Oscillations and waves

Introduction to simple harmonic motion

In simple harmonic motion, displacement varies sinusoidally with time; velocity and acceleration are set by its phase, angular frequency, and amplitude.

Oscillation is repeated motion around an equilibrium position. When the restoring force is proportional to displacement and points toward equilibrium, the motion is simple harmonic: its displacement–time graph is sinusoidal.

Velocity and acceleration

Differentiating gives $v=-A\omega\sin(\omega t+\varphi)$ and $a=-A\omega^2\cos(\omega t+\varphi)$. Thus $v_{\max}=A\omega$ and $a_{\max}=A\omega^2$; acceleration always has the opposite sign to displacement.

Quick check

Phase specifies the instantaneous state: at two times with equal phase, the oscillator has the same displacement and velocity direction. One period corresponds to a phase increase of $2\pi$; over $\Delta t$, the phase changes by $\omega\Delta t$. Choosing $t=0$ at the positive endpoint gives $\varphi=0$, while a passage through equilibrium requires an initial phase consistent with the direction of motion.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.