The equation of a transverse wave travelling along a string is given by
`y=0.4sin[pi(0.5x-200t)]` where `y` and `x` are measured in cm and `t` in seconds. Suppose that you clamp the string at two points `L` cm apart and you observe a standing wave of the same wavelength as above transverse wave. For what values of `L` less than `10 cm` is this possible?

Options

A. `2cm`
B. `4cm`
C. `9cm`
D. `8cm`

Explanation

To solve the problem, we need to analyze the given wave equation and determine the possible values of \( L \) for which a standing wave can be formed.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The equation of the transverse wave is given as:
\[ y = 0.4 \sin[\pi(0.5x - 200t)] \]

2. **Extract the Wave Number \( k \)**:
From the wave equation, we can identify the wave number \( k \):
\[ k = 0.5\pi \]

3. **Calculate the Wavelength \( \lambda \)**:
The relationship between wave number \( k \) and wavelength \( \lambda \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.5\pi \):
\[ \lambda = \frac{2\pi}{0.5\pi} = 4 \text{ cm} \]

4. **Understand the Condition for Standing Waves**:
When the string is clamped at two points \( L \) cm apart, standing waves can form. The distance \( L \) must be a multiple of half the wavelength:
\[ L = n \frac{\lambda}{2} \]
where \( n \) is a positive integer.

5. **Substitute the Wavelength**:
Since \( \lambda = 4 \text{ cm} \), we have:
\[ L = n \frac{4}{2} = 2n \text{ cm} \]

6. **Determine Possible Values of \( L \)**:
We need to find values of \( L \) that are less than 10 cm:
- For \( n = 1 \): \( L = 2 \times 1 = 2 \text{ cm} \)
- For \( n = 2 \): \( L = 2 \times 2 = 4 \text{ cm} \)
- For \( n = 3 \): \( L = 2 \times 3 = 6 \text{ cm} \)
- For \( n = 4 \): \( L = 2 \times 4 = 8 \text{ cm} \)
- For \( n = 5 \): \( L = 2 \times 5 = 10 \text{ cm} \) (not valid since \( L < 10 \))

Thus, the possible values of \( L \) that are less than 10 cm are:
\[ L = 2 \text{ cm}, 4 \text{ cm}, 6 \text{ cm}, 8 \text{ cm} \]

### Final Answer:
The possible values of \( L \) less than 10 cm are:
- \( 2 \text{ cm} \)
- \( 4 \text{ cm} \)
- \( 6 \text{ cm} \)
- \( 8 \text{ cm} \)