3.1 Euler’s formula and the phasor

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 first lesson develops Euler’s formula and the phasor representation that the rest of the chapter (and the rest of the Sound book) uses. The next two lessons take the same machinery into damped oscillations and into wave propagation, in turn.

Euler’s formula

The starting point is

\boxed{\;e^{i\theta} \;=\; \cos\theta \;+\; i\sin\theta,\;}

equivalently

\cos\theta \;=\; \tfrac{1}{2}\bigl(e^{i\theta} + e^{-i\theta}\bigr), \qquad \sin\theta \;=\; \tfrac{1}{2i}\bigl(e^{i\theta} - e^{-i\theta}\bigr).

The identity has a one-page derivation from the Taylor series of $e^x$ , $\cos\theta$ , and $\sin\theta$ , but is best memorised: it is the most-used identity in this project after $E = \tfrac12 m v^2$ .

θ = 0.00 rad (0°)

e^iθ = +1.00 + i·+0.00

The interactive is the entire content of Euler’s formula in one picture. As $\theta$ winds around the unit circle, the complex number $e^{i\theta}$ traces it; its real part is $\cos\theta$ , its imaginary part is $\sin\theta$ . The two side panels build up the cosine and sine curves in synchrony with the rotation. Scrub the slider or hit spin — the entire machinery of phasors is in that single rotating arrow.

▶ Euler's formula from Taylor series

The Taylor series of the three functions around zero:

e^x \;=\; 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots

\cos\theta \;=\; 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots

\sin\theta \;=\; \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots

Substitute $x = i\theta$ in the exponential. Use $i^2 = -1$ , so $i^3 = -i$ , $i^4 = 1$ , $i^5 = i$ , $i^6 = -1$ , and the powers cycle with period 4:

e^{i\theta} \;=\; 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \frac{(i\theta)^5}{5!} + \cdots

\;=\; 1 + i\theta - \frac{\theta^2}{2!} - i\,\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\,\frac{\theta^5}{5!} - \cdots.

Group real and imaginary parts:

\;=\; \underbrace{\left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right)}_{\cos\theta} \;+\; i\,\underbrace{\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)}_{\sin\theta}.

Therefore $e^{i\theta} = \cos\theta + i\sin\theta$ . ✓

The derivation is purely formal — it rests on the Taylor series of $e^x$ also converging for $x = i\theta$ , which is a real theorem of complex analysis (the series for $e^x$ has infinite radius of convergence). The identity is the cleanest single instance of why complex numbers are useful: a single analytic function $e^{i\theta}$ packages the two real trigonometric functions into one object.

⏳ The history — Euler 1748, Steinmetz 1893

Leonhard Euler stated the identity $e^{i\theta} = \cos\theta + i\sin\theta$ in his 1748 Introductio in analysin infinitorum. He derived it from the Taylor series, much as above, treating the substitution $x \to i\theta$ in the exponential series as a formal manipulation. At the time the legitimacy of complex numbers was contested — some mathematicians regarded $\sqrt{-1}$ as a meaningless symbol — and Euler’s identity was one of the strongest arguments for taking them seriously. The special case $\theta = \pi$ gives $e^{i\pi} + 1 = 0$ , often cited as the most beautiful equation in mathematics for the way it ties together five fundamental constants.

The use of complex exponentials as phasors for engineering analysis came nearly 150 years later. Charles Proteus Steinmetz, a German-American engineer at General Electric, introduced the phasor method in an 1893 paper to handle AC-circuit analysis. Before Steinmetz, the equations of alternating-current networks were solved by trigonometric identities — slow, error-prone, and unscalable. Steinmetz’s phasor representation collapsed the algebra into single-line formulas, and within a decade AC power systems were the standard. The same trick reaches into acoustics through the wave-equation phasor solutions you’ll meet in Sound Ch 5.

Phasors

A phasor is a complex amplitude that encodes both amplitude and phase. Given a real oscillation

x(t) \;=\; A \cos(\omega t + \varphi),

the corresponding phasor is

\boxed{\;\tilde X \;\equiv\; A\, e^{i\varphi}, \qquad x(t) \;=\; \operatorname{Re}\bigl[\, \tilde X\, e^{i\omega t}\, \bigr].\;}

The real signal is recovered by multiplying the phasor by $e^{i\omega t}$ and taking the real part. The phasor captures the constant information about the oscillation — amplitude and phase — while $e^{i\omega t}$ carries the time evolution. The advantage: time derivatives become multiplication by $i\omega$ , and time integrals become division by $i\omega$ . The equations of motion become algebraic.

|Z₁| = 1.00 arg Z₁ = 0°

|Z₂| = 0.80 arg Z₂ = 90°

Z₁ = +1.00 +0.00i | Z₂ = +0.00 +0.80i | Z₁ + Z₂ = +1.00 +0.80i (magnitude 1.28, phase 39°)

The interactive shows the addition of two phasors $Z_1 + Z_2$ as vector addition in the complex plane. Tip-to-tail: the orange phasor starts at the origin; the blue one is translated so it starts at the orange’s tip; the resultant (black) goes from origin to the final tip. Slide the magnitudes and phases to feel how the sum’s magnitude and phase depend on both inputs — in particular, two phasors of equal magnitude can sum to anything from $2|Z|$ (in phase) to $0$ ( $180°$ out of phase). The latter case is the algebraic signature of destructive interference.

Useful identities

A few that come up constantly:

$|e^{i\theta}| = 1$ for real $\theta$ . The complex exponential of a real argument is a unit-magnitude phasor.
$e^{i\theta} \cdot e^{i\varphi} = e^{i(\theta + \varphi)}$ — products of phasors add their phases. The reason ${(e^{i\theta})}^n = e^{in\theta}$ (de Moivre’s theorem).
$(e^{i\theta})^* = e^{-i\theta}$ — complex conjugation flips the sign of the phase.
$\operatorname{Re}[z] = \tfrac{1}{2}(z + z^*)$ , $\operatorname{Im}[z] = \tfrac{1}{2i}(z - z^*)$ .

A first application: solving the driven oscillator in one line

Consider the damped driven oscillator

m \ddot x + b \dot x + k x \;=\; F_0 \cos(\omega t).

Replace the real driving force by the complex one $F_0 e^{i\omega t}$ (taking real parts at the end). Try $x(t) = \tilde X e^{i\omega t}$ as an ansatz. Substituting and using $\dot x = i\omega \tilde X e^{i\omega t}$ , $\ddot x = -\omega^2 \tilde X e^{i\omega t}$ :

\bigl[-m\omega^2 + i b \omega + k\bigr]\, \tilde X \, e^{i\omega t} \;=\; F_0 \, e^{i\omega t}.

Cancel the exponentials and solve:

\tilde X \;=\; \frac{F_0}{k - m\omega^2 + i b \omega}.

The amplitude is $|\tilde X|$ and the phase relative to the drive is $\arg(\tilde X)$ . We just dispatched a second-order ODE with one line of algebra. The cost was carrying around an $i$ .

The same trick — substitute a complex exponential, do algebra, take real parts — runs through every linear acoustic problem in the bookshelf. The damped oscillator is one example; plane-wave solutions of the wave equation, transfer functions of linear filters, impedance of circuits, are all variations.

Cautions

Two things to watch:

Complex amplitudes can be added (they are linear), but their magnitudes cannot: $|z_1 + z_2| \neq |z_1| + |z_2|$ in general. Use the parallelogram-rule addition for phasors, not the scalar one.
The “take the real part at the end” rule only works for linear operations. For energy and intensity (which are quadratic), you must work with the real signal or apply the right time-average rule: the time-average of $\operatorname{Re}[\tilde A e^{i\omega t}]\,\operatorname{Re}[\tilde B e^{i\omega t}]$ is $\tfrac{1}{2} \operatorname{Re}[\tilde A \tilde B^*]$ — the famous one-half real-part-of-product-with-conjugate formula that runs through acoustic intensity calculations.

What’s next

The next lesson, 3.2 — Damped oscillations as phasors, extends the phasor picture to complex arguments: $e^{(-\gamma + i\omega) t}$ traces a logarithmic spiral inward in the complex plane, with the real part being a damped sinusoid. This is the eigenfunction of every linear damped system on the bookshelf.

The lesson after that, 3.3 — Plane waves and complex impedance, extends the phasor framework to functions of space and time, producing plane waves $e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}$ and the complex-impedance picture that runs through every filter, every Bode plot, every acoustic-loudspeaker analysis.