A plane progressive wave is given by `y=0.3 sin ((220)/(7) t -25.12x)`
Find the wavelength and the phase difference between two points at r= 0.3 m and r=0.425 m. Also find the maximum particle velocity.

Answer

To solve the problem step by step, we will analyze the given wave equation and extract the necessary information.

### Given Wave Equation:
\[ y = 0.3 \sin\left(\frac{220}{7} t - 25.12 x\right) \]

### Step 1: Find the Wavelength
The general form of a plane progressive wave is:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

From the given equation, we can identify:
- \( k = 25.12 \)

The wavelength \( \lambda \) is related to the wave number \( k \) by the formula:
\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{25.12} \]

Calculating:
\[ \lambda \approx \frac{6.2832}{25.12} \approx 0.25 \, \text{m} \]

### Step 2: Find the Phase Difference
To find the phase difference between two points at \( r_1 = 0.3 \, \text{m} \) and \( r_2 = 0.425 \, \text{m} \), we use the formula:
\[ \Delta \phi = -k \Delta r \]
where:
\[ \Delta r = r_2 - r_1 = 0.425 - 0.3 = 0.125 \, \text{m} \]

Substituting the values:
\[ \Delta \phi = -25.12 \times 0.125 \]

Calculating:
\[ \Delta \phi = -3.14 \]

### Step 3: Find the Maximum Particle Velocity
The maximum particle velocity \( V_{\text{max}} \) can be found by differentiating the wave equation with respect to time \( t \):
\[ V = \frac{dy}{dt} = \omega A \cos(\omega t - kx) \]

From the wave equation, we have:
- \( \omega = \frac{220}{7} \)
- \( A = 0.3 \)

Thus,
\[ V_{\text{max}} = \omega A = \left(\frac{220}{7}\right) \times 0.3 \]

Calculating:
\[ V_{\text{max}} \approx \frac{220 \times 0.3}{7} \approx \frac{66}{7} \approx 9.428 \, \text{m/s} \]

### Final Answers:
- Wavelength \( \lambda \approx 0.25 \, \text{m} \)
- Phase difference \( \Delta \phi \approx -3.14 \)
- Maximum particle velocity \( V_{\text{max}} \approx 9.428 \, \text{m/s} \)

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