The Intuition: When Two Almost-Identical Notes Clash
Imagine you're tuning a guitar. You pluck the string you want to tune, and simultaneously strike the reference note from a tuning fork. If the two notes are exactly the same pitch, you hear a single, steady tone. But if they are almost the same — say one is 440 Hz and the other is 442 Hz — something strange happens. The sound doesn't stay steady. Instead, it swells in loudness, fades away, swells again, fades again, in a slow, throbbing rhythm. That rhythmic pulsing is what we call beats.
Why does this happen? Because the two waves, travelling together, sometimes add up constructively (making a loud sound) and sometimes cancel each other out (making a near-silence). They are constantly shifting in and out of phase.
The Precise Physics: Superposition of Two Slightly Different Frequencies
Let two waves of equal amplitude A but slightly different angular frequencies ω1 and ω2 arrive at your ear. Their displacements at a point can be written as:
y1=Asin(ω1t)
y2=Asin(ω2t)
By the principle of superposition, the resultant displacement is:
y=y1+y2=A[sin(ω1t)+sin(ω2t)]
Using the sum-to-product identity:
sinP+sinQ=2sin(2P+Q)cos(2P−Q)
we get:
y=2Acos(2ω1−ω2t)sin(2ω1+ω2t)
This is the key result. The resultant wave has two parts:
- A fast oscillation at the average frequency 2ω1+ω2 (which is nearly the same as the original frequencies). This is what your ear hears as the pitch.
- A slowly varying amplitude given by 2Acos(2ω1−ω2t). This envelope modulates the loudness.
fbeat=∣f1−f2∣
The beat frequency is the absolute difference of the two original frequencies. Your ear perceives one loud-soft cycle for every complete cycle of the cosine envelope. Since the cosine goes through a full cycle when its argument changes by 2π, the time period of one beat is:
Tbeat=∣ω1−ω2∣/22π=∣ω1−ω2∣2π
And since f=ω/2π, the beat frequency in hertz is simply:
fbeat=∣f1−f2∣
What You Actually Hear
Your ear does not follow the rapid (ω1+ω2)/2 oscillations individually — that's just the pitch you perceive. What you notice is the envelope: the amplitude rises and falls at the beat frequency. So with 440 Hz and 442 Hz, you hear a note of roughly 441 Hz that grows louder and softer 2 times every second. …