Which of the following functions is a solution to the 1D wave equation ∂²u/∂t² = v²∂²u/∂x²?
Au(x,t) = A·sin(x)·cos(t) only if v=1
Bu(x,t) = A·exp(−x²)
Cu(x,t) = A·sin(kx − ωt) for any k,ω with ω/k = v
Du(x,t) = A·(x² + t²)
Any function of the form f(x − vt) or g(x + vt) satisfies the wave equation. A·sin(kx − ωt) = A·sin(k(x − (ω/k)t)) has the form f(x − vt) provided v = ω/k. The Gaussian exp(−x²) has no t-dependence and does not satisfy the equation. The polynomial x² + t² fails because its second derivatives in x and t give 2 and 2, but 2 ≠ v²·2 in general.
Question 2 True / False
The 1D wave equation and the simple harmonic oscillator equation are essentially the same — both describe oscillatory motion and have sinusoidal solutions.
TTrue
FFalse
Answer: False
The SHO equation is an ODE: d²x/dt² = −ω²x, describing a single point oscillating in time. The wave equation is a PDE with second derivatives in both space and time: ∂²u/∂t² = v²∂²u/∂x². The spatial derivative encodes coupling between neighboring locations — disturbances propagate. A pulse that is zero at t=0 can arrive at a distant point later, which the SHO equation cannot describe at all.
Question 3 Short Answer
In the derivation of the wave equation from a vibrating string, what physical quantity determines the wave speed v, and what does this tell you about wave propagation in stiffer vs. heavier strings?
Think about your answer, then reveal below.
Model answer: The wave speed is v = sqrt(T/μ), where T is the string tension and μ is the mass per unit length. A stiffer (higher tension) string carries waves faster; a heavier (larger μ) string carries waves slower. Wave speed is set by the medium's restoring force and inertia, not by the wave's amplitude or shape.
This connects the mathematical wave equation to the physical derivation. Force balance on a small string element gives the restoring force proportional to tension times the curvature ∂²u/∂x², while Newton's second law gives the inertia term μ·∂²u/∂t². Dividing by μ and identifying v² = T/μ produces the wave equation. The same structure — restoring force / inertia — appears in all linear wave systems.