The Wave Equation and d'Alembert's Solution

The Idea in Plain Language

Imagine a long, taut string. If you pluck a section of the string, the disturbance does not stay in one place; it travels along the string. The wave equation tells us that the acceleration of any point on the string is proportional to the curvature of the string at that point. The remarkable result of d'Alembert is that the most general motion of the string is simply the sum of two "shapes" that move rigidly in opposite directions. One shape moves to the right at speed \(c\), and the other moves to the left at speed \(c\). They do not change shape as they travel; they simply translate. If you have a complex wave, it is just a superposition of these two rigidly moving profiles. This is a profound simplification: instead of solving a complex differential equation for every point, we just need to track two arbitrary functions moving at constant speed.

The General Solution

Consider the one-dimensional wave equation on an infinite domain, where \(x \in \mathbb{R}\): \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \] Here, \(u(x,t)\) is the displacement of the string at position \(x\) and time \(t\), and \(c\) is the wave speed. The general solution to this equation is given by d'Alembert's formula. It states that any twice-differentiable function \(u(x,t)\) that satisfies the wave equation can be written as the sum of two arbitrary functions: \[ u(x,t) = f(x - ct) + g(x + ct) \] The term \(f(x - ct)\) represents a wave profile \(f\) that moves to the right with constant velocity \(c\). To see this, consider a specific point on the wave, say where the argument \(x - ct\) is constant. Differentiating \(x - ct = \text{const}\) with respect to time gives \(v = dx/dt = c\). Similarly, the term \(g(x + ct)\) represents a wave profile \(g\) that moves to the left with velocity \(-c\), since \(x + ct = \text{const}\) implies \(dx/dt = -c\).

This form is powerful because \(f\) and \(g\) are completely arbitrary functions (provided they are twice differentiable). Strictly they must be twice differentiable for the equation to hold point by point; physically, kinked profiles (such as a plucked string) are still admitted as generalised (weak) solutions. This means the wave equation does not restrict the shape of the wave, only how it propagates.

Initial Value Problem

In physical problems, we usually do not know the functions \(f\) and \(g\) directly. Instead, we know the initial state of the system: the initial displacement \(\phi(x) = u(x,0)\) and the initial velocity \(\psi(x) = \partial_t u(x,0)\). We can determine \(f\) and \(g\) from these initial conditions.

At \(t=0\), the general solution becomes: \[ u(x,0) = f(x) + g(x) = \phi(x) \] Differentiating the general solution with respect to time: \[ \frac{\partial u}{\partial t} = -c f'(x - ct) + c g'(x + ct) \] At \(t=0\): \[ \frac{\partial u}{\partial t}\bigg|_{t=0} = -c f'(x) + c g'(x) = \psi(x) \] We now have a system of two equations for the unknown functions \(f\) and \(g\). By integrating the second equation and substituting into the first, we can solve for \(f\) and \(g\) explicitly. The result is d'Alembert's explicit formula for the initial value problem: \[ u(x,t) = \frac{1}{2} \left[ \phi(x - ct) + \phi(x + ct) \right] + \frac{1}{2c} \int_{x - ct}^{x + ct} \psi(s) \, ds \] This formula is unique. It tells us that the displacement at any point \((x,t)\) depends only on the initial conditions within the interval \([x-ct, x+ct]\). This region is known as the domain of dependence.

Physical Interpretation

The explicit formula has a clear physical interpretation. The first term, \(\frac{1}{2} [\phi(x - ct) + \phi(x + ct)]\), represents the initial shape \(\phi\) splitting into two identical halves, each travelling in opposite directions at speed \(c\). The second term, \(\frac{1}{2c} \int_{x - ct}^{x + ct} \psi(s) \, ds\), accounts for the initial velocity. It is the average of the initial velocity over the interval \([x-ct, x+ct]\), scaled by time. This term vanishes if the initial velocity is zero everywhere.

The speed \(c\) is determined by the physical properties of the medium. For a string with tension \(T\) and linear mass density \(\mu\), the wave speed is \(c = \sqrt{T/\mu}\). This means that heavier strings (larger \(\mu\)) or looser strings (smaller \(T\)) support slower waves. The wave equation itself does not specify \(c\); it is a parameter that must be determined by the physics of the medium.