A sine wave travels through a medium. At a specific instant, a...

The equation of a sine wave travelling through a medium can be represented as $y = A \sin(kx - \omega t + \phi)$.

At a specific instant $t$, the displacement of a particle is zero when the phase is an integral multiple of $\pi$.

Let the position of the first particle be $x_1$. For zero displacement, $kx_1 - \omega t + \phi = n\pi$, where $n$ is an integer.

The closest particle that also has zero displacement will be at a position $x_2$ such that its phase differs by $\pi$.

Therefore, $k(x_2 - x_1) = \pi$.

Since the wave number $k = \dfrac{2\pi}{\lambda}$, we get:

$\dfrac{2\pi}{\lambda} \Delta x = \pi$

$\Delta x = \dfrac{\lambda}{2}$

Thus, the distance to the closest particle with zero displacement is $\lambda/2$.