4.1 Why four derivations?

We are about to derive the same equation — the acoustic wave equation — four different times, from four different starting points. This is unusual in a physics textbook: normally a derivation is done once, cleanly, and the chapter moves on. There is a reason to belabour it here.

The wave equation has a single form, but its physical meaning depends on which laws you put in. Each derivation tells us something the others do not. Working through all four is how you stop being surprised that the equation has the form it has.

The four routes, in one paragraph each

Route 1 — fluid mechanics (lessons 4.2–4.5). Conservation of mass, Newton’s second law for a fluid (Euler’s equation), and an equation of state, each linearised around equilibrium. Combine, eliminate $\rho'$ and $\mathbf{v}'$, and out drops $\partial_t^2 p' = c^2 \nabla^2 p'$ with $c^2 = (\partial p / \partial \rho)_s$. This is the canonical derivation and the spine of the chapter.

Route 2 — lattice limit (lesson 4.6). A 1-D chain of mass-springs (like chapter 3) but with longitudinal compressions instead of transverse displacements. Take the continuum limit and the wave equation falls out, with $c^2$ set by spring stiffness and lattice spacing. Shows that “the wave equation” is what a continuous, locally restoring medium does, regardless of the microscopic details.

Route 3 — kinetic theory (lesson 4.7). Pressure is a flux of molecular momentum. A pressure perturbation is a local imbalance in that flux. Track the imbalance through Boltzmann-like reasoning and the wave equation emerges with $c \sim \sqrt{k_B T / m}$. Connects the wave equation to molecular motion explicitly — the bridge back to chapter 1.

Route 4 — Hamilton’s principle (lesson 4.8). Write down a Lagrangian density for the perturbation field and apply the Euler–Lagrange equation. The wave equation falls out as the condition for stationary action. As a bonus, Noether’s theorem gives you the conserved energy and momentum currents for free — currents we will need in chapter 5.

What the four agree on

All four routes produce the same PDE with the same numerical value of $c$ for air at given $p_0$, $\rho_0$, $T_0$. They differ in:

• Starting point. $F = ma$ for a slab (route 1), Newton on discrete masses (route 2), molecular momentum flux (route 3), stationary action (route 4).
• What is computed for free. Route 1 emphasises the three fluid conservation laws. Route 2 emphasises the continuum limit and dispersion-free propagation. Route 3 gives $c \sim \sqrt{k_B T / m}$ from molecular speeds. Route 4 gives Noether currents for energy and momentum.
• What is easier in this framework. Linearisation is easiest in route 1. Mode counting and dispersion are easiest in route 2. Connection to thermodynamics is easiest in routes 1 and 3. Symmetry and conservation are easiest in route 4.

How to read the rest of the chapter

If you only have time for one derivation, read route 1 (lessons 4.2–4.5). It is the standard treatment and the form in which the rest of the book uses the wave equation.

If you want intuition for why the wave equation looks the way it does and not some other way, read all four. The repetition is the point. After the fourth derivation you will not be able to imagine the wave equation having a different form.