Question

A wave disturbance in a medium is described by $y(x,t)=0.02cos(50\pi t+\frac{\pi }{2})cos\left(10\pi x\right)$ where x and y are in meter and t is in seconds. Which of the following is correct?

• A. A node occurs at $x=0.15m$
• B. An antinode occurs at $x=0.3m$
• C. The speed wave is $5m{s}^{-1}$
• D. The wavelength is $0.3m$

Answer 1

The correct option is C The speed wave is $5m{s}^{-1}$

Comparing the given wave equation with
$y(x,t)=A\cos (\omega t+\pi /2)\cos \left(x\right)$, we get
$kx=10\pi x$
$\Rightarrow k=10\pi$
Nodes are at $kx=\pi /2,3\pi /2$ etc. (see figure).
Hence, node position, $x=\frac{\pi /2}{k}=\frac{\pi /2}{10\pi }=0.05m$
Antinodes are at $kx=0,\pi ,2\pi$ etc. (see figure).
Hence, anti-node position $x=\frac{\pi }{k}=\frac{\pi }{10\pi }=0.1m$
Again, $k=\frac{2\pi }{\lambda }$
$\lambda =0.2m$

Comparing the wave equations, we get
$\omega =50\pi$

Wave speed $v=\frac{\omega }{k}=\frac{50\pi }{10\pi }=5$ $m/s$