Question

A wave disturbance in a medium is described by y(x , t)= 0.02 cos (50 $\pi$ t + $\frac{\pi }{2}$) cos (10 $\pi$x) where x and y are in metre and t in second

• A. A displacement node occurs at x = 0.5 m
• B. An antinode occurs at x = 0.3 m
• C. The wavelength of the wave is 0.1m
• D. The speed of the wave is 50.0 m/s

Answer 1

The correct option is B

An antinode occurs at x = 0.3 m

y(x , t)= 0.02 cos (10 $\pi$ x) cos (50 $\pi$ t + $\frac{\pi }{2}$)

At node , amplitude = 0

$\Rightarrow$ cos(10 $\pi$x) = 0 $\Rightarrow$ 10 $\pi$x = $\frac{\pi }{2}$ , $\frac{3\pi }{2}$

$\Rightarrow$ x = $\frac{1}{20}$ = 0.05 m , 0.15 m ..

At antinode , amplitude is maximum

$\Rightarrow$ cos (10 $\pi$x) = $\pm$ 1 $\Rightarrow$ x = 0 , $\pi$, 2$\pi$

$\Rightarrow$ x = 0 , 0.1m, 0.2 m ..

Now $\lambda$ = 2 $\times$ Distance between two nodes or antinodes

= 2 $\times$ 0.1= 0.2 m and $\frac{2\pi vt}{\lambda }$ = 50 $\pi$ t

v = 25 $\lambda$ = 25 $\times 0.2=5m{s}^{-1}$