The third overtone produced by a vibrating string 0.5m long is 1200Hz. The speed of propagation of the wave in `ms^(-1)` is

Options

A. 400
B. 300
C. 600
D. 1200

Answer

To find the speed of propagation of the wave produced by a vibrating string, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Length of the string (L) = 0.5 m
- Frequency of the third overtone (f) = 1200 Hz

2. **Determine the Wavelength for the Third Overtone**:
- The third overtone corresponds to the fourth harmonic of the string. The formula for the wavelength (λ) of the nth harmonic in a string fixed at both ends is given by:
\[ \lambda_n = \frac{2L}{n} \]
- For the third overtone (n = 4):
\[ \lambda_4 = \frac{2 \times 0.5}{4} = \frac{1}{4} = 0.25 \, \text{m} \]

3. **Use the Wave Speed Formula**:
- The speed of a wave (v) is given by the formula:
\[ v = f \times \lambda \]
- Substitute the values of frequency and wavelength:
\[ v = 1200 \, \text{Hz} \times 0.25 \, \text{m} \]

4. **Calculate the Speed**:
- Performing the multiplication:
\[ v = 1200 \times 0.25 = 300 \, \text{m/s} \]

5. **Final Answer**:
- The speed of propagation of the wave is **300 m/s**.