A progressive wave is given by Y = 6 (3t - 2x) m. On this wave, how...

The given equation of the progressive wave is $Y = 6 \sin (3t - 2x)$.

Comparing this with the standard wave equation $Y = A \sin (\omega t - kx)$, we get the wave number $k = 2$.

The phase difference $\Delta \phi$ between two points separated by a distance $\Delta x$ is given by $\Delta \phi = k \Delta x$.

Given \Delta \phi = 45^{}^{\circ} = \dfrac{\pi}{4} rad.

Substituting the values, we get:

$\dfrac{\pi}{4} = 2 \Delta x$

$\Delta x = \dfrac{\pi}{8}$ m.

Answer: $\dfrac{\pi}{8}\text{m}$