2.2 Beats and Resonance

Attach the mass to a shaker and drive it. With no damping, and with the spring tuned so the system's natural frequency is $2$ , the equation is

$x^{″} + 4 x = \cos (a t),$

where $a$ is the frequency you choose to push at. The method of undetermined coefficients says: guess a response of the same shape as the input, $x_{p} = A \cos (a t)$ . Substituting gives $- a^{2} A \cos (a t) + 4 A \cos (a t) = \cos (a t)$ , so $A = 1 / (4 - a^{2})$ — provided $a not equal to 2$ . Adding the homogeneous solution and imposing rest initial conditions $x (0) = x^{'} (0) = 0$ produces

$x (t) = \frac{\cos (a t) - \cos (2 t)}{4 - a^{2}} .$

The denominator is the whole story in miniature. Push at a frequency far from $2$ and the response is small; push near $2$ and $4 minus a squared$ is tiny, so the response is enormous. But the shape is stranger than "big" — as the figure shows, the motion swells and collapses in slow packets.

A red wave along the time axis whose amplitude swells and shrinks in slow packets: fast oscillations inside a slowly waving envelope, the classic picture of beats. Dragging the slider a toward the natural frequency 2 stretches the packets longer and taller, so near a = 1.95 a single enormous swell fills the window - the approach to resonance. — The rest-start response of $x^{″} + 4 x = \cos (a t)$ , an undamped oscillator with natural frequency $2$ . Dragging the forcing frequency $a$ toward $2$ lengthens and heightens the beat packets — the run-up to resonance.

Explore

Observe. Set $a equals 1$ , well away from resonance. The motion is bounded and busy. Estimate the largest height it ever reaches, and compare it to the coefficient $1 / (4 - a^{2}) = 1 / 3$ .
Predict. Move $a$ to $1.6$ . Will the packets get longer, shorter, taller, or shorter and taller? Write your guess down first.
Verify. Drag to $a equals 1.6$ , then $a equals 1.8$ , then $a equals 1.9$ . Record roughly how many packets fit in the window at each setting. The packet count is falling — toward what, as $a → 2$ ?
Observe. At $a equals 1.95$ a single swell nearly fills the window. The amplitude scale has changed dramatically. Compute $1 / open parenthesis 4 minus a squared close parenthesis$ at this setting and see whether it matches the height you observe.
Predict, then reason. What does the formula do at exactly $a equals 2$ ? The slider stops just short — explain from the algebra why the figure cannot draw the resonant case, and what the true resonant solution must look like instead.

Connect

The product-to-sum identity turns the response into

$x (t) = \frac{2}{4 - a^{2}} \sin (\frac{2 + a}{2} t) \sin (\frac{2 - a}{2} t),$

and now the two time scales are visible separately. The first sine oscillates at roughly the average frequency — that is the fast wiggle you see. The second oscillates at half the difference $2 minus a$ , which is small when $a$ is near $2$ , and it acts as a slowly varying amplitude: the envelope. This is the phenomenon of beats, the same effect a piano tuner hears as a slow throb when a string is close to but not on pitch. As $a → 2$ the envelope period $4 π / open parenthesis 2 minus a close parenthesis$ grows without bound and the coefficient $2 / open parenthesis 4 minus a squared close parenthesis$ grows with it.

At exactly $a equals 2$ the guess $A cos 2 t$ fails, because $cos 2 t$ already solves the homogeneous equation — the standard repair is to multiply by $t$ , and the true resonant solution is $x (t) = \frac{1}{4} t \sin 2 t$ . The envelope is no longer a slow sine but the straight line $± t / 4$ : amplitude that grows forever. Real systems have damping, which caps the growth at a large but finite value, and real structures are designed to keep their natural frequencies away from the frequencies they will be driven at.

An original work of XYZ Homework, built around interactive XYZ 3D figures, following the standard first-course sequence (first-order equations through systems and the phase plane). Distinct from the LibreTexts-sourced Differential Equations for Engineers (Lebl) edition, which has its own attribution. License: CC-BY-NC-SA-4.0.