Question

The equation of a wave is, $\frac{\pi }{2}(10x-40t)-\frac{\pi }{4}$. Find :
(a) The wavelength, the frequency and the wave velocity
(b) The particle velocity and acceleration at $x=0.5m$ and $t=0.05s$.

Answer 1

a) The equation of wave is $y=f\left(kx-\omega t\right)+c$

where $k=\frac{2\pi }{\lambda },\omega =2\pi f$ and $v=\frac{\omega }{k}$

Comparing with the equation, we get

$\omega =40,k=10$ Hence, $\lambda =\frac{2\pi }{k}=\frac{\pi }{5}$ and $f=\frac{\omega }{2\pi }=\frac{20}{\pi }$

Also $v=\frac{\omega }{k}=\frac{40}{10}=4m/s$

b)The particle's velocity and acceleration are given by

$v=\frac{dy}{dt},a=\frac{dv}{dt}$

Hence, $v=\frac{d}{dt}\left(\frac{\pi }{2}\right(10x-\omega t)-\frac{\pi }{4})=-\omega \frac{\pi }{2}=-20\pi$

And $a=\frac{d}{dt}(-\omega \frac{\pi }{2})=0$