Question
A wave disturbance in a medium is described by $$y\left( {x,t} \right) = 0.02\cos \left( {50\,\pi t + \frac{\pi }{2}} \right)\cos \left( {10\,\pi x} \right)$$ where $$x$$ and $$y$$ are in metre and $$t$$ is in second
A.
A node occurs at $$x = 0.15\,m$$
B.
An antinode occurs at $$x = 0.3\,m$$
C.
The speed wave is $$5\,m{s^{ - 1}}$$
D.
The wave length is $$0.3\,m$$
Answer :
The speed wave is $$5\,m{s^{ - 1}}$$
Solution :
Comparing it with
$$\eqalign{
& y\left( {x,t} \right) = A\cos \left( {\omega t + \frac{\pi }{2}} \right)\cos k\,x \cr
& {\text{If, }}k\,x = \frac{\pi }{2},\,{\text{a node occurs;}} \cr
& \therefore \,\,{\text{10}}\,\pi x = \frac{\pi }{2} \cr
& \Rightarrow \,\,x = 0.05\,m \cr
& {\text{If }}k\,x = \pi ,\,{\text{an antinode occurs}} \cr
& \Rightarrow \,\,{\text{10}}\,\pi x = \pi \cr
& \Rightarrow \,\,x = 0.1\,m \cr} $$
Also speed of wave
$$\eqalign{
& = \frac{\omega }{k} = \frac{{50\,\pi }}{{10\,\pi }} = 5\,m/s \cr
& {\text{and, }}\lambda = \frac{{2\,\pi }}{k} = \frac{{2\,\pi }}{{10\,\pi }} = 0.2\,m \cr} $$