Oscillations

A motion is periodic if it repeats itself, identically, after equal intervals of time. That interval is the period $T$. The Earth going around the Sun, a clock’s hour hand, a swinging pendulum, a vibrating guitar string, even the Earth spinning on its own axis, are all periodic in this sense, even though almost nothing else about them is alike.

A narrower and more useful category is oscillatory (or vibratory) motion: periodic motion that is also to-and-fro about a fixed mean (equilibrium) position, a pendulum swinging left and right of vertical, a block on a spring sliding past and back through its rest point, an atom in a solid jiggling about its lattice site. Every oscillation is periodic, but the reverse is not true. A satellite in uniform circular motion repeats its position every orbit, so it is periodic, yet it never reverses direction or retraces its path back through a single equilibrium point, so it is not oscillatory in this specific sense.

The reciprocal of the period is the frequency, $\nu = 1/T$, measured in hertz (Hz), one complete repetition per second. Together, $T$ and $\nu$ say nothing about what is oscillating, position, pressure, voltage, the electric field in a light wave, only how quickly the repetition happens, which is exactly why the same two numbers describe a pendulum and a radio wave equally well.

Among all periodic motions, one family is both the simplest to describe and, remarkably, the one nature keeps reproducing whenever a system is nudged slightly away from a stable equilibrium: simple harmonic motion (SHM). A particle executes SHM if its displacement from the mean position varies with time as

Here $A$, the amplitude, is the greatest displacement on either side of the mean position (always taken positive); the full argument $(\omega t+\varphi)$ is the phase, which fixes exactly where in the cycle the particle is at time $t$; and $\varphi$, the phase constant (or initial phase), is just the phase’s value at $t=0$, fixed by wherever and however the oscillation was started. The constant $\omega$ is the angular frequency, related to the period and frequency by $\omega = 2\pi/T = 2\pi\nu$, since cosine repeats whenever its argument advances by exactly $2\pi$.

Differentiating the displacement once gives the velocity, and again gives the acceleration:

Speed is greatest, $A\omega$, exactly at the mean position, and falls to zero at the two extremes; acceleration behaves exactly oppositely, zero at the mean position and greatest, $A\omega^2$, at the extremes. The relation $a=-\omega^2x$ is worth isolating on its own, because it is really the defining property of SHM, stripped of any reference to time:

acceleration is always directly proportional to the displacement from the mean position, and always directed opposite to it, back toward the mean position. This statement runs both ways: any system whose acceleration obeys $a=-\omega^2x$ for some constant $\omega^2$, whatever the underlying physics, is guaranteed to execute SHM with that $\omega$.

An oscillator is constantly trading kinetic energy for potential energy and back. Using $v(t)$ from above, the kinetic energy is

and the potential energy stored against the restoring force $F=-kx$, taking $U=0$ at the mean position, is the familiar spring potential energy evaluated at the instantaneous displacement,

Add them, and the time-dependence cancels completely, since $\sin^2+\cos^2=1$:

constant, for all time, and fixed entirely by the amplitude and the force constant. In words: $K(t)$ and $U(t)$ each oscillate between $0$ and $\tfrac12kA^2$, but at twice the angular frequency of $x(t)$ itself (because they depend on squares of sine and cosine), and they are exactly out of step with each other: $U$ peaks at the two extremes $x=\pm A$, where the particle is momentarily at rest, and $K$ peaks at the mean position, where speed is greatest. The two curves cross, each equal to half the total energy, at $x=\pm A/\sqrt2$. None of this requires any outside energy input, it is simply energy shuttling between two forms, and it is exactly the conservation law you would expect from a force that does no net work over a full cycle.

SHM is not a special trick a spring performs, it is what any system with a linear restoring force does. Two standard examples from the chapter make the point from opposite directions: one where the force law is exactly linear, and one where it only approximately is.

Every real oscillator loses some energy, to air resistance, internal friction, or some other drag. Model that loss as a damping force proportional to velocity, $F_d=-bv$, acting alongside the restoring force, and the amplitude no longer stays fixed, it decays exponentially with time:

with a slightly lower angular frequency $\omega'=\sqrt{k/m-b^2/4m^2}$ than the undamped value. Larger $b$ means faster decay; the mechanical energy, tracking the square of the amplitude, falls off as $E(t)=\tfrac12kA^2e^{-bt/m}$. A swing left alone, or a plucked guitar string fading out, are both damped oscillators.

Push a damped oscillator with an external periodic force at some driving frequency $\omega_d$ and, once the initial transient dies away, it settles into steady oscillation at $\omega_d$ itself, not at its own natural frequency $\omega_0=\sqrt{k/m}$, with an amplitude that depends on how close $\omega_d$ is to $\omega_0$ and how strong the damping is. As $\omega_d$ approaches $\omega_0$, the driving force pumps energy in step with the system’s own rhythm on every cycle, and the amplitude grows sharply, a phenomenon called resonance, limited in practice only by however much damping is present. Soldiers breaking step while crossing a bridge, rather than marching in time, exist as a rule precisely because a matched rhythm can pump a bridge’s own oscillation to dangerous amplitude; a sustained note at just the right pitch can do the same to a wine glass, shattering it, and a child’s swing climbs highest when each push lands in step with the swing’s own natural frequency rather than at some arbitrary rhythm.

Two of the chapter’s own exercise questions, with original worked solutions.