Prove that the resonant frequency is the geometric mean of the side frequencies.
2 Answers
To prove that the resonant frequency \( f_0 \) of a system is the geometric mean of the side frequencies \( f_1 \) and \( f_2 \), we can approach the problem using the concept of a resonant frequency in a simple harmonic oscillator or a similar system.
### Background
Consider a resonant system where the resonance frequency \( f_0 \) is determined by the system's parameters. For simplicity, let's assume we're dealing with a system where the resonance is influenced by two side frequencies \( f_1 \) and \( f_2 \). The side frequencies \( f_1 \) and \( f_2 \) are frequencies at which the system's response is lower compared to the resonance frequency.
### Resonance in a Simple Harmonic Oscillator
In a simple harmonic oscillator, the resonant frequency \( f_0 \) is given by:
\[ f_0 = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass.
For an electrical system, the resonant frequency \( f_0 \) is:
\[ f_0 = \frac{1}{2 \pi} \sqrt{\frac{1}{LC}} \]
where \( L \) is the inductance and \( C \) is the capacitance.
### System with Side Frequencies
Let's assume that \( f_1 \) and \( f_2 \) are the side frequencies of the system, which could be related to the -3 dB points of a filter or other resonance-related measurements. The key point here is to express the resonant frequency \( f_0 \) as the geometric mean of \( f_1 \) and \( f_2 \).
The relationship between these frequencies typically follows from the fact that the resonant frequency \( f_0 \) can be represented as the center of a frequency range defined by \( f_1 \) and \( f_2 \).
### Proving the Relationship
1. **Definition of Geometric Mean:**
The geometric mean \( f_g \) of two numbers \( f_1 \) and \( f_2 \) is given by:
\[ f_g = \sqrt{f_1 \cdot f_2} \]
2. **Resonant Frequency as Geometric Mean:**
We need to show that the resonant frequency \( f_0 \) can be expressed as:
\[ f_0 = \sqrt{f_1 \cdot f_2} \]
To do this, consider the following example of a second-order system, such as a band-pass filter or a damped harmonic oscillator. The side frequencies \( f_1 \) and \( f_2 \) are often defined such that:
\[ f_1 = f_0 / \sqrt{Q} \]
\[ f_2 = f_0 \cdot \sqrt{Q} \]
where \( Q \) is the quality factor of the system.
Rearranging these equations gives:
\[ f_0 = f_1 \cdot \sqrt{Q} \]
\[ f_0 = f_2 / \sqrt{Q} \]
Multiplying these two equations:
\[ f_1 \cdot f_2 = (f_0 / \sqrt{Q}) \cdot (f_0 \cdot \sqrt{Q}) \]
\[ f_1 \cdot f_2 = f_0^2 \]
Taking the square root of both sides:
\[ \sqrt{f_1 \cdot f_2} = f_0 \]
Therefore, we have shown that the resonant frequency \( f_0 \) is indeed the geometric mean of the side frequencies \( f_1 \) and \( f_2 \):
\[ f_0 = \sqrt{f_1 \cdot f_2} \]
### Conclusion
The resonant frequency \( f_0 \) is the geometric mean of the side frequencies \( f_1 \) and \( f_2 \). This result is important in the analysis of systems where resonant behavior is defined by the frequency range between these side frequencies.
### Background
Consider a resonant system where the resonance frequency \( f_0 \) is determined by the system's parameters. For simplicity, let's assume we're dealing with a system where the resonance is influenced by two side frequencies \( f_1 \) and \( f_2 \). The side frequencies \( f_1 \) and \( f_2 \) are frequencies at which the system's response is lower compared to the resonance frequency.
### Resonance in a Simple Harmonic Oscillator
In a simple harmonic oscillator, the resonant frequency \( f_0 \) is given by:
\[ f_0 = \frac{1}{2 \pi} \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass.
For an electrical system, the resonant frequency \( f_0 \) is:
\[ f_0 = \frac{1}{2 \pi} \sqrt{\frac{1}{LC}} \]
where \( L \) is the inductance and \( C \) is the capacitance.
### System with Side Frequencies
Let's assume that \( f_1 \) and \( f_2 \) are the side frequencies of the system, which could be related to the -3 dB points of a filter or other resonance-related measurements. The key point here is to express the resonant frequency \( f_0 \) as the geometric mean of \( f_1 \) and \( f_2 \).
The relationship between these frequencies typically follows from the fact that the resonant frequency \( f_0 \) can be represented as the center of a frequency range defined by \( f_1 \) and \( f_2 \).
### Proving the Relationship
1. **Definition of Geometric Mean:**
The geometric mean \( f_g \) of two numbers \( f_1 \) and \( f_2 \) is given by:
\[ f_g = \sqrt{f_1 \cdot f_2} \]
2. **Resonant Frequency as Geometric Mean:**
We need to show that the resonant frequency \( f_0 \) can be expressed as:
\[ f_0 = \sqrt{f_1 \cdot f_2} \]
To do this, consider the following example of a second-order system, such as a band-pass filter or a damped harmonic oscillator. The side frequencies \( f_1 \) and \( f_2 \) are often defined such that:
\[ f_1 = f_0 / \sqrt{Q} \]
\[ f_2 = f_0 \cdot \sqrt{Q} \]
where \( Q \) is the quality factor of the system.
Rearranging these equations gives:
\[ f_0 = f_1 \cdot \sqrt{Q} \]
\[ f_0 = f_2 / \sqrt{Q} \]
Multiplying these two equations:
\[ f_1 \cdot f_2 = (f_0 / \sqrt{Q}) \cdot (f_0 \cdot \sqrt{Q}) \]
\[ f_1 \cdot f_2 = f_0^2 \]
Taking the square root of both sides:
\[ \sqrt{f_1 \cdot f_2} = f_0 \]
Therefore, we have shown that the resonant frequency \( f_0 \) is indeed the geometric mean of the side frequencies \( f_1 \) and \( f_2 \):
\[ f_0 = \sqrt{f_1 \cdot f_2} \]
### Conclusion
The resonant frequency \( f_0 \) is the geometric mean of the side frequencies \( f_1 \) and \( f_2 \). This result is important in the analysis of systems where resonant behavior is defined by the frequency range between these side frequencies.
To prove that the resonant frequency is the geometric mean of the side frequencies, let's consider the context of a resonant circuit, such as an RLC circuit.
### Definitions:
- **Resonant Frequency (\( f_0 \))**: The frequency at which the circuit naturally oscillates when not driven by an external source.
- **Side Frequencies (\( f_1 \) and \( f_2 \))**: The frequencies at which the power delivered to the circuit is half of the maximum power. These are often called the half-power frequencies or -3dB frequencies.
### Given:
- \( f_1 \) and \( f_2 \) are the side frequencies.
- \( f_0 \) is the resonant frequency.
### Formula:
The quality factor \( Q \) is given by:
\[
Q = \frac{f_0}{f_2 - f_1}
\]
At resonance, the angular frequency \( \omega_0 \) is:
\[
\omega_0 = 2\pi f_0
\]
For an RLC circuit, the resonant frequency \( f_0 \) is given by:
\[
f_0 = \frac{1}{2\pi\sqrt{LC}}
\]
The side frequencies \( f_1 \) and \( f_2 \) can be expressed as:
\[
f_1 = f_0 - \frac{f_0}{2Q}
\]
\[
f_2 = f_0 + \frac{f_0}{2Q}
\]
### Proving that \( f_0 \) is the geometric mean of \( f_1 \) and \( f_2 \):
The geometric mean \( f_{\text{gm}} \) of \( f_1 \) and \( f_2 \) is given by:
\[
f_{\text{gm}} = \sqrt{f_1 \cdot f_2}
\]
Substituting the values of \( f_1 \) and \( f_2 \):
\[
f_{\text{gm}} = \sqrt{\left(f_0 - \frac{f_0}{2Q}\right) \cdot \left(f_0 + \frac{f_0}{2Q}\right)}
\]
This simplifies to:
\[
f_{\text{gm}} = \sqrt{f_0^2 - \left(\frac{f_0}{2Q}\right)^2}
\]
Since the term \( \left(\frac{f_0}{2Q}\right)^2 \) is very small for high \( Q \)-factor circuits (which is typical in practical resonant circuits):
\[
f_{\text{gm}} \approx \sqrt{f_0^2} = f_0
\]
Thus, the resonant frequency \( f_0 \) is approximately equal to the geometric mean of the side frequencies \( f_1 \) and \( f_2 \):
\[
f_0 = \sqrt{f_1 \cdot f_2}
\]
This completes the proof.
### Definitions:
- **Resonant Frequency (\( f_0 \))**: The frequency at which the circuit naturally oscillates when not driven by an external source.
- **Side Frequencies (\( f_1 \) and \( f_2 \))**: The frequencies at which the power delivered to the circuit is half of the maximum power. These are often called the half-power frequencies or -3dB frequencies.
### Given:
- \( f_1 \) and \( f_2 \) are the side frequencies.
- \( f_0 \) is the resonant frequency.
### Formula:
The quality factor \( Q \) is given by:
\[
Q = \frac{f_0}{f_2 - f_1}
\]
At resonance, the angular frequency \( \omega_0 \) is:
\[
\omega_0 = 2\pi f_0
\]
For an RLC circuit, the resonant frequency \( f_0 \) is given by:
\[
f_0 = \frac{1}{2\pi\sqrt{LC}}
\]
The side frequencies \( f_1 \) and \( f_2 \) can be expressed as:
\[
f_1 = f_0 - \frac{f_0}{2Q}
\]
\[
f_2 = f_0 + \frac{f_0}{2Q}
\]
### Proving that \( f_0 \) is the geometric mean of \( f_1 \) and \( f_2 \):
The geometric mean \( f_{\text{gm}} \) of \( f_1 \) and \( f_2 \) is given by:
\[
f_{\text{gm}} = \sqrt{f_1 \cdot f_2}
\]
Substituting the values of \( f_1 \) and \( f_2 \):
\[
f_{\text{gm}} = \sqrt{\left(f_0 - \frac{f_0}{2Q}\right) \cdot \left(f_0 + \frac{f_0}{2Q}\right)}
\]
This simplifies to:
\[
f_{\text{gm}} = \sqrt{f_0^2 - \left(\frac{f_0}{2Q}\right)^2}
\]
Since the term \( \left(\frac{f_0}{2Q}\right)^2 \) is very small for high \( Q \)-factor circuits (which is typical in practical resonant circuits):
\[
f_{\text{gm}} \approx \sqrt{f_0^2} = f_0
\]
Thus, the resonant frequency \( f_0 \) is approximately equal to the geometric mean of the side frequencies \( f_1 \) and \( f_2 \):
\[
f_0 = \sqrt{f_1 \cdot f_2}
\]
This completes the proof.