Complex exponentials and phasors

Euler’s formula and why we use it for oscillations.

Almost every formula in acoustics that looks like $\cos(\omega t - k x + \varphi)$ is really a complex exponential in disguise. The reason is purely pragmatic: complex exponentials make linear ODEs and PDEs into algebra. Time derivatives become multiplication by $i\omega$; spatial derivatives become multiplication by $-i\mathbf{k}$; the wave equation collapses to the algebraic dispersion relation $\omega^2 = c^2 |\mathbf{k}|^2$. The “complex” is a misnomer — using complex exponentials makes everything simpler.

This chapter is three lessons developing the same machinery in increasing generality:

• 3.1 Euler’s formula and the phasor — $e^{i\theta} = \cos\theta + i\sin\theta$ derived from Taylor series; phasor addition; a worked one-line solution of the driven oscillator.
• 3.2 Damped oscillations as phasors — what happens with a complex argument $\lambda = -\gamma + i\omega$: a logarithmic spiral inward in the complex plane, with the real part being a damped sinusoid. The eigenfunctions of every linear damped system on the bookshelf.
• 3.3 Plane waves and complex impedance — the same machinery in space and time, producing plane waves $e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}$ and the complex-valued impedance $Z(\omega)$ that characterises every linear acoustic, electrical, or mechanical element.