Physics - Wave on String Question with Solution | TestHub

The given equation of the progressive wave is $Y=3\sin \left[\pi \left(\frac{t}{3}-\frac{x}{5}\right)+\frac{\pi }{4}\right]$.

Rewriting the equation by multiplying $\pi$ inside the bracket, we get:
$Y=3\sin \left(\frac{\pi t}{3}-\frac{\pi x}{5}+\frac{\pi }{4}\right)$

Comparing this with the standard equation of a progressive wave $Y=A\sin (\omega t-kx+\varphi )$, we get:
Amplitude $A=3$ m (since $x$ and $y$ are in meters)
Angular frequency $\omega =\frac{\pi }{3}$ rad/s
Wave number $k=\frac{\pi }{5}$ rad/m

Now, calculating the wavelength $\lambda$:
$k=\frac{2\pi }{\lambda }$
$\frac{\pi }{5}=\frac{2\pi }{\lambda }$
$\lambda =10$ m

Calculating the wave velocity $v$:
$v=\frac{\omega }{k}=\frac{\pi /3}{\pi /5}=\frac{5}{3}\approx 1.67$ m/s

Calculating the frequency $f$:
$f=\frac{\omega }{2\pi }=\frac{\pi /3}{2\pi }=\frac{1}{6}\approx 0.167$ Hz

Thus, the only correct statement is that the wavelength is $10$ m.

Answer: Wavelength $=10\mathrm{m}$