Waves as physical objects

The plane-wave ansatz and dispersion, phase and group velocity, wave impedance, energy and radiation pressure, reflection and transmission, and the WKB approximation.

The wave equation and its mathematics live in the Foundations book; this chapter is its physical companion — the properties that waves carry regardless of what is waving, and that recur across acoustics, elasticity, electromagnetism, and quantum mechanics. From the single plane-wave building block come the dispersion relation, the two velocities, the impedance that couples a wave’s two fields, the energy and momentum it transports, its behaviour at a boundary, and its adaptation to a slowly-varying medium.

• 10.1 The plane-wave ansatz and dispersion relations — the elementary solution, and how substituting it turns a PDE into $\omega(k)$.
• 10.2 Phase and group velocity — crest speed versus envelope speed, and the stationary-phase derivation of $v_g = d\omega/dk$.
• 10.3 Wave impedance — $Z = \rho_0 c$ derived from the wave equation, and the general mechanical impedance.
• 10.4 Energy, intensity, and radiation pressure — the energy density and its equipartition, intensity, and the force a wave exerts.
• 10.5 Reflection and transmission at interfaces — the impedance-contrast coefficients, power conservation, and impedance matching.
• 10.6 The WKB approximation — waves in slowly-varying media and the $A \propto 1/\sqrt{k}$ amplitude rule.