A string that is stretched between fixed supports separated by 75.0 cm has resonant frequencies of 450 and 308 Hz, with no intermediate resonant frequencies. What are (a) the lowest resonant frequency and (b) the wave speed?

Answer

To solve the problem step by step, we will first identify the given information and then apply the relevant formulas for resonant frequencies in a stretched string.

### Given:
- Length of the string, \( L = 75.0 \, \text{cm} = 0.75 \, \text{m} \)
- Two resonant frequencies: \( f_1 = 450 \, \text{Hz} \) and \( f_2 = 308 \, \text{Hz} \)

### Step 1: Identify the relationship between the frequencies
Since the problem states that there are no intermediate resonant frequencies, we can conclude that:
- \( f_2 \) corresponds to the \( n \)-th harmonic
- \( f_1 \) corresponds to the \( (n+1) \)-th harmonic

### Step 2: Write the equations for the resonant frequencies
The resonant frequency for a string fixed at both ends can be expressed as:
\[ f_n = \frac{n v}{2L} \]
where \( v \) is the wave speed.

For the \( n \)-th harmonic:
\[ f_2 = \frac{n v}{2L} = 308 \, \text{Hz} \quad \text{(1)} \]
For the \( (n+1) \)-th harmonic:
\[ f_1 = \frac{(n+1) v}{2L} = 450 \, \text{Hz} \quad \text{(2)} \]

### Step 3: Set up the equations
From equation (1):
\[ n v = 2L \cdot 308 \]
From equation (2):
\[ (n+1) v = 2L \cdot 450 \]

### Step 4: Solve for \( v \)
Subtract equation (1) from equation (2):
\[ (n+1)v - nv = 2L \cdot 450 - 2L \cdot 308 \]
This simplifies to:
\[ v = 2L \cdot (450 - 308) = 2L \cdot 142 \]
Now substitute \( L = 0.75 \, \text{m} \):
\[ v = 2 \cdot 0.75 \cdot 142 \]
Calculating this gives:
\[ v = 1.5 \cdot 142 = 213 \, \text{m/s} \]

### Step 5: Find the lowest resonant frequency
The lowest resonant frequency (fundamental frequency) corresponds to \( n = 1 \):
\[ f_1 = \frac{1 \cdot v}{2L} = \frac{v}{2L} \]
Substituting \( v = 213 \, \text{m/s} \) and \( L = 0.75 \, \text{m} \):
\[ f_1 = \frac{213}{2 \cdot 0.75} = \frac{213}{1.5} = 142 \, \text{Hz} \]

### Final Answers:
(a) The lowest resonant frequency is \( 142 \, \text{Hz} \).

(b) The wave speed is \( 213 \, \text{m/s} \).