The equation of progressive wave is `y=0.01sin(100t-x)` where `x,y` are in meter and t in second, then
`(a)` Velocity of wave is `50m//s`
`(b)` Maximum velocity of particle is `1m//s`
`(c )` Wave length of wave is `2pi` meter

Options

A. only `a,c` are true
B. only `a,b` are true
C. only `b,c` are true
D. `a,b,c` are true

Explanation

To solve the question, we need to analyze the given wave equation and check the three statements regarding the wave's properties.

### Given Wave Equation:
\[ y = 0.01 \sin(100t - x) \]

### Step 1: Identify Parameters
From the wave equation, we can identify:
- Amplitude \( A = 0.01 \, \text{m} \)
- Angular frequency \( \omega = 100 \, \text{rad/s} \)
- Wave number \( k \)

### Step 2: Calculate Wave Number \( k \)
The general form of a progressive wave is:
\[ y = A \sin(\omega t - kx) \]
Comparing with the given equation, we have \( k = 1 \, \text{rad/m} \).

### Step 3: Calculate Velocity of the Wave
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{100 \, \text{rad/s}}{1 \, \text{rad/m}} = 100 \, \text{m/s} \]

### Step 4: Check Statement (a)
The first statement claims that the velocity of the wave is \( 50 \, \text{m/s} \). Since we calculated the velocity to be \( 100 \, \text{m/s} \), this statement is **incorrect**.

### Step 5: Calculate Maximum Velocity of the Particle
The maximum velocity \( V_{\text{max}} \) of a particle in the wave can be calculated using:
\[ V_{\text{max}} = \omega A \]
Substituting the values:
\[ V_{\text{max}} = 100 \, \text{rad/s} \times 0.01 \, \text{m} = 1 \, \text{m/s} \]

### Step 6: Check Statement (b)
The second statement claims that the maximum velocity of the particle is \( 1 \, \text{m/s} \). Since we calculated it to be \( 1 \, \text{m/s} \), this statement is **correct**.

### Step 7: Calculate Wavelength \( \lambda \)
The relationship between wave number \( k \) and wavelength \( \lambda \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
We know \( k = 1 \, \text{rad/m} \), so:
\[ 1 = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = 2\pi \, \text{m} \]

### Step 8: Check Statement (c)
The third statement claims that the wavelength is \( 2\pi \, \text{m} \). Since we calculated it to be \( 2\pi \, \text{m} \), this statement is **correct**.

### Conclusion
- Statement (a) is **incorrect**.
- Statement (b) is **correct**.
- Statement (c) is **correct**.

### Final Answers:
- (a) False
- (b) True
- (c) True

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