Periodic Motion and Oscillation Variables

What makes motion repeat?

A guitar string, a swing and a loudspeaker diaphragm move back and forth, yet a planet can repeat its orbit without moving back and forth along a line. Repetition is the shared idea. Oscillations matter because vibrating strings and membranes produce sound, pressure changes transmit sound through air, atoms vibrate about equilibrium sites in solids, and an alternating voltage repeatedly changes sign. In the classical description of a solid, increasing temperature increases the average vibrational energy; the relation has limits at low temperatures. The same mathematical language can describe a position, an angle or an electrical quantity.

A periodic motion repeats its entire state of motion after a fixed time interval. A planet on an ideal closed orbit and Earth rotating uniformly are periodic, but are not linear to-and-fro oscillations. An oscillation is a repeated excursion about an equilibrium position. Vibration is another name, often used for faster oscillations; there is no sharp frequency separating the words. Simple harmonic motion is a special sinusoidal oscillation. A repeating motion with a triangular or flat-topped displacement graph can be periodic without being simple harmonic.

Equilibrium and the restoring tendency

An equilibrium position is one where the net force is zero. Left there at rest, an ideal body stays there. For oscillation about a stable equilibrium, a small displacement must produce a tendency to return: a ball near the bottom of a smooth bowl is an example. Inertia carries the moving ball through the bottom, so a restoring force does not make it stop immediately at equilibrium. It overshoots, slows on the other side and returns. A point where force vanishes is not automatically stable; a ball balanced at the top of a hill moves away after a small disturbance.

Exact periodicity is an idealisation. Real friction and resistance dissipate mechanical energy, so a freely oscillating swing usually loses amplitude and is not strictly periodic. A repeated external push can supply the lost energy. These are damped and driven oscillations, respectively; a full theory of damping, driving and resonance is outside this chapter’s supplied scope. Coupled oscillators can transmit disturbances as waves: neighbouring particles of a medium can exchange forces as they oscillate. Electromagnetic waves also have oscillating fields, but do not require a material medium.

Period and frequency are reciprocal quantities

The period T is the smallest positive time after which a nonconstant motion repeats completely. A multiple such as 2T is a repeat interval, but is not the fundamental period. Returning to one position does not establish a complete cycle: a swing passes the centre twice, in opposite directions. The direction of motion and the subsequent history must repeat as well. Frequency f, also written ν, is the number of cycles per second. It can be fractional: one cycle in four seconds means 0.25 cycles each second on average.

Choose time units appropriate to the motion, then convert consistently. A quartz vibration may have a period of microseconds, while Mercury’s orbit takes about 88 Earth days and Halley’s comet returns on a timescale of about 76 years. Those astronomical intervals are approximations to real motions. Counting N complete oscillations and dividing the total time by N reduces the relative effect of a stopwatch reaction delay compared with timing just one.

An insect repeatedly ascending a ramp and returning gives a sloping pattern; a child repeatedly stepping up, pausing and stepping down gives a pattern with flat portions. For a repeatedly bounced ball, flight segments are parabolic because gravitational acceleration is constant during flight. With upward positive, height is h = h₀ + ut − gt²/2; a downward-positive coordinate reverses the gravity sign. Impacts or hand contact change the motion between segments. The graph can repeat without being a sine curve. Its horizontal axis is time, and its vertical axis is the chosen height, not velocity.

Choose the variable and its zero before writing an equation

For a block attached to a spring, choose x = 0 at its equilibrium position and positive x to the right. Then x = −3 cm is a position 3 cm left of equilibrium; it is not a negative distance travelled. For a pendulum, choose angular displacement θ from the downward vertical, with one rotational sense positive. θ is measured in radians in equations. More generally a displacement variable can be the voltage of a capacitor relative to its mean, a pressure variation relative to ambient pressure, or electric and magnetic field components relative to their mean values. Its unit follows the variable: metres, radians, volts, pascals or field units.

Sine and cosine supply the basic repeating patterns

A cosine repeats when its argument advances by 2π radians. For q(t) = C cos(ωt), where C is a nonzero amplitude and ω > 0, the condition q(t + T) = q(t) for every t is met when ωT = 2π. Hence T = 2π/ω. A sine of the same ω has the same period. The trigonometric argument must be dimensionless; ω has unit rad s⁻¹, and ωt is an angle in radians. A numerical argument without a degree sign is understood in radians. For example, sin(15) and sin(15°) are different.

A sine and cosine with the same angular frequency combine into one sinusoid. Let q = a sin(ωt) + b cos(ωt). The addition identity gives D sin(ωt + δ) = D cos δ sin(ωt) + D sin δ cos(ωt). Matching coefficients requires a = D cos δ and b = D sin δ. Squaring and adding uses sin²δ + cos²δ = 1 to obtain D = √(a² + b²). Both coefficient signs are needed to place δ in the correct quadrant; the ratio b/a alone is insufficient. If a = b = 0 the result is constant and phase is undefined.

Different frequencies can combine periodically when they are commensurate: their ratio is rational, so each completes an integer number of cycles in one common interval. Frequencies 2ω and 3ω provide a useful counterexample to an integer-multiple-only rule: their ratio is 3/2, yet the sum repeats after 2π/ω. An irrational ratio such as √2 has no exact common period for two nonzero independent sinusoidal components. Fourier analysis reverses the construction: ordinary well-behaved periodic signals can be built from sine and cosine harmonics with suitable coefficients. This explains why the basic sinusoid is useful even for a waveform that is not itself SHM; a Fourier proof is not needed here.

Read an entire graph, not just a return to zero

In graph (a), a continuously increasing displacement cannot repeat. In (b), a nonsinusoidal pattern repeats after 2 s, so it is periodic but not SHM. In (c), the particle repeatedly returns to zero, but the sizes and timing of the following excursions differ; a repeated position is not a repeated motion. In (d), a smooth sinusoid repeats after 2 s and is compatible with SHM about its central position. Match peaks, troughs, zero crossings and direction together. Curves that only look similar in a short window need not be exactly periodic.

A polyatomic molecule has several possible normal modes, each approximately harmonic for small vibrations. A general motion can contain several frequencies and is therefore not usually one SHM. It is exactly periodic only when the participating frequencies are commensurate. Without that extra condition it is generally quasiperiodic, although each individual normal mode repeats. This distinction separates “built from harmonic motions” from “one simple harmonic motion”.

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