A simple harmonic progressive wave is represented as Y = 0.04 π (2t...

The given equation of the wave is $Y = 0.04 \sin(\pi(2t - 0.02x))$.

The phase of the wave at position $x$ and time $t$ is $\phi = 2\pi t - 0.02\pi x$.

Comparing this with the standard wave equation $Y = A \sin(\omega t - kx)$, the propagation constant is $k = 0.02\pi$.

At a given instant of time, the phase difference $\Delta \phi$ between two particles separated by a distance $\Delta x$ is given by $\Delta \phi = k \Delta x$.

Given the distance between the particles is $\Delta x = 10$ m.

Substituting the values, we get $\Delta \phi = 0.02\pi (10) = 0.2\pi = \dfrac{\pi}{5}$ radian.

Answer: $\dfrac{\pi}{5}$