Progressive simple harmonic wave is moving in air along the x-axis. The part of this wave at a given point $X = X 0$ from the source and at a certain instant $t = t 0$ has the waveform shown below in the displacement-time $(y − t)$ graph and velocity-time $(v − t)$ graph respectively.
Velocity of the wave has value $v 0$ and its angular velocity is $ω (= 2 π / T) .$ Which of the following equations will correctly represent the complete wave at $x_{0}$ agreeing with above wave forms ?

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$y = - a [\cos \{\frac{2 π}{T} (t - t_{0}) - \frac{π}{2}\}]$

$y = - a [\sin \{\frac{2 π}{T} (t - t_{0}) + \frac{π}{2}\}]$

$y = a [\sin \{\frac{2 π}{T} (t - t_{0}) + \frac{π}{2}\}]$

$y = a [\cos \{\frac{2 π}{T} (t - t_{0}) - \frac{π}{2}\}]$

answer is C.

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Detailed Solution

From graph we see that $y = amplitude ' a' at x = x_{0} and t = t_{0}$
$& \frac{dy}{dt} = V = 0 at x = x_{0} and t = t_{0}$
$\frac{dy}{dt} = V_{t} is negative for t < t_{0} and is + ve for t > t_{0}$
Only equation $y = a sin 2 π T t − t 0 + π / 2$ is satisfying the conditions.

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