Question

Frequency of the sinusoidal wave, $ y=0.40\cos \left( 2000t+0.080 \right) $ would be:-
(A) $ 100Hz $
(B) $ 200Hz $
(C) $ 20Hz $
(D) $ 318Hz $

Answer 1

Hint:
We first use the comparison of the general form of sinusoidal wave equation with the wave equation given in the question and find the angular velocity from there. Then using the angular velocity we can calculate the frequency of the sinusoidal wave easily.
Formula Used: The relation between angular velocity and frequency is given as,
 $ \omega =2\pi \nu $
Here $ \omega $ is the angular velocity and $ \nu $ is the frequency of the given wave.

Complete step by step answer:
In this problem, the wave equation given to us is sinusoidal wave equation, whose general form is given as,
 $ y=A\cos \left( wt+kx \right) $
Here $ y $ is the wave displacement, $ A $ is the amplitude of the wave $ w $ is the angular velocity of the wave and $ k $ is the wave vector; $ t $ and $ x $ are simply the time and displacement variables respectively.
Thus from the wave equation, we can easily infer angular velocity to be,
 $ \omega =2000 $
Thus, we get the frequency as,
 $ \omega =2\pi \nu $
 $ \Rightarrow \nu =\dfrac{\omega }{2\pi } \\ $
Now putting in the values we get the frequency to be,
 $ \nu =\dfrac{2000}{2\pi }=\dfrac{1000}{\pi } $
$ \Rightarrow \nu \approx 318Hz \\ $
$ \therefore $ Option (D) is the correct option out of the given options.

Additional Information
The ‘k’ in the wave equation refers to the wave vector and is given as the ratio of velocity of the wave to its wavelength.

Note:
In such a question we should be very careful with the terms used. Here frequency is given as inverse of the time period of the wave whereas the variable t in the equation is actually a time variable. It should not be confused with the Time Period of the wave. When variable t equals the time period, it is after this value that the wave repeats itself or becomes periodic.