Question

The equation of a simple harmonic wave is given by $y=3 \sin \frac{\pi}{2}(50t-x),$ where $x$ and $y$ are in metres and $t$ is in seconds. The ratio of maximum particle velocity to the wave velocity is

• A. $2\pi$
• B. $\frac{3}{2}\pi$
• C. $3\pi$
• D. $\frac{2}{3}\pi$

Answer 1

Answer: (B)

$\frac{3}{2}\pi$

Answer 2

The given wave equation is $y=3 \sin \frac{\pi}{2}(50t -x)$ y=3\sin\Big(25\pi t-\frac{\pi}{2}x\Big)\hspace25mm ...(i) The standard wave equation is y = A \sin(\omega t - kx)\hspace25mm ...(ii) Comparing (i) and (ii), we get $\omega=25\pi, k=\frac{\pi}{2}$ Wave velocity, $\upsilon=\frac{\omega}{k}=\frac{25\pi}{(\pi/2)}=50 \,ms^{-1}$ Particle velocity, $\upsilon_p=\frac{gy}{dt}\Bigg(3 \sin\Bigg((25\pi t-\frac{\pi}{2}\Bigg)\Bigg)$ $=75\pi \cos \Bigg(25\pi t-\frac{\pi}{2}\Bigg)$ Maximum particle velocity,$(\upsilon_p)_{max}=75 \pi \,ms^{-1}$ $\therefore \frac{(\upsilon_p)_{max}}{\upsilon}=\frac{75 \pi}{50}=\frac{3}{2}\pi$