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: Identifying the wave equation.
The general form of a progressive wave equation is:
\[ Y = A \sin(kx - \omega t + \phi), \]
where:
- \( A \) is the amplitude of the wave,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.
Comparing the given wave equation \( Y = 3 \sin(kx - \frac{\pi}{3}) + \frac{1}{2} \), we can identify:
- Amplitude \( A = 3 \, \text{cm} \),
- The phase constant \( \frac{\pi}{3} \),
- The displacement term has no time dependence, so it's a spatial wave.

Step 2: Determining the wavelength.
The wave number \( k \) is related to the wavelength \( \lambda \) by:
\[ k = \frac{2\pi}{\lambda}. \]
However, we don’t have enough information here to directly calculate the wavelength, but we can see that the amplitude is directly given in the wave equation.

Step 3: Determining the frequency.
The frequency \( f \) is related to the angular frequency \( \omega \) by:
\[ f = \frac{\omega}{2\pi}. \]
However, the frequency is not directly given in the problem and we are not required to calculate it.

Step 4: Conclusion.
The correct value for the amplitude of the wave is 3 cm, as directly derived from the wave equation.
Final Answer:
Thus, the correct answer is:
\[ \boxed{\text{Amplitude} = 3 \, \text{cm}}. \]