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The wave equation ∂²u/∂t² = v²∂²u/∂x² describes how wave displacement evolves in space and time. Any function of the form u(x,t) = f(x - vt) + g(x + vt) satisfies this equation, representing waves traveling in both directions at speed v. This fundamental equation emerges from Newton's laws applied to continuous media and is the basis for understanding all linear wave phenomena.
Derive it for a vibrating string using force balance on a small element. Then verify that sinusoidal and simple linear functions satisfy it.
The wave equation is not the same as the equation of motion for a simple harmonic oscillator—the double spatial derivative encodes how neighboring regions couple.
You already know how to take partial derivatives and how the chain rule handles composite functions. The 1D wave equation, ∂²u/∂t² = v²∂²u/∂x², is a partial differential equation (PDE) that connects the second derivative of a field u (displacement, pressure, electric field) in time to its second derivative in space. The constant v is the wave speed — a property of the medium, not the wave itself.
The cleanest way to understand the equation is to derive it from a physical model. Imagine a flexible string under tension T with mass per unit length μ. Consider a tiny segment of the string at position x. The net upward force on the segment comes from the difference in tension pulling on its two ends — proportional to the curvature ∂²u/∂x² of the string times the tension T. By Newton's second law, this equals the segment's mass (μ·dx) times its acceleration ∂²u/∂t². Rearranging gives ∂²u/∂t² = (T/μ)·∂²u/∂x², which is the wave equation with v = sqrt(T/μ). The double spatial derivative is not an accident — it captures how neighboring string segments are coupled through tension.
The general solution is d'Alembert's formula: u(x, t) = f(x − vt) + g(x + vt) for any twice-differentiable functions f and g. You can verify this by direct substitution: the chain rule gives ∂²f(x − vt)/∂t² = v²f''(x − vt) and ∂²f(x − vt)/∂x² = f''(x − vt), so the equation is satisfied. The function f(x − vt) represents a disturbance of any shape traveling in the +x direction at speed v — a Gaussian pulse, a triangular bump, or a sine wave all work equally well. The function g(x + vt) travels in the −x direction. The superposition of both represents the most general wave pattern.
A key misconception to avoid: the wave equation is not the simple harmonic oscillator (SHO) equation written in two variables. The SHO, d²x/dt² = −ω²x, governs a single point bobbing up and down — it has no spatial structure. The wave equation describes how disturbances at one location propagate to neighboring locations because of the ∂²u/∂x² coupling term. A localized pulse at x=0 at t=0 can arrive at x=10 later; this spatial propagation is precisely what the double spatial derivative encodes, and it is entirely absent from the SHO equation.
Sinusoidal solutions — harmonic waves of the form A·sin(kx − ωt) — are special cases of the general solution where f(x − vt) = A·sin(k(x − vt)) with ω = kv. These will be the building blocks for analyzing reflection, interference, and standing waves in subsequent topics. The wave equation's linearity means solutions can be superposed freely — two waves can pass through each other without interaction, each obeying the equation independently.