2.2 The complex-exponential trick

The SHM solution $x(t) = X \cos(\omega_0 t + \varphi)$ has two real numbers buried in it — amplitude $X$ and phase $\varphi$ . Carrying them around explicitly through every derivation is unpleasant. The complex-exponential trick (also called phasor analysis) packages both into a single complex number and turns time derivatives into multiplication. Almost every equation in the rest of the book is easier in this notation.

If you have already worked through the foundations chapter on complex exponentials, this lesson is a quick re-application; otherwise, read it first.

The trick

Given a real oscillation

x(t) \;=\; X \cos(\omega_0 t + \varphi),

define the complex amplitude (phasor)

\tilde X \;\equiv\; X\, e^{i\varphi},

and write

x(t) \;=\; \operatorname{Re}\!\left[\tilde X\, e^{i \omega_0 t}\right].

The real signal is recovered by multiplying by $e^{i\omega_0 t}$ and taking the real part. The complex number $\tilde X$ carries both amplitude $|\tilde X| = X$ and phase $\arg(\tilde X) = \varphi$ in a single object.

Why this is so useful

Time derivatives of $e^{i\omega_0 t}$ are easy: $\frac{d}{dt} e^{i\omega_0 t} = i\omega_0 e^{i\omega_0 t}$ . So if we work with the complex form $z(t) = \tilde X e^{i\omega_0 t}$ , the equation $\ddot z + \omega_0^2 z = 0$ becomes

(i\omega_0)^2 \tilde X + \omega_0^2 \tilde X \;=\; 0 \;\;\Longrightarrow\;\; -\omega_0^2 \tilde X + \omega_0^2 \tilde X \;=\; 0,

trivially satisfied. The differential equation has been converted to an algebraic identity.

More usefully, in the driven oscillator (next lesson but one), with damping and a sinusoidal forcing term, the SHM equation becomes

\ddot z + 2\gamma \dot z + \omega_0^2 z \;=\; \tilde F_0\, e^{i\omega t},

and substituting $z = \tilde Z e^{i\omega t}$ gives

\big[-\omega^2 + 2i\gamma\omega + \omega_0^2\big]\, \tilde Z \;=\; \tilde F_0,

a one-line algebraic equation for the amplitude. No characteristic equation, no real-part juggling, no separate amplitude-and-phase calculations. Acoustics happens at the level of $\tilde Z$ .

A small example

Two oscillations of the same frequency:

x_1(t) = 3 \cos(\omega t), \qquad x_2(t) = 4 \sin(\omega t) = 4 \cos(\omega t - \pi/2).

Their phasors are $\tilde X_1 = 3$ and $\tilde X_2 = -4i$ . Their sum, also a sinusoid at frequency $\omega$ , has phasor

\tilde X_{1+2} \;=\; 3 - 4i,

with magnitude $|3 - 4i| = 5$ and phase $\arctan(-4/3) \approx -0.93\,$ rad. So

x_1(t) + x_2(t) \;=\; 5 \cos(\omega t - 0.93).

Adding two sinusoids became adding two complex numbers. This is a small example; the saving compounds dramatically in problems with many sources, reflections, or modes.

A caveat

The “take the real part at the end” rule works only for linear operations. Sums, differences, scalar multiples, time derivatives, integrals — all fine. Products of fields (intensity, energy, power) are not linear, and you cannot just multiply complex amplitudes and take a real part. The correct rule for the time-average of a product of two real sinusoidal signals is

\big\langle\, \operatorname{Re}[\tilde A e^{i\omega t}]\, \operatorname{Re}[\tilde B e^{i\omega t}]\, \big\rangle \;=\; \tfrac12\, \operatorname{Re}[\tilde A \tilde B^*],

which we will invoke when we compute acoustic intensity in chapter 5.

For everything that is linear, working in complex exponentials is essentially free. The rest of the chapter does so without further comment.