The stationary waves set up on a string have the equation :
`y = ( 2 mm) sin [ (6.28 m^(-1)) x] cos omega t`
The stationary wave is created by two identical waves , of amplitude `A` each , moving in opposite directions along the string . Then :

Options

A. `A = 2 mm`
B. `A = 1 mm`
C. the smallest length of the string is `50 cm`
D. the smallest length of the string is `2 m`

Explanation

To solve the problem step by step, we will analyze the given equation of the stationary wave and derive the required values.

### Step 1: Identify the given equation
The equation of the stationary wave is given as:
\[ y = (2 \, \text{mm}) \sin(6.28 \, \text{m}^{-1} \, x) \cos(\omega t) \]

### Step 2: Compare with the general form of stationary wave
The general equation for a stationary wave formed by two identical waves moving in opposite directions is:
\[ y = 2A \sin(kx) \cos(\omega t) \]
where \( A \) is the amplitude and \( k \) is the wave number.

### Step 3: Extract the amplitude
From the given equation, we can see that:
\[ 2A = 2 \, \text{mm} \]
Thus, we can find \( A \) by dividing both sides by 2:
\[ A = 1 \, \text{mm} \]

### Step 4: Identify the wave number
The wave number \( k \) is given as:
\[ k = 6.28 \, \text{m}^{-1} \]
We can also express this as:
\[ k = 2\pi \, \text{m}^{-1} \]

### Step 5: Relate wave number to string length
The wave number \( k \) is related to the length of the string \( L \) by the formula:
\[ k = \frac{n\pi}{L} \]
where \( n \) is a positive integer representing the mode of vibration.

### Step 6: Solve for the length of the string
From the equation \( k = 2\pi \):
\[ 2\pi = \frac{n\pi}{L} \]
Cancelling \( \pi \) from both sides gives:
\[ 2 = \frac{n}{L} \]
Rearranging this gives:
\[ L = \frac{n}{2} \]

### Step 7: Find the smallest length of the string
To find the smallest length, we take the smallest integer value of \( n \), which is 1:
\[ L = \frac{1}{2} \, \text{m} = 0.5 \, \text{m} = 50 \, \text{cm} \]

### Summary of Results
- The amplitude \( A \) is \( 1 \, \text{mm} \).
- The smallest length of the string \( L \) is \( 50 \, \text{cm} \).

### Final Answers
- Amplitude \( A = 1 \, \text{mm} \) (Option 2 is correct)
- Smallest length of the string \( L = 50 \, \text{cm} \) (Option C is correct)