Questions: The Wave Equation and Vibrating Strings

Question 1

At time t=0, a violin string is displaced only near its midpoint and is undisturbed everywhere else. A student claims that the entire string, including the endpoints, will begin moving at the same instant t=0⁺ because it is a connected, continuous object. Is this consistent with the wave equation?

• A. Yes — on a connected string, any local displacement instantaneously propagates everywhere.
• B. No — the wave equation has finite propagation speed c; points far from the disturbance are unaffected until the wavefront arrives at time t = distance/c.
• C. No — the wave equation predicts that disturbances do not propagate at all in a bounded medium.
• D. Yes, but only for an infinitely long string; on a finite string, reflections create instantaneous coupling.

Answer: (B)

The wave equation is a hyperbolic PDE with finite propagation speed c. D'Alembert's formula makes this explicit: u(x,t) = φ(x+ct) + ψ(x−ct), meaning information travels at exactly speed c. A disturbance at x₀ at time 0 only reaches position x at time t = |x−x₀|/c. This is fundamentally different from the heat equation (parabolic), where a disturbance instantaneously affects all points — a common source of confusion between hyperbolic and parabolic PDEs.

Question 2

D'Alembert's formula writes the general solution to the wave equation as u(x,t) = φ(x+ct) + ψ(x−ct). What is the correct physical interpretation of the two terms?

• A. φ represents the amplitude envelope and ψ represents the frequency content of the wave.
• B. φ(x+ct) is a wave traveling in the negative x-direction and ψ(x−ct) is a wave traveling in the positive x-direction.
• C. Both terms represent standing waves that oscillate in place without net movement.
• D. φ and ψ are determined entirely by boundary conditions and carry no information about initial conditions.

Answer: (B)

In φ(x+ct), as time t increases, the argument stays constant when x decreases — meaning the pattern moves in the negative x-direction (leftward) at speed c. In ψ(x−ct), the argument is constant when x increases with t — a rightward-traveling wave. Every solution to the wave equation is a superposition of these two counter-propagating traveling waves. Initial conditions (displacement and velocity at t=0) determine φ and ψ; boundary conditions then cause reflections.

Question 3

A disturbance introduced at position x=0 at time t=0 in a medium governed by the wave equation with speed c cannot affect the displacement at position x=L until time t = L/c.

• T. True
• F. False

Answer: True

This is the finite propagation speed property of the wave equation, encoded directly in d'Alembert's formula. The wavefront travels at exactly speed c. Before time L/c, the wave has not yet reached x=L, so the displacement there remains zero. This locality property distinguishes the wave equation (hyperbolic) from the heat equation (parabolic), where disturbances propagate instantaneously.

Question 4

Like the heat equation, the wave equation predicts that a local disturbance propagates to most other positions instantaneously, though the effect diminishes rapidly with distance.

• T. True
• F. False

Answer: False

This describes the heat equation (parabolic), not the wave equation (hyperbolic). The heat equation exhibits infinite propagation speed — a temperature change at one point mathematically affects all other points at any positive time t, though the effect decays exponentially with distance. The wave equation, by contrast, has strictly finite propagation speed c: a point at distance d is completely unaffected until time t = d/c. This distinction between hyperbolic and parabolic PDEs is one of the most important in mathematical physics.

Question 5

What does d'Alembert's formula reveal about the structure of wave solutions on an infinite domain, and why is this more physically transparent than the normal mode (separation of variables) decomposition?

Model answer: D'Alembert's formula u(x,t) = φ(x+ct) + ψ(x−ct) shows that every solution is a superposition of exactly two traveling waves — one moving left, one moving right — each maintaining its shape at speed c. The solution at (x,t) depends only on initial data in the interval [x−ct, x+ct], making the finite propagation speed and causal structure immediately visible. The normal mode decomposition expresses solutions as standing waves (sinusoids in space times sinusoids in time), which are useful for resonance and frequency analysis but obscure the fact that disturbances travel. D'Alembert's approach is better for understanding signal propagation and causality; normal modes are better for studying the harmonic content of bounded systems.

The key contrast is traveling waves (d'Alembert) vs. standing waves (separation of variables). Both are valid representations of the same solutions, but the traveling-wave picture makes causality and finite propagation speed transparent.