question_answer

The equation of plane wave is given by as: \[y=2\,\sin \,\pi \left( 200\,t\frac{x}{150} \right)\] When displacement \[y\] is given in cms and time \[t\] in second, then the velocity of the wave is:

A)  \[30000\,\,cm/s\]                          

B)  \[200\,\,cm/s\]

C)  \[150\,\,cm/s\]                               

D)  \[2\,\,cm/s\]

Correct Answer: A

Solution :

Key Idea: compare the given equation with standard one. The standard equation of a plane progressive wave is \[y=a\,\sin \frac{2\,\pi }{\lambda }\left( v\,t-x \right)\]                  ?(1) Where \[\lambda \]is wavelength, \[v\] is velocity, \[t\] is time and a is amplitude. \[y=2\,\sin \,\pi \left( 200\,t-\frac{x}{150} \right)\]                 ?(2) Comparing Eqs. (1) and (2)                                                 \[\frac{2\pi }{\lambda }=\frac{\pi }{150}\] \[\Rightarrow \]                               \[\lambda =300\,\,cm\] And                        \[\frac{2\pi }{T}=200\pi \] \[\Rightarrow \]                               \[T=\frac{1}{100}s\] Frequency \[\left( n \right)\]=\[\frac{1}{T}=\frac{1}{1/100}=100\,Hz\]. Also velocity = frequency\[\times \]wavelength                                                 \[=100\times 300\]                                                 \[=30000\,cm/s.\]