A wave is propagating along the positive direction of X-axis. If the amplitude of wave is 10 cm, frequency 200 Hz and velocity 400 m/s, then write the equation of progressive wave.

Answer

To find the equation of a progressive wave propagating in the positive direction of the X-axis, we can follow these steps:

### Step 1: Identify the given parameters
- Amplitude (A) = 10 cm = 0.1 m (convert to meters)
- Frequency (f) = 200 Hz
- Velocity (v) = 400 m/s

### Step 2: Write the general equation of a progressive wave
The general equation for a wave traveling in the positive direction of the X-axis is given by:
\[ y(x, t) = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 3: Calculate the angular frequency (\( \omega \))
The angular frequency is related to the frequency by the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 200 = 400\pi \, \text{rad/s} \]

### Step 4: Calculate the wave number (\( k \))
The wave number is related to the wavelength (\( \lambda \)) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
To find \( \lambda \), we can use the relationship between velocity, frequency, and wavelength:
\[ v = f \lambda \]
Substituting the known values:
\[ 400 = 200 \lambda \]
Solving for \( \lambda \):
\[ \lambda = \frac{400}{200} = 2 \, \text{m} \]
Now, substituting \( \lambda \) into the wave number formula:
\[ k = \frac{2\pi}{2} = \pi \, \text{rad/m} \]

### Step 5: Substitute the values into the wave equation
Now, we can substitute \( A \), \( \omega \), and \( k \) into the wave equation:
\[ y(x, t) = 0.1 \sin(400\pi t - \pi x) \]

### Final Equation
Thus, the equation of the progressive wave is:
\[ y(x, t) = 0.1 \sin(400\pi t - \pi x) \]

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