation?
Vibrating strings and sound. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Wave Equation?
The key formula is uₜₜ = c² uₓₓ.
Where is Wave Equation used in real life?
Real-world examples: Modelling guitar strings, sound in air and seismic waves through the ground.
What should I learn before Wave Equation?
Make sure you are comfortable with Heat Equation, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Wave Equation?
Forgetting that the solution depends on both initial position and velocity.
How can I practise Wave Equation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Wave Equation?
Move on to Laplace’s Equation next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.