8. Find the frequencies for the $$ 2^{ ext {nd }} $$ and $$ 3^{ ext {rd }} $$ harmonics. (keep in mind the note from the previous problem)

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8. Find the frequencies for the $$ 2^{	ext {nd }} $$ and $$ 3^{	ext {rd }} $$ harmonics. (keep in mind the note from the previous problem)

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To determine the frequencies of the $$ 2^{\text{nd}} $$ and $$ 3^{\text{rd}} $$ harmonics, we'll first need to understand the concept of harmonics in the context of wave phenomena, such as vibrations on a string or sound waves in a pipe.

### **Fundamental Frequency (First Harmonic)**
The **fundamental frequency** ($$ f_1 $$) is the lowest frequency at which a system naturally vibrates. It sets the base for all other harmonics.

### **Harmonic Frequencies**
Harmonics are integer multiples of the fundamental frequency. Specifically:
- **Second Harmonic ($$ f_2 $$)**: This is twice the fundamental frequency.
  $$
  f_2 = 2 \times f_1
  $$
- **Third Harmonic ($$ f_3 $$)**: This is three times the fundamental frequency.
  $$
  f_3 = 3 \times f_1
  $$

### **Example Calculation**
Suppose the fundamental frequency ($$ f_1 $$) of a vibrating string is **200 Hz**. Then:
- **Second Harmonic ($$ f_2 $$)**:
  $$
  f_2 = 2 \times 200\, \text{Hz} = 400\, \text{Hz}
  $$
- **Third Harmonic ($$ f_3 $$)**:
  $$
  f_3 = 3 \times 200\, \text{Hz} = 600\, \text{Hz}
  $$

### **General Formula**
If the fundamental frequency is $$ f_1 $$, the $$ n^{\text{th}} $$ harmonic ($$ f_n $$) can be calculated as:
$$
f_n = n \times f_1
$$
where $$ n $$ is an integer representing the harmonic number (e.g., $$ n = 2 $$ for the second harmonic, $$ n = 3 $$ for the third harmonic, etc.).

### **Considerations from Previous Problems**
Since you mentioned a note from the previous problem, ensure that:
- **Boundary Conditions**: The type of harmonic (e.g., whether the ends are fixed or free) can affect the harmonic frequencies.
- **Medium Properties**: Factors like tension, length, and mass per unit length (for strings) or temperature and pressure (for air columns) can influence the fundamental frequency and, consequently, the harmonics.
- **Nonlinear Effects**: In some systems, especially at high amplitudes, harmonics might not be perfect integer multiples due to nonlinearities.

If the previous problem provided specific information about the system (like the fundamental frequency, length, tension, etc.), you should use that data to calculate the exact frequencies of the $$ 2^{\text{nd}} $$ and $$ 3^{\text{rd}} $$ harmonics using the formulas above.

### **Summary**
- **Second Harmonic**: $$ f_2 = 2 \times f_1 $$
- **Third Harmonic**: $$ f_3 = 3 \times f_1 $$

Make sure to apply any specific conditions or formulas provided in your previous problem to obtain accurate results.
The $$ 2^{\text{nd}} $$ harmonic is twice the fundamental frequency, and the $$ 3^{\text{rd}} $$ harmonic is three times the fundamental frequency.

Find the frequencies for the and harmonics. (keep in mind the note from the previous
problem)

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Find the frequencies for the and harmonics. (keep in mind the note from the previous problem)