To solve the problem of how many overtones can be distinctly heard from a closed organ pipe with a fundamental frequency of 1.5 kHz, we can follow these steps:
### Step 1: Understand the Harmonics of a Closed Organ Pipe
A closed organ pipe produces only odd harmonics. The fundamental frequency (first harmonic) is given as \( f_1 = 1.5 \, \text{kHz} \). The frequencies of the overtones can be expressed as:
- First harmonic (fundamental): \( f_1 = \frac{V}{4L} \)
- Third harmonic (first overtone): \( f_3 = 3f_1 \)
- Fifth harmonic (second overtone): \( f_5 = 5f_1 \)
- Seventh harmonic (third overtone): \( f_7 = 7f_1 \)
- And so on...
### Step 2: Calculate the Frequencies of the Overtones
Using the fundamental frequency:
- \( f_1 = 1.5 \, \text{kHz} \)
- \( f_3 = 3 \times 1.5 \, \text{kHz} = 4.5 \, \text{kHz} \)
- \( f_5 = 5 \times 1.5 \, \text{kHz} = 7.5 \, \text{kHz} \)
- \( f_7 = 7 \times 1.5 \, \text{kHz} = 10.5 \, \text{kHz} \)
- \( f_9 = 9 \times 1.5 \, \text{kHz} = 13.5 \, \text{kHz} \)
- \( f_{11} = 11 \times 1.5 \, \text{kHz} = 16.5 \, \text{kHz} \)
- \( f_{13} = 13 \times 1.5 \, \text{kHz} = 19.5 \, \text{kHz} \)
### Step 3: Determine the Maximum Frequency Heard
The maximum frequency a person can hear is given as \( 20,000 \, \text{Hz} \) or \( 20 \, \text{kHz} \).
### Step 4: Identify the Highest Overtone Frequency Below 20 kHz
From our calculations:
- The frequencies of the overtones are:
- \( f_1 = 1.5 \, \text{kHz} \)
- \( f_3 = 4.5 \, \text{kHz} \)
- \( f_5 = 7.5 \, \text{kHz} \)
- \( f_7 = 10.5 \, \text{kHz} \)
- \( f_9 = 13.5 \, \text{kHz} \)
- \( f_{11} = 16.5 \, \text{kHz} \)
- \( f_{13} = 19.5 \, \text{kHz} \)
The next overtone would be \( f_{15} = 22.5 \, \text{kHz} \), which exceeds the hearing limit of 20 kHz.
### Step 5: Count the Distinct Overtones
The overtones that can be distinctly heard are:
- First overtone: \( f_3 \)
- Second overtone: \( f_5 \)
- Third overtone: \( f_7 \)
- Fourth overtone: \( f_9 \)
- Fifth overtone: \( f_{11} \)
- Sixth overtone: \( f_{13} \)
Thus, the total number of overtones that can be distinctly heard is 6.
### Final Answer
The number of overtones that can be distinctly heard by a person with this organ pipe is **6**.
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