A transverse wave propagating along $x-$ axis is represented by $y (x,t)= 8.0 sin$ $\left( {0.5\pi x - 4\pi t - \frac{\pi }{4}} \right)$ where $x$ is in metres and $t$ is in seconds. The speed of the wave is ..... $m/s$
AIPMT 2006, Medium
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$y(x, t)=8.0 \sin \left(0.5 \pi x-4 \pi t-\frac{\pi}{4}\right)$
Compare with a standard wave equation,
$y=a \sin \left(\frac{2 \pi x}{\lambda}-\frac{2 \pi t}{T}+\phi\right)$
we get $\frac{2 \pi}{\lambda}=0.5 \pi$ or, $\lambda=\frac{2 \pi}{0.5 \pi}=4 \mathrm{m}$
$\frac{2 \pi}{T}=4 \pi$ or $, T=\frac{2 \pi}{4 \pi}=\frac{1}{2} \sec$
$v=1 / T=2 \mathrm{Hz}$
Wave velocity, $v=\lambda v=4 \times 2=8 \mathrm{m} / \mathrm{sec}$
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