A wave disturbance in a medium is described by $y(x,t)=0.02\cos\left(50\pi t+\frac{\pi}{2}\right)\cos(10\pi…
A wave disturbance in a medium is described by $y(x,t)=0.02\cos\left(50\pi t+\frac{\pi}{2}\right)\cos(10\pi x)$, where $x$ and $y$ are in metre and $t$ in second. Choose the correct statement.
(a) A node occurs at x = 0.15 m
(b) An antinode occurs at x = 0.3 m
(c) The wave velocity is 5 ms −1
(d) All of the above
Solution
Given $y=0.02\cos(10\pi x)\cos\big(50\pi t+\dfrac{\pi}{2}\big)$.
At a node amplitude $=0$ so $\cos(10\pi x)=0\Rightarrow 10\pi x=\dfrac{\pi}{2},\dfrac{3\pi}{2}\Rightarrow x=\dfrac{1}{20}=0.05\ \mathrm{m},\ \dfrac{3}{20}=0.15\ \mathrm{m}$.
At an antinode $\cos(10\pi x)=\pm1\Rightarrow x=0,0.1,0.2,0.3\ldots\ \mathrm{m}$.
Wave speed $v=\dfrac{\omega}{k}=\dfrac{50\pi}{10\pi}=5\ \mathrm{m\,s^{-1}}$.
(These are the node and antinode positions and the wave speed.)