The Wave Equation

When Disturbances Travel

The heat equation models diffusion: disturbances smooth out and disappear. But not all physical phenomena dissipate. Pluck a guitar string, and the vibration persists, bouncing back and forth between the fixed ends. Strike a drum, and the sound propagates outward, carrying energy without losing it. Drop a stone in a pond, and ripples spread in expanding circles.

These phenomena are governed by the wave equation:

The crucial difference from the heat equation is the second derivative in time. Heat flow is a first-order process: the rate of change of temperature depends on how heat is distributed now. Wave motion is a second-order process: how a string accelerates depends on how it is curved.

This distinction has profound consequences. Heat dissipates; waves conserve energy. Heat smooths; waves maintain their shape as they travel.

Deriving the Equation from Physics

Consider a vibrating string stretched between two fixed points. Take a small segment of the string and analyze the forces acting on it.

Let $u(x,t)$ be the vertical displacement of the string at position $x$ and time $t$. The string has mass density $\rho$ (mass per unit length) and is under tension $T$.

For a small segment of length $\Delta x$:

• The mass of the segment is $\rho \Delta x$
• The tension pulls at both ends, nearly horizontal for small displacements
• The vertical components of tension are approximately $T \frac{\partial u}{\partial x}$ at each end

Newton's second law for the segment gives:

Dividing by $\Delta x$ and taking the limit as $\Delta x \to 0$:

Writing $c^2 = T/\rho$, we get the wave equation. The quantity $c = \sqrt{T/\rho}$ is the wave speed: higher tension means faster waves, greater mass density means slower waves. This matches intuition: tighten a guitar string and the pitch goes up.

The Boundary Value Problem

For a string of length $L$ with both ends fixed, we have:

The PDE:

Boundary conditions (fixed ends):

Initial conditions:

Notice that the wave equation requires two initial conditions: the initial shape and the initial velocity. This is because the equation is second-order in time. Compare this to the heat equation, which only needs the initial temperature distribution.

Physically, knowing where the string is at $t=0$ is not enough. You also need to know how fast each point is moving. A string pulled aside and released from rest behaves differently from a string given an initial push.

Separation of Variables

We seek solutions of the form $u(x,t) = X(x)T(t)$. Substituting into the wave equation:

Dividing by $c^2 X(x)T(t)$:

The separation constant must be the same for both sides. Following the reasoning from the heat equation chapter, $\lambda$ must be positive for bounded solutions.

This gives us two ordinary differential equations:

The Spatial Eigenfunctions

The boundary conditions $u(0,t) = 0$ and $u(L,t) = 0$ translate to $X(0) = 0$ and $X(L) = 0$.

This is the same eigenvalue problem we solved for the heat equation. The solutions are:

Each $X_n$ is a normal mode of vibration. The fundamental mode ($n=1$) has no interior nodes. The second harmonic ($n=2$) has one node at the center. The $n$th mode has $n-1$ interior nodes.

The Temporal Factor

With $\lambda_n = n^2\pi^2/L^2$, the temporal equation becomes:

where $\omega_n = \frac{n\pi c}{L}$ is the angular frequency of the $n$th mode.

This is the simple harmonic oscillator equation. The general solution is:

The solution oscillates forever, unlike the exponentially decaying temporal factor in the heat equation. Energy is conserved in wave motion.

Standing Waves

Each separated solution

is called a standing wave. The spatial shape $\sin(n\pi x/L)$ stays fixed; only the amplitude oscillates in time.

Points where $\sin(n\pi x/L) = 0$ are called nodes. They never move. Points where $|\sin(n\pi x/L)| = 1$ are antinodes, oscillating with maximum amplitude.

The fundamental mode ($n=1$) has the lowest frequency $\omega_1 = \pi c/L$. Higher modes have frequencies that are integer multiples: $\omega_n = n\omega_1$. This is why musical strings produce harmonious overtones. The frequencies are in the ratio $1:2:3:4:\ldots$

General Solution by Superposition

The general solution is a superposition of all normal modes:

The coefficients $A_n$ and $B_n$ are determined by the initial conditions.

From $u(x,0) = f(x)$:

These are the Fourier sine coefficients of $f$:

From $u_t(x,0) = g(x)$:

So:

The actual vibration of a string is a combination of many modes. The relative amplitudes determine the timbre of the sound. A violin, a guitar, and a piano playing the same note have the same fundamental frequency but different mixtures of overtones.

Plucking a String

When you pluck a guitar string, you pull it aside to some shape $f(x)$ and release it from rest. The initial velocity is zero: $g(x) = 0$.

With $g(x) = 0$, all the $B_n$ coefficients vanish. The solution is purely a sum of cosines:

The coefficients $A_n$ depend on the shape of the pluck. A triangular pluck at position $p \cdot L$ (where $0 < p < 1$) has:

where $h$ is the pluck height.

Notice something remarkable: when you pluck at the center ($p = 0.5$), the formula $\sin(n\pi/2)$ is zero for even $n$. Only odd harmonics are excited. This is why the sound of a center-plucked string is hollow compared to one plucked near the end, which excites a richer mixture of overtones.

Traveling Waves: D'Alembert's Solution

There is a completely different way to understand the wave equation, discovered by d'Alembert in 1747.

Consider the wave equation on an infinite domain (no boundaries):

Introduce new variables:

Using the chain rule to transform the derivatives (a calculation we omit here), the wave equation becomes:

This remarkably simple equation says that $\frac{\partial u}{\partial \xi}$ is independent of $\eta$. Therefore:

for any twice-differentiable functions $F$ and $G$.

The function $F(x - ct)$ represents a wave traveling to the right at speed $c$. As time increases, the argument $x - ct$ stays constant when $x$ increases by $ct$. The shape moves to the right without changing.

Similarly, $G(x + ct)$ is a wave traveling to the left at speed $c$.

Any solution to the wave equation is a superposition of left-traveling and right-traveling waves. The shape of each wave is preserved as it travels. This is fundamentally different from diffusion, where shapes inevitably smooth out.

D'Alembert's Formula

For the initial value problem on an infinite string:

the solution is given by d'Alembert's formula:

The first term shows the initial shape splitting into two halves, each traveling in opposite directions. The second term accounts for any initial velocity.

This formula makes the physics transparent. An initial disturbance does not stay put or spread uniformly. It splits and travels. Information propagates at the finite speed $c$.

Standing Waves vs. Traveling Waves

Standing waves and traveling waves are two perspectives on the same phenomenon.

A standing wave $\sin(kx)\cos(\omega t)$ can be written as:

The right side is the sum of two traveling waves moving in opposite directions. When they superpose, they create the illusion of a wave that stands still, with nodes and antinodes.

Conversely, two standing waves that are $90°$ out of phase combine to form a traveling wave. This is how rotating machines work: two perpendicular oscillations create circular motion.

On a finite string with fixed ends, a right-traveling wave hits the boundary and reflects back as a left-traveling wave. The continuous reflection back and forth, combined with superposition, creates standing wave patterns.

Energy Conservation

The wave equation conserves energy. Define:

The first term is kinetic energy (mass times velocity squared). The second term is potential energy (tension times stretch squared).

Taking the time derivative and using the wave equation, one can show that $\frac{dE}{dt} = 0$. Energy sloshes back and forth between kinetic and potential forms, but the total remains constant.

This is why a guitar string keeps vibrating long after you pluck it. In reality, energy is slowly lost to air resistance and internal friction in the string, but the wave equation itself has no dissipation.

Key Takeaways

• The wave equation $u_{tt} = c^2 u_{xx}$ models vibrations and wave propagation
• Unlike heat, waves conserve energy and maintain their shape as they travel
• Separation of variables gives standing waves: $\sin(n\pi x/L) \cdot [A_n\cos(\omega_n t) + B_n\sin(\omega_n t)]$
• Modal frequencies are $\omega_n = n\pi c/L$, forming the harmonic series $1:2:3:\ldots$
• D'Alembert's solution $u = f(x-ct) + g(x+ct)$ shows waves as traveling disturbances
• Initial conditions require both position $f(x)$ and velocity $g(x)$
• The pluck position determines which harmonics are excited, affecting timbre
• Standing waves are superpositions of traveling waves reflecting between boundaries