Two timpani (tunable drums) are played at the same time. One is correctly tuned so that when it is struck, sound is produced that has a wavelength of 2.20 m. The second produces sound with a wavelength of 2.08 m. If the speed of sound is 343 m/s, what beat frequency is heard?

Options

A. 7 beats/s
B. 9 beats/s
C. 11 beats/s
D. 13 beats/s

Answer

To find the beat frequency heard when two timpani are played simultaneously, we can follow these steps:

### Step 1: Identify the Wavelengths
We have two drums producing sound with the following wavelengths:
- Drum 1: \( \lambda_1 = 2.20 \, \text{m} \)
- Drum 2: \( \lambda_2 = 2.08 \, \text{m} \)

### Step 2: Use the Speed of Sound
The speed of sound is given as:
- \( v = 343 \, \text{m/s} \)

### Step 3: Calculate the Frequencies
The frequency \( f \) of a wave is calculated using the formula:
\[ f = \frac{v}{\lambda} \]

#### For Drum 1:
Using \( \lambda_1 \):
\[ f_1 = \frac{343 \, \text{m/s}}{2.20 \, \text{m}} \]
\[ f_1 = 156.82 \, \text{Hz} \]

#### For Drum 2:
Using \( \lambda_2 \):
\[ f_2 = \frac{343 \, \text{m/s}}{2.08 \, \text{m}} \]
\[ f_2 = 165.87 \, \text{Hz} \]

### Step 4: Calculate the Beat Frequency
The beat frequency \( f_b \) is given by the absolute difference between the two frequencies:
\[ f_b = |f_2 - f_1| \]
\[ f_b = |165.87 \, \text{Hz} - 156.82 \, \text{Hz}| \]
\[ f_b = 9.05 \, \text{Hz} \]

### Final Answer
The beat frequency heard is approximately:
\[ f_b \approx 9 \, \text{Hz} \]

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