CLASS 11 · CHAPTER 14 · WAVES
Waves
Two waves can add up, cancel out, or do something in between depending on how their timing lines up. Beats are what that timing looks like when you can't quite decide which.
Watch it happen
Set f₁ and f₂ close together and watch the sum trace swell and fade, that pulsing envelope is a beat, repeating 0.8 times a second.
Set f₁ and f₂ far apart and the sum just looks messy. Bring them close together and a slow, unmistakable pulse appears, that’s two nearly-matched waves drifting in and out of phase with each other.
Where the formula comes from
A wave’s speed on a stretched string has to be built from tension and linear mass density . Checking dimensions first:
Dividing gives exactly the dimensions of velocity squared, which is why the full mechanics derivation lands on:
For two waves of nearly equal frequency at the same point in space, superposition just adds them. The sum-to-product identity turns that sum into a product:
The factor oscillates at the average frequency, that’s the pitch you hear. The factor is the envelope riding on top of it.
Where the shortcut stops working
Look at that envelope term again: . Read off its frequency naively and you’d say the beats happen times a second, half the textbook answer. This is one of the most common errors in this chapter, and it’s a subtle one.
Loudness doesn’t care about the sign of the envelope, only its size, . A plain cosine hits its maximum magnitude twice in every cycle, once at and once at , and both are equally loud. So the ear perceives a loud moment twice as often as the cosine’s own mathematical period suggests, which exactly doubles the frequency back to the familiar result:
Apply it under exam conditions
Q1. Tuning fork A (256 Hz) and fork B produce 4 beats per second. Loading fork B with a touch of wax, which only ever lowers a fork’s frequency, brings the beat rate down to 2 per second. Find B’s original frequency.
B is either 252 Hz or 260 Hz. Wax can only lower B’s frequency: if B started at 252 Hz, lowering it would pull it further from 256 Hz, increasing the beats, not decreasing them. Only starting at 260 Hz is consistent, lowering it toward 256 Hz shrinks the gap, and the beats drop to 2 per second exactly as observed.
Q2. A string has linear mass density 0.05 kg/m under 20 N of tension. Find the wave speed, and the wavelength of a 100 Hz wave travelling on it.
Quick answers
What exactly is a beat?+
It's the slow swelling and fading in loudness you hear when two tones of nearly equal frequency play together, caused by their waves drifting in and out of step with each other.
Why does the beat frequency equal f₁ − f₂, not half of it?+
The mathematical envelope term does oscillate at (f₁−f₂)/2, but loudness depends on the size of that envelope regardless of sign, and |cos| hits its peak twice as often as cos itself. That doubling is exactly what recovers the familiar formula.
Is the speed of a wave on a string the same as the speed of the vibrating particles?+
No, and mixing these up is a common mistake. The wave pattern itself travels at v = √(T/μ) along the string, while each individual bit of string only moves up and down, its own speed depends on amplitude and frequency, not on tension or mass density directly.
Do you need two different instruments to hear beats?+
No, just two sources close enough in frequency, two tuning forks, two guitar strings, even the same note played slightly out of tune on two instruments.
How are beats actually used in practice?+
Tuning instruments, mostly: musicians and technicians listen for the beat frequency to fall to zero, which means the two frequencies have converged exactly.
Related concepts
Physics doesn’t stay inside chapter boundaries. Neither should you.