4.8 Route 4 — from Hamilton’s principle

The fourth and final route to the acoustic wave equation. Instead of Newton’s second law for a slab (route 1), or coupled oscillators (route 2), or molecular momentum flux (route 3), we start from a variational principle: the action

S \;=\; \int dt \int d^3 r\; \mathcal{L}

is stationary on physical trajectories. Out falls the wave equation, and — by Noether’s theorem — the conserved energy and momentum currents of the field. This is a payoff routes 1–3 cannot give.

This is the deepest route, and intentionally last. It plants the seeds for chapter 5 (energy and momentum carried by sound waves) and shows how acoustics fits into the same framework as classical field theory, electromagnetism, and general relativity.

The acoustic Lagrangian density

For a linearised compressible fluid with velocity potential $\phi$ (where $\mathbf{v}' = \nabla \phi$ and $p' = -\rho_0 \partial_t \phi$ ), the Lagrangian density is

\mathcal{L} \;=\; \tfrac12 \rho_0 \big(\partial_t \phi\big)^2 \;-\; \tfrac12 \rho_0 c^2 \big(\nabla \phi\big)^2.

Two terms. The first is the kinetic energy density of the local fluid motion ( $\tfrac12 \rho_0 v'^2$ with $v' = \partial_x \phi$ , but actually $\tfrac12 \rho_0 (\partial_t \phi)^2$ in this potential formulation — see derivation). The second is the potential energy density stored in the compression. Subtracting potential from kinetic, as Lagrangian mechanics always does.

The Euler–Lagrange equation

For a field $\phi(\mathbf{r}, t)$ with Lagrangian density $\mathcal{L}(\phi, \partial_t \phi, \nabla \phi)$ , the Euler–Lagrange equation is

\frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial (\partial_t \phi)} \;+\; \nabla \cdot \frac{\partial \mathcal{L}}{\partial (\nabla \phi)} \;-\; \frac{\partial \mathcal{L}}{\partial \phi} \;=\; 0.

Apply to our $\mathcal{L}$ :

$\partial \mathcal{L} / \partial (\partial_t \phi) = \rho_0 \partial_t \phi$ . Its time derivative is $\rho_0 \partial_t^2 \phi$ .
$\partial \mathcal{L} / \partial (\nabla \phi) = -\rho_0 c^2 \nabla \phi$ . Its divergence is $-\rho_0 c^2 \nabla^2 \phi$ .
$\partial \mathcal{L} / \partial \phi = 0$ .

Putting it together:

\rho_0 \partial_t^2 \phi \;-\; \rho_0 c^2 \nabla^2 \phi \;=\; 0,

i.e.

\boxed{\;\;\partial_t^2 \phi \;=\; c^2\, \nabla^2 \phi.\;\;}

The acoustic wave equation, for the velocity potential. Since $p' = -\rho_0 \partial_t \phi$ , the same equation holds for $p'$ (apply $-\rho_0 \partial_t$ to both sides).

▶ Where the Lagrangian density comes from

A self-contained justification: the linearised continuity and Euler equations are equivalent to a single equation for the velocity potential $\phi$ . From Euler, $\rho_0 \partial_t \mathbf{v}' = -\nabla p'$ , and substituting $\mathbf{v}' = \nabla \phi$ and $p' = -\rho_0 \partial_t \phi$ makes the equation an identity (both sides become $-\rho_0 \nabla \partial_t \phi$ ). From continuity, $\partial_t \rho' = -\rho_0 \nabla \cdot \mathbf{v}' = -\rho_0 \nabla^2 \phi$ . Use $p' = c^2 \rho'$ to get $\partial_t \rho' = c^{-2} \partial_t p' = -c^{-2} \rho_0 \partial_t^2 \phi$ . Combining, $\rho_0 \nabla^2 \phi = c^{-2} \rho_0 \partial_t^2 \phi$ , i.e. the wave equation for $\phi$ .

A Lagrangian density that produces this wave equation via the Euler–Lagrange equation is, up to total time derivatives,

\mathcal{L} \;=\; \tfrac12 \rho_0 (\partial_t \phi)^2 \;-\; \tfrac12 \rho_0 c^2 (\nabla \phi)^2.

Physically the two terms are kinetic and potential energy densities of the perturbation field. The verification: compute the equations of motion, recover the wave equation.

What Noether gives us for free

Hamilton’s principle is the gateway to symmetries and conservation laws via Noether’s theorem. Three immediate fruits:

Energy conservation (from time-translation invariance):

\mathcal{E} \;=\; \tfrac12 \rho_0 (\partial_t \phi)^2 + \tfrac12 \rho_0 c^2 (\nabla \phi)^2.

This is the acoustic energy density — kinetic plus potential. Its time evolution satisfies a continuity equation $\partial_t \mathcal{E} + \nabla \cdot \mathbf{S} = 0$ , with energy flux $\mathbf{S} = -\rho_0 c^2 (\partial_t \phi)(\nabla \phi) = p' \mathbf{v}'$ . The product $p' \mathbf{v}'$ is the acoustic intensity (we will use this name in chapter 5).

Momentum conservation (from space-translation invariance): the wave carries momentum, with density and flux derivable from the same Lagrangian. Radiation pressure on an obstacle (lesson 5.6) is the macroscopic consequence.
Lorentz-like symmetries. The wave equation has the form of the d’Alembertian operator from special relativity, with $c$ replacing the speed of light. Acoustic plane waves transform under “Lorentz boosts at speed $c$ ” in a formal sense, and many derivations in moving media (chapter 9) become cleaner in those frames.

We will not pursue all of these. The point of route 4 is to make their existence visible, not to compute every consequence.

Why this is non-overlapping

Routes 1, 2, 3 are different starting points but they all eventually invoke $F = ma$ (route 1 directly, routes 2 and 3 through the chain or molecular bath). Route 4 starts from a different principle: that physical trajectories extremise an action functional. Newton’s law is a derived consequence, not the foundation. The equivalence of the two formulations is one of the deep facts of classical mechanics — but the variational form generalises in directions that $F = ma$ does not (gauge theories, general relativity, path-integral quantisation). For acoustics, the variational form’s payoff is the conservation laws.

Next lesson: all four routes give the same speed of sound — but each route gives that speed a different meaning.