Waves & soundUndergraduate & beyond
One-dimensional linear wave equation
∂²y/∂x² = (1/v²)∂²y/∂t²
Differential equation for a nondispersive one-dimensional linear wave.
Symbols, conditions & references
When this applies
Linear restoring response, constant medium properties and no damping or source term.
- y
- wave disturbancequantity-dependent
- x
- position coordinatem
- v
- constant wave speedm s⁻¹
- t
- times
Worked check question
Show that y = A cos(kx − ωt) solves the wave equation when ω = vk.
Show worked answer
Differentiate twice: ∂²y/∂x² = −k²y and ∂²y/∂t² = −ω²y. Substitution requires k² = ω²/v², hence ω = vk for positive frequency.