Sound WavesHardBloom L3
Question
In a resonance tube experiment, an 80 cm air column is in resonance with a turning fork in first overtone. Which equation can represent correct pressure variation in the air column (x = 0 is the top point of the tube, neglect end correction, speed of sound = 320 m/sec)
Options
A.A sin$\frac{15\pi}{8}$ x cos 600πt
B.A cos$\frac{15\pi}{8}$ x sin 600 πt
C.A cos $\frac{15\pi}{8}$x sin 300 πt
D.A sin $\frac{15\pi}{8}$x sin 300 πt
Solution
Sol. For first overtone
L = $\frac{\lambda}{4} + \frac{\lambda}{2}$
⇒ 80 cm = $\frac{3\lambda}{4}$
⇒ λ = $\frac{3.2}{3}m$
⇒ K = $\frac{2\pi}{\lambda}$ = $\frac{2\pi}{3.2} \times 3$ = $\frac{6\pi}{3.2}$ = $\frac{60\pi}{32}$ = $\frac{15\pi}{8}$
f = $\frac{v}{\lambda}$ = $\frac{320}{(3.2/3)}$ = 300 Hz
⇒ ω = 2πf = 600 π
Hence equation
y = A sin Kx cos ωt = $A\sin\left( \frac{15\pi}{8}x \right)\cos(600\pi t)$
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