Question

The equation of a progressive wave is \( Y = 3 \sin\left(kx - \frac{\pi}{3}\right) + \frac{1}{2} \), where \(x\) and \(y\) are in meter and time is in second. Which of the following is correct? 

• A. Wavelength = 10 m
• B. Velocity = 1.5 m/s
• C. Amplitude = 3 cm
• D. Frequency = 0.2 Hz

Hint:

For a progressive wave, the amplitude is the coefficient of the sine term. It represents the maximum displacement from the equilibrium position.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to compare the given wave equation with the standard progressive wave form \( Y = A \sin(\omega t - kx + \phi) \) to find its parameters.
Step 2: Key Formula or Approach:
\[ Y = 3 \sin \left[ \frac{\pi t}{3} - \frac{\pi x}{5} + \frac{\pi}{4} \right] \]
1. \( A = 3\text{ m} \).
2. \( \omega = \frac{\pi}{3} = 2\pi f \).
3. \( k = \frac{\pi}{5} = \frac{2\pi}{\lambda} \).
Step 3: Detailed Explanation:
Checking wavelength:
\[ \frac{\pi}{5} = \frac{2\pi}{\lambda} \implies \lambda = 10\text{ m} \]
Statement (A) is correct.
Checking velocity:
\[ v = \frac{\omega}{k} = \frac{\pi / 3}{\pi / 5} = \frac{5}{3} \approx 1.67\text{ m/s} \neq 1.5\text{ m/s} \].
Checking frequency:
\[ \frac{\pi}{3} = 2\pi f \implies f = \frac{1}{6} \approx 0.166\text{ Hz} \neq 0.2\text{ Hz} \].
Checking amplitude:
\( A = 3\text{ m} = 300\text{ cm} \neq 3\text{ cm} \).
Step 4: Final Answer:
The correct statement is Wavelength \( = 10\text{ m} \).