Why oscillations and waves matter

One equation, many disguises. Whenever a small displacement $x$ from stable equilibrium produces a restoring force proportional to $-x$, Newton's second law reduces to the same linear differential equation regardless of what $x$ physically represents:

where: $m$ is the inertia associated with the coordinate $x$ (mass, or an effective inertia for a non-mechanical system), $k$ is the restoring-force constant (stiffness, or an effective stiffness), and $x(t)$ is the displacement from equilibrium as a function of time $t$. A mass on a spring, a pendulum swinging through small angles, a charge oscillating in a molecule, and — once many such oscillators are coupled — a wave equation for a field variable, all reduce to this form or its close cousins with damping and driving terms added.

Why complex exponentials, not just sines and cosines. The standard oscillatory solution is written using Euler's formula,

so that $x(t) = \mathrm{Re}\big(Ae^{i(\omega_0 t + \phi)}\big)$ represents the same physical motion as $x(t) = A\cos(\omega_0 t + \phi)$, but with algebra that is dramatically simpler. where: $A$ is the amplitude, $\omega_0$ the angular frequency, $\phi$ the phase, and $\mathrm{Re}(\cdot)$ extracts the real part, which is the only part with physical meaning here.

Two reasons this toolkit is indispensable. First, linearity: because $m\ddot x + kx = 0$ is linear, superposition holds, and adding oscillations becomes ordinary addition of complex numbers (phasors) instead of repeated use of trigonometric sum-to-product identities. Second, universality: the same differential equation structure reappears for pressure oscillations in acoustics, electric-field oscillations in optics, and probability-amplitude oscillations in quantum mechanics — only the physical quantity that oscillates changes, never the mathematics.

Problem. A quantity obeys $\ddot y + 400\,y = 0$, with $y$ in millimetres and $t$ in seconds. At $t = 0$, $y = 5$ mm and the quantity is momentarily at rest. Find $\omega_0$, the period $T$, and the full solution $y(t)$; then add a second oscillation $x_2(t) = 4\cos(\omega t + \pi/2)$ mm of the same frequency to $x_1(t) = 3\cos(\omega t)$ mm using phasors.

Step 1. Comparing $\ddot y + 400\,y = 0$ with the universal form $\ddot y + \omega_0^2 y = 0$ gives $\omega_0^2 = 400\ \text{s}^{-2}$, so $\omega_0 = \sqrt{400} = 20\ \text{rad/s}$, and the period is $T = 2\pi/\omega_0 = 2\pi/20 \approx 0.314\ \text{s}$.

Step 2. The general solution is $y(t) = A\cos(\omega_0 t + \phi)$. The condition $\dot y(0) = -A\omega_0\sin\phi = 0$ forces $\sin\phi = 0$, and $y(0) = A\cos\phi = 5$ mm with $A\neq 0$ selects $\phi = 0$, giving $A = 5$ mm, so $y(t) = 5\cos(20t)\ \text{mm}$. Nothing here depended on whether $y$ was a spring displacement, a voltage, or a pendulum angle.

Step 3. Now apply the phasor method of the derivation above. Write $z_1 = 3e^{i\omega t}$ and $z_2 = 4e^{i\pi/2}e^{i\omega t} = 4i\,e^{i\omega t}$, since $e^{i\pi/2} = \cos(\pi/2) + i\sin(\pi/2) = i$. The complex amplitude is $\tilde A = 3 + 4i$.

Step 4. The resultant amplitude and phase follow from the magnitude and argument of $\tilde A$:

The full solution is $y(t) = 5\cos(20t)\ \text{mm}$, and the combined oscillation is $x_1(t)+x_2(t) = 5\cos(\omega t + 53.1^\circ)$ mm — the same amplitude value, 5, arising twice from two different phasor calculations.

Competitive stretch

If a third oscillation $x_3(t) = 5\cos(\omega t + 180^\circ)$ mm of the same frequency is added to $x_1$ and $x_2$ above, its phasor is $5e^{i\pi} = -5$, so the new total complex amplitude is $\tilde A' = (3+4i) + (-5) = -2 + 4i$, giving resultant amplitude $|\tilde A'| = \sqrt{4+16} = \sqrt{20} \approx 4.47$ mm — showing that phasor addition scales to any number of same-frequency oscillations by simply summing more complex terms, with no extra trigonometric work at each step.

• Treating complex numbers as decoration for "advanced" physics rather than an algebraic shortcut — the physical displacement is always $\mathrm{Re}(\cdot)$; the imaginary part carries no separate physical meaning in this context.
• Applying phasor addition to oscillations of different angular frequencies; it only works when every term shares the same $\omega$, since only then does $e^{i\omega t}$ factor out cleanly.
• Assuming the universal equation $\ddot x + \omega_0^2 x = 0$ only describes mechanical systems, and being surprised when the identical structure reappears for electric fields or quantum wavefunctions.
• Confusing angular frequency $\omega$ (rad/s) with ordinary frequency $f$ (Hz); always convert with $\omega = 2\pi f$ before comparing the two.

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