The equation `Y=0.02 sin (500 pi t) cos (4.5x)` represents

Options

A. progressive wave of frequency 250 Hz along x-axis
B. a stationary wave of wavelength of 1.4 m
C. a transverse progressive wave of amplitude 0.02m
D. progressive wave of speed of about `350 ms^-1`

Explanation

To analyze the given equation \( Y = 0.02 \sin(500 \pi t) \cos(4.5x) \) and determine what it represents, we can follow these steps:

### Step 1: Identify the form of the wave equation
The equation is in the form \( Y = A \sin(\omega t) \cos(kx) \), which is characteristic of a stationary (or standing) wave. Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

### Step 2: Determine the amplitude
From the equation, the coefficient of the sine function gives the amplitude of the wave:
- \( A = 0.02 \)
- The amplitude of the stationary wave is \( \frac{A}{2} = \frac{0.02}{2} = 0.01 \).

### Step 3: Find the angular frequency \( \omega \)
The angular frequency \( \omega \) can be directly read from the equation:
- \( \omega = 500 \pi \)
- To find the frequency \( f \), we use the relation \( \omega = 2\pi f \):
\[ 500 \pi = 2\pi f \implies f = \frac{500 \pi}{2\pi} = 250 \text{ Hz} \]

### Step 4: Determine the wave number \( k \)
The wave number \( k \) is also given in the equation:
- \( k = 4.5 \)
- The relationship between \( k \) and the wavelength \( \lambda \) is given by \( k = \frac{2\pi}{\lambda} \):
\[ 4.5 = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{4.5} \approx 1.4 \text{ m} \]

### Step 5: Conclusion
Since the equation represents a stationary wave, we can conclude that:
- The equation \( Y = 0.02 \sin(500 \pi t) \cos(4.5x) \) represents a stationary wave with:
- Amplitude = 0.01 m
- Frequency = 250 Hz
- Wavelength = 1.4 m

### Final Answer
The equation represents a stationary wave with a wavelength of 1.4 meters.

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