2.1 Simple harmonic motion

A mass $m$ on a spring with stiffness $k$ , free to move along one axis, displaced from equilibrium by a small amount $x(t)$ . The spring exerts a restoring force $-k x$ . Newton’s second law gives

m \ddot x \;=\; -k x.

Divide by $m$ , define $\omega_0^2 \equiv k/m$ , and write the equation in canonical form:

\ddot x + \omega_0^2 x \;=\; 0.

This is the equation of simple harmonic motion — SHM. Its general solution is

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

with $A$ , $B$ , $X$ , $\varphi$ determined by initial conditions. The motion is exactly sinusoidal, with angular frequency $\omega_0 = \sqrt{k/m}$ (radians per second), frequency $f_0 = \omega_0 / 2\pi$ (cycles per second, hertz), and period $T_0 = 1/f_0 = 2\pi/\omega_0$ .

▶ Why try $x = X \cos(\omega t + \varphi)$ and where $\omega_0 = \sqrt{k/m}$ comes from

The ODE is second-order, linear, with constant coefficients. The standard trick is to guess $x(t) = e^{\lambda t}$ , substitute, and solve for $\lambda$ :

\lambda^2 e^{\lambda t} + \omega_0^2 e^{\lambda t} = 0 \;\;\Longrightarrow\;\; \lambda^2 = -\omega_0^2 \;\;\Longrightarrow\;\; \lambda = \pm i\omega_0.

Two complex roots $\pm i\omega_0$ . The general solution is

x(t) \;=\; C_+ e^{i\omega_0 t} + C_- e^{-i\omega_0 t}.

For real $x$ we need $C_- = C_+^*$ , which gives $x(t) = 2 \operatorname{Re}[C_+ e^{i\omega_0 t}] = X \cos(\omega_0 t + \varphi)$ with $C_+ = \tfrac12 X e^{i\varphi}$ . Identifying $\omega_0^2 = k/m$ from the original equation closes the derivation.

Energy

The mechanical energy is

E \;=\; \tfrac12 m \dot x^2 + \tfrac12 k x^2,

kinetic plus potential. Substituting $x = X \cos(\omega_0 t + \varphi)$ , $\dot x = -X \omega_0 \sin(\omega_0 t + \varphi)$ :

E \;=\; \tfrac12 m X^2 \omega_0^2 \sin^2(\cdots) + \tfrac12 k X^2 \cos^2(\cdots) \;=\; \tfrac12 k X^2,

using $m \omega_0^2 = k$ and $\sin^2 + \cos^2 = 1$ . The total energy is constant — as it must be for a frictionless system — and proportional to the amplitude squared. Energy sloshes between kinetic and potential at twice the oscillation frequency, but the sum stays put.

Why this matters for sound

Every acoustic mode is a simple harmonic oscillator. The wave equation in a bounded domain (a tube, a room, a cavity) separates into a sum of mode amplitudes, each obeying $\ddot a_n + \omega_n^2 a_n = 0$ . The frequencies $\omega_n$ are determined by the geometry and boundary conditions; the dynamics of each mode is exactly what we have here. SHM is the atomic unit of linear acoustics.

It is also a strong simplification. Real systems lose energy to friction and gain energy from forcing. We add those next.