The equation of a progressive wave is $Y=3 \sin \left[\pi\left(\frac{t}{3}-\frac{x}{5}\right)+\frac{\pi}{4}\right]$ where $x$ and $y$ are in meter and time in second. Which of the following is correct?

The wave equation is given as $Y = 3 \sin \left[\pi\left(\frac{t}{3} - \frac{x}{5}\right) + \frac{\pi}{4}\right]$, which simplifies to $Y = 3 \sin \left(\frac{\pi t}{3} - \frac{\pi x}{5} + \frac{\pi}{4}\right)$.

Comparing with the standard form $Y = A \sin(\omega t - kx + \phi)$, the amplitude is $A = 3$ m, angular frequency $\omega = \frac{\pi}{3}$ rad/s, and wave number $k = \frac{\pi}{5}$ rad/m.

The wavelength is given by $\lambda = \frac{2\pi}{k} = \frac{2\pi}{\pi/5} = 10$ m, matching option A.

The wave velocity is $v = \frac{\omega}{k} = \frac{\pi/3}{\pi/5} = \frac{5}{3} \approx 1.67$ m/s, not $1.5$ m/s as in option B.

The frequency is $f = \frac{\omega}{2\pi} = \frac{1}{6} \approx 0.167$ Hz, not $0.2$ Hz as in option D.

The correct choice is A.