Q.22 Which one of the following equations represents a one-dimensional wave equation?
(A) ∂u/∂t = c2 ∂2u/∂x2
(B) ∂2u/∂t2 = c2 ∂2u/∂x2
(C) ∂2u/∂t2 = c2 ∂u/∂x
(D) ∂2u/∂t2 + ∂2u/∂x2 = 0
Correct Answer: Option C
The standard 1D wave equation is:
∂2u/∂t2 = c2 ∂2u/∂x2
This governs wave propagation in vibrating strings, sound pulses, and electromagnetic waves.
Option Analysis
(A) ∂u/∂t = c2 ∂u/∂x
This is a first-order transport (advection) equation, not oscillatory wave motion.
(B) ∂u/∂t = ∂2u/∂x2
This is the heat equation, describing diffusion rather than wave propagation.
(C) ∂2u/∂t2 = c2 ∂2u/∂x2
Correct — second-order derivatives allow solutions like
u(x,t) = f(x − ct) + g(x + ct), representing waves moving both directions.
(D) ∂u/∂t + ∂2u/∂x2 + u = 0
Modified reaction-diffusion type equation, not a pure 1D wave equation.
Derivation Overview
For a stretched string (tension T, density ρ), net force leads to:
ρΔx ∂2u/∂t2 = T (∂2u/∂x2Δx)
Simplifying gives the wave equation with wave speed:
c = √(T/ρ)
Solutions propagate left and right at speed c. Boundary conditions determine standing modes.
Common Exam Confusions
| Equation | Form | Physical Meaning |
|---|---|---|
| Wave | ∂2u/∂t2 = c2∂2u/∂x2 | Oscillatory propagation |
| Heat | ∂u/∂t = k ∂2u/∂x2 | Diffuses and smooths behaviour |
| Advection | ∂u/∂t + c ∂u/∂x = 0 | Pure transport motion |
Option (D) adds a +u term, implying decay or growth, not undamped waves.
