5 · Beats — Theory + 3 Solved PYQs

This page covers superposition of two collinear SHMs. It has three short theory sections, two figures, and three solved PYQs.

1. Same frequency → resultant amplitude and phase

Two vibrations along the same line with the same frequency join into one vibration. Like two people pushing a swing together: the pushes add.

For $y_1 = a_1\sin\omega t$ and $y_2 = a_2\sin(\omega t+\phi)$, the result is $y = A\sin(\omega t+\theta)$ with

Derivation in one line

Expand $y_2 = a_2\sin\omega t\cos\phi + a_2\cos\omega t\sin\phi$. Collect $\sin\omega t$ and $\cos\omega t$ terms, then match with $A\sin(\omega t+\theta) = A\sin\omega t\cos\theta + A\cos\omega t\sin\theta$. Compare coefficients, square and add to eliminate $\theta$: gives $A^2$. Divide to get $\tan\theta$.

Special cases

• $\phi = 0$: $A = a_1 + a_2$ (loudest).
• $\phi = \pi$: $A = |a_1-a_2|$ (quietest).
• $\phi = \pi/2$, equal $a$: $A = a\sqrt{2}$.

2. Different frequencies → beats + mine-gas use

Two nearly equal frequencies go in and out of step. Sound gets loud, then soft, then loud again. Each loud-soft cycle is one beat. Like two fans nearly in sync: the hum swells and fades.

For equal amplitudes $y_1 = a\sin\omega_1 t$, $y_2 = a\sin\omega_2 t$, the sum-to-product identity gives

$$y = 2a\cos!\left(\frac{\Delta\omega,t}{2}\right)\sin(\bar{\omega}t)$$

where $\bar{\omega} = (\omega_1+\omega_2)/2$ is the fast carrier and $\Delta\omega = \omega_1-\omega_2$ is the slow envelope. The amplitude $2a\cos(\Delta\omega t/2)$ swells to $2a$ and sinks to $0$. Intensity $\propto (\text{amplitude})^2$ peaks twice per envelope cycle, so

Mine-gas detector (2022)

A reference tuning fork sounds with a resonator filled with mine air. Clean air gives no beats. Poisonous gas changes the air density, which shifts the resonator frequency by a tiny amount. Even a 1–2 Hz shift gives audible beats. So beats detect gas before instruments show it (the ear hears $|f_1-f_2|$ easily).

Fig 1. Eq. 2 drawn — fast $\sin(\bar{\omega}t)$ inside a slow $\cos(\Delta\omega t/2)$ envelope. Loudness peaks $|f_1-f_2|$ times per second.

3. Sonometer frequency law

A stretched string sings higher when pulled tighter, shorter, thinner, or lighter. One formula joins all four effects.

Ratio form

Drop constants ($\pi/4$ and $1/2$ cancel in ratios): $n \propto \sqrt{T}/(l,d\sqrt{\rho})$. For two strings: $\dfrac{n_1}{n_2} = \dfrac{l_2}{l_1}\sqrt{\dfrac{T_1}{T_2}}\dfrac{d_2}{d_1}\sqrt{\dfrac{\rho_2}{\rho_1}}$.

Fig 2. Sonometer setup — wire length $l$ between bridges, tension $T$ from weights, frequency $n$ matched to a fork.

Solved PYQs

Reference. Subrahmanyam & Brij Lal — Waves & Oscillations Ch-4 (superposition, beats, sonometer).