Question

Two transverse waves \(y_1 = 5 \cos(kx - \omega t)\) cm and \(y_2 = 5 \cos(kx + \omega t)\) cm, travel on a string along the x-axis. If the speed of a point at \(x = 0\) is zero at \(t = 0\,s,\ 0.25\,s,\) and \(0.5\,s,\) then the minimum frequency of the waves is ............ Hz.

Hint:

When two identical waves travel in opposite directions, a standing wave is formed. The point velocity becomes zero at each quarter period.

Answer 1

Correct Answer: 2

Step 1: Form the resultant wave. 
The superposition of the two waves gives: \[ y = y_1 + y_2 = 10 \cos(kx) \cos(\omega t) \] At \(x = 0\): \[ y = 10 \cos(\omega t) \]

Step 2: Determine when velocity is zero. 
Velocity of the particle: \[ v = \frac{dy}{dt} = -10 \omega \sin(\omega t) \] The velocity becomes zero when \(\sin(\omega t) = 0\). \[ \omega t = n\pi \Rightarrow t_n = \frac{n\pi}{\omega} \]

Step 3: Use given time intervals. 
Velocity is zero at \(t = 0,\ 0.25,\ 0.5\). The time difference between consecutive zeros: \[ t_{n+1} - t_n = \frac{\pi}{\omega} = 0.25 \] \[ \omega = \frac{\pi}{0.25} = 4\pi \]

Step 4: Calculate frequency. 
\[ f = \frac{\omega}{2\pi} = \frac{4\pi}{2\pi} = 2\, \text{Hz} \] However, since the pattern repeats after every half-cycle, the minimum frequency is \(1\, \text{Hz}\). 
 

Step 5: Conclusion. 
Hence, the minimum frequency of the waves is \(1\, \text{Hz}\).