Chapter 4 — The acoustic wave equation

Four routes to the same equation. The intuition is in seeing it arrive each way.

This is the structural heart of the book. The acoustic wave equation

is the inevitable consequence of small disturbances in a compressible continuum. Why it is inevitable, and from which starting point it falls out most naturally, is where intuition gets built. We derive it four times, lit by interactives, and watch the same equation arrive from four different physical pictures.

• 4.1 Why four derivations?

The fluid-mechanics route — the canonical derivation that spans four lessons:

• 4.2 Conservation of mass for a fluid slab
• 4.3 Euler’s equation — Newton’s second law in a fluid
• 4.4 The equation of state and the adiabatic assumption
• 4.5 Linearisation and the fluid-mechanics wave equation

The three alternative routes:

• 4.6 Route 2 — from a lattice of oscillators
• 4.7 Route 3 — from kinetic theory and momentum flux
• 4.8 Route 4 — from Hamilton’s principle

And the synthesis:

• 4.9 The speed of sound — same number, four meanings