Q.N tuning forks are arranged in order of increasing frequency and any two successive tuning forks give n beats per second when sounded together. If the last fork gives double the frequency of the first (called an octave), show that the frequency of the first tuning fork is $f = (N-1)n$.

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 $\omega_1$ and $\omega_2$ arrive at your ear. Their displacements at a point can be written as:

By the principle of superposition, the resultant displacement is:

Using the sum-to-product identity:

we get:

This is the key result. The resultant wave has two parts:

• A fast oscillation at the average frequency $\frac{\omega_1 + \omega_2}{2}$ (which is nearly the same as the original frequencies). This is what your ear hears as the pitch.
• A slowly varying amplitude given by $2A \cos\left(\frac{\omega_1 - \omega_2}{2} t\right)$. This envelope modulates the loudness.

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 $2\pi$, the time period of one beat is:

And since $f = \omega/2\pi$, the beat frequency in hertz is simply:

What You Actually Hear

Your ear does not follow the rapid $(\omega_1 + \omega_2)/2$ 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. …