A `75 cm` string fixed at both ends produces resonant frequencies `384 Hz and 288 Hz` without there being any other resonant frequency between these two . Wave speed for the string is

Options

A. `144 m//s`
B. `216 m//s`
C. `108 m//s`
D. `72 m//s`

Explanation

To find the wave speed for the string, we can follow these steps:

### Step 1: Identify the given frequencies
We are given two resonant frequencies of the string:
- \( f_1 = 384 \, \text{Hz} \)
- \( f_2 = 288 \, \text{Hz} \)

### Step 2: Find the ratio of the frequencies
To understand the relationship between these frequencies, we can calculate the ratio:
\[ \frac{f_1}{f_2} = \frac{384}{288} \]
Simplifying this gives:
\[ \frac{384 \div 96}{288 \div 96} = \frac{4}{3} \]

### Step 3: Determine the fundamental frequency
From the ratio \( \frac{f_1}{f_2} = \frac{4}{3} \), we can express \( f_1 \) and \( f_2 \) in terms of the fundamental frequency \( f_0 \):
- Let \( f_0 = 96 \, \text{Hz} \)
- Then, \( f_1 = 4f_0 = 384 \, \text{Hz} \)
- And \( f_2 = 3f_0 = 288 \, \text{Hz} \)

### Step 4: Relate the frequency to the wavelength
For a string fixed at both ends, the fundamental frequency corresponds to the first harmonic, where the length of the string \( L \) is related to the wavelength \( \lambda \) by:
\[ L = \frac{\lambda}{2} \]
Given that the length of the string is \( L = 75 \, \text{cm} = 0.75 \, \text{m} \), we can find the wavelength:
\[ \lambda = 2L = 2 \times 0.75 = 1.5 \, \text{m} \]

### Step 5: Calculate the wave speed
The wave speed \( v \) can be calculated using the formula:
\[ v = f_0 \cdot \lambda \]
Substituting the values we found:
\[ v = 96 \, \text{Hz} \times 1.5 \, \text{m} = 144 \, \text{m/s} \]

### Final Answer
The wave speed for the string is \( 144 \, \text{m/s} \).

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