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Q.
In a sine wave, position of different particles at time $t=0$ is shown in figure. The equation for this wave travelling along positive $x$ -axis can be
Waves
Solution:
As, it is clear from figure, at $t=0, x=0$, displacement $y=0$.
Therefore, options (a) or (d) may be correct. In case of (d);
$y=A \sin (k x-\omega t)$
$\frac{d y}{d t}=A \cos (k x-\omega t)[-\omega]$
(after differentiating w.r.t. $t$ )
and $\frac{d y}{d x}=A \cos (k x-\omega t)[k]$
(after differentiating w.r.t. $x$ )
$\therefore \frac{\frac{d y}{d t}}{d y / d x} =\frac{-\omega A \cos (k x-\omega t)}{k A \cos (k x-\omega t)}=-\frac{\omega}{k}=-v$
$\therefore \frac{d y}{d t} =-v\left(\frac{d y}{d x}\right)$
i.e., particle velocity $=-$ (wave speed) $\times$ slope
And slope at $x=0$ and $t=0$ is positive, in figure.
Therefore, particle velocity is in negative $y$ - direction.
