Recipe

Write the beat solution for an undamped near-resonant drive

Use when: you have an undamped forced equation $x''+\omega_0^2x=F_0\cos(\omega t)$ with rest initial conditions and a driving frequency $\omega$ close to, but not equal to, the natural frequency $\omega_0$

Goal: the full solution $x(t)$ written in envelope form, with the slow envelope frequency and fast carrier frequency identified separately

Taught in Beats

Quick recipe

Confirm the beat conditions apply
Compute the coefficient $2F_0/(\omega_0^2-\omega^2)$
Compute the envelope (slow) frequency $(\omega_0-\omega)/2$
Compute the carrier (fast) frequency $(\omega_0+\omega)/2$
Assemble $x(t)$

1
Check that the system is undamped ( $c=0$ , or the equation is already given as $x''+\omega_0^2x=F_0\cos(\omega t)$ ) and that $\omega\ne\omega_0$ .
- $\omega=\omega_0$ exactly — this is resonance, not beats; the resonance modification rule applies instead.
- $\omega\ne\omega_0$ , however close — beats occur no matter how small the gap; continue to Step 2.
Why: the non-resonant undetermined-coefficients solution this recipe rewrites is exactly what breaks down at $\omega=\omega_0$ , since its denominator $\omega_0^2-\omega^2$ hits zero there.
2
This constant multiplies both sine factors in the final envelope form.

Example

For $\omega_0=8$ , $\omega=7$ , $F_0=3$ : $\omega_0^2-\omega^2=64-49=15$ , so the coefficient is $2(3)/15=6/15=0.4$ .

Why: this carries over, unchanged, from the non-resonant particular solution the sum-to-product identity started from.
3
Subtract and halve the two frequencies directly.

Example

$(8-7)/2=0.5$ .

Why: this is the frequency of the slowly swelling-and-fading sine factor, small exactly because $\omega$ sits close to $\omega_0$ .
4
Add and halve the two frequencies directly.

Example

$(8+7)/2=7.5$ .

Why: this is the frequency of the fast oscillation the envelope modulates, close to both $\omega$ and $\omega_0$ since they're close to each other.
5
Combine the coefficient from Step 2 and both frequencies from Steps 3-4 into $x(t)=\big(\text{coefficient}\big)\sin\big(\text{envelope freq}\cdot t\big)\sin\big(\text{carrier freq}\cdot t\big)$ .

Example

$x(t)=0.4\sin(0.5t)\sin(7.5t)$

Why: this is the same solution rewritten in the form that makes the beat pattern visible (a slow envelope multiplying a fast carrier) instead of the difference-of-cosines form it started as.

Common mistakes

Swapping which of the two frequencies is the envelope: the smaller of $(\omega_0-\omega)/2$ and $(\omega_0+\omega)/2$ is always the slow envelope.
Dropping the factor of $2$ in $2F_0/(\omega_0^2-\omega^2)$ , a leftover of the sum-to-product identity that's easy to forget when converting from the cosine-difference form.
Applying this recipe when $\omega=\omega_0$ exactly, dividing by zero instead of recognizing that resonance, not beats, applies there.
Forgetting the rest-initial-conditions assumption. A nonzero initial displacement or velocity changes the solution, and this envelope form no longer applies as written.