Vardaan Institute - Detailed Solutions: Master Sheet (Waves)

Section A: Wave Equation Solutions

$y(x,t) = 0.005 \cos(80 x - 3 t)$. Find $A, \lambda, f, v$.

Sol: $A = 0.005\text{ m}$. $k = 80 \implies \lambda = \frac{2\pi}{80} = \frac{\pi}{40} \approx 0.0785\text{ m}$. $\omega = 3 \implies f = \frac{3}{2\pi} \approx 0.477\text{ Hz}$. $v = \frac{\omega}{k} = \frac{3}{80} = 0.0375\text{ m/s}$.

Explain Laplace's correction ($v = \sqrt{\frac{\gamma P}{\rho}}$) at STP.

Sol: Laplace recognized sound waves cause rapid, adiabatic pressure changes ($B_{ad} = \gamma P$). $v = \sqrt{\frac{1.4 \times 1.013 \times 10^5}{1.293}} \approx 331.3\text{ m/s}$.

Section B: Organ Pipes & Beats Solutions

Compare organ pipes.

Sol: Closed pipe produces only odd harmonics ($1:3:5:...$) with fundamental $f_1 = \frac{v}{4L}$. Open pipe produces all harmonics ($1:2:3:...$) with fundamental $f_1 = \frac{v}{2L}$.

Two tuning forks A and B produce 4 beats/s. On filing A, beats become 6 beats/s ($f_B = 320\text{ Hz}$).

Sol: Initial $f_A = 320 \pm 4 = 324\text{ Hz}$ or $316\text{ Hz}$. Filing fork A increases its frequency. If $f_A = 324\text{ Hz}$, increasing it increases beat count ($326 - 320 = 6$). Hence $f_A = 324\text{ Hz}$.

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