A standing wave is produced on a string on a string clamped at one end and free at the other. The length of the string

Options

A. must be an integral multiple `lamda/4`
B. must be an integral multiple of `lamda/2`
C. must be an integral multiple of `lamda`
D. may be an integral multiple of `lamda/2`

Answer

To solve the problem of finding the length of a string that produces a standing wave, we can follow these steps:

### Step 1: Understand the Setup
We have a string that is clamped at one end and free at the other. This setup allows for the formation of standing waves with specific boundary conditions.

### Step 2: Identify the Wave Equation
The general equation for a standing wave on a string can be expressed as:
\[ y(x, t) = 2a \sin\left(\frac{2\pi x}{\lambda}\right) \cos(2\pi \nu t) \]
where:
- \( a \) is the amplitude,
- \( \lambda \) is the wavelength,
- \( \nu \) is the frequency,
- \( x \) is the position along the string.

### Step 3: Determine the Condition for Antinodes
At the free end of the string (where \( x = l \)), we have an antinode. The condition for an antinode is that the sine function reaches its maximum value (which is 1):
\[ \sin\left(\frac{2\pi l}{\lambda}\right) = 1 \]

### Step 4: Solve for the Position of Antinode
The sine function equals 1 at specific points:
\[ \frac{2\pi l}{\lambda} = \frac{\pi}{2} + n\pi \]
where \( n \) is an integer (0, 1, 2, ...).

### Step 5: Rearrange the Equation
From the equation above, we can isolate \( l \):
\[ 2\pi l = \lambda\left(\frac{\pi}{2} + n\pi\right) \]
\[ l = \frac{\lambda}{4} + \frac{n\lambda}{2} \]

### Step 6: Express Length in Terms of Wavelength
This can be simplified to:
\[ l = \frac{(2n + 1)\lambda}{4} \]
where \( n \) is an integer. This indicates that the length of the string is an integral multiple of \( \frac{\lambda}{4} \).

### Step 7: Conclusion
Thus, the length of the string can be expressed as:
\[ l = \frac{(2n + 1)\lambda}{4} \]
This means that the length of the string is proportional to \( \frac{\lambda}{4} \).

### Final Answer
The length of the string is given by:
\[ l = \frac{(2n + 1)\lambda}{4} \]

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