Question

For the given wave equation \(y=10\,\cos \pi \left(\frac{x}{4}+\frac{t}{2}\right)\), the CORRECT statement is ( where x is in metre and time t is in second.)

• A. frequency \(=0.25\ \text{Hz}\)
• B. Wavelength \(=10\ \text{m}\)
• C. velocity \(=20\ \text{cm/s}\)
• D. angular velocity \(=2\pi\)

Hint:

Expand the phase to read off the angular frequency and wave number, then find frequency, wavelength and speed.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Compare with \(y=A\cos(kx+\omega t)\). Expanding, \(y=10\cos\left(\dfrac{\pi x}{4}+\dfrac{\pi t}{2}\right)\).

Step 2: Read the constants
\(A=10\ \text{m}\), \(k=\dfrac{\pi}{4}\ \text{rad/m}\), \(\omega=\dfrac{\pi}{2}\ \text{rad/s}\).

Step 3: Derived quantities
\(f=\dfrac{\omega}{2\pi}=0.25\ \text{Hz}\), \(\lambda=\dfrac{2\pi}{k}=8\ \text{m}\), \(v=\dfrac{\omega}{k}=2\ \text{m/s}\).

Step 4: Check the options
Only the frequency of \(0.25\ \text{Hz}\) matches. The wavelength is 8 m, not 10 m, the speed is 2 m/s, not 20 cm/s, and the angular velocity is \(\dfrac{\pi}{2}\), not \(2\pi\).

Final Answer:
The correct statement is that the frequency is 0.25 Hz. \[ \boxed{\text{(A) }\text{frequency } = 0.25\ \text{Hz}} \]