Foundations — Beats — derivation, applications

Before you can derive beats, you need to be fluent in the small pile of symbols and pictures the parent note quietly leans on. This page builds each one from absolute zero, in an order where every idea rests on the one before it. Nothing here is assumed — if you can read a clock, you can finish this page.

1. A wave at one point: displacement $y$ and time $t$

Imagine a single speck of air sitting in front of a tuning fork. As the sound passes, that speck doesn't fly away — it just jiggles back and forth around one home spot, like a bead on a wire being nudged left and right.

Why track only $t$ and not position in space? We plant ourselves at ONE fixed point (say, exactly where your ear is) and just watch the speck there wobble over time. That is the whole reason the parent note writes "$y$ as a function of $t$ only" — we froze the location and are filming the motion.

This up-and-down wobble is exactly Simple Harmonic Motion — smooth, repeating, sinusoidal. That link becomes important when the parent page loads a tuning fork with wax: a spring-mass oscillator has $f=\frac{1}{2\pi }\sqrt{k/m}$, so adding mass $m$ lowers $f$. We will not need that formula on this foundations page, but keep in mind that "a fork is an SHM oscillator" is why its pitch can be nudged at all.

2. What "cosine" is really drawing

The parent writes $y=a\cos (\dots )$. To a newcomer $\cos$ looks like a mystery button on a calculator. Here is what it is.

Why cosine and not some other shape? Because a mass wobbling under a spring — and an air speck under a passing pure tone — moves exactly like the shadow of circular motion. Nature hands us cosines for free whenever something oscillates smoothly. We use the tool that matches the motion.

3. The three dials on a cosine: $a$, frequency $f$, and the angle inside

A cosine wave has three things you can adjust. The parent uses all three, so let's name each with its picture.

3a. Amplitude $a$ — how tall

Why the topic needs it: loudness of a sound grows with amplitude. The parent deliberately makes both waves have the same $a$ so that when they cancel, they cancel completely — total silence. Different amplitudes only ever give a partial dip.

3b. Frequency $f$ — how often

Picture: a fast wave is scrunched together (many crests packed into a second); a slow wave is stretched out. Higher $f$ → higher pitch to your ear.

3c. Why $2\pi ft$ appears inside the cosine

Here is the piece that trips everyone up. Why is the parent's argument $2\pi ft$ and not just $ft$?

4. Superposition: waves just add

Two forks play at once. What does our single air speck do — obey fork 1, or fork 2?

Picture: if wave 1 wants to push the speck $+3$ and wave 2 wants $-1$, the speck goes to $+2$. Just add the arrows. That's it — no wave "wins."

Why the topic needs it: this single rule is the entire engine of beats. It is the Superposition Principle, and beats are one of its cleanest consequences — see also Interference of Waves.

5. The trig identity that unlocks the algebra

Adding two cosines looks ugly. There's a well-known identity that rewrites a sum of cosines as a product:

Why does the topic reach for THIS tool and not something else? We want a product, because a product of a slow cosine and a fast cosine literally reads as "a slowly-changing height ($2a\cos$ of the small term) multiplied by a fast wobble." The slow factor becomes the swelling loudness; the fast factor becomes the pitch. The identity is the exact lever that separates "how loud" from "what pitch" — which is precisely the physical split we hear.

• $\frac{C-D}{2}$ carries the difference $f_1-f_2$ → tiny → slow → the envelope.
• $\frac{C+D}{2}$ carries the sum $f_1+f_2$ → the average pitch.

6. Absolute value $|\cdot |$ and why loudness ignores sign

The final formula is $f_{beat}=|f_1-f_2|$, and the "twice" argument uses $|\cos |$. So what do the bars mean?

Why the topic needs it in two places:

1. $|f_1-f_2|$: it doesn't matter which fork is higher; the gap between them is what counts, and a gap is never negative.
2. $|A(t)|$: your ear hears loudness, and a speck slammed hard to the left ($y$ very negative) is just as loud as slammed hard to the right ($y$ very positive). Loudness cares about size, not direction.

How these foundations feed the topic

Read it top to bottom: each box only uses ideas from the boxes above it, and they all funnel into the one boxed result of the parent page.

Equipment checklist

Test yourself — cover the right side. If any answer is fuzzy, re-read that section before tackling the parent derivation.