How do you calculate Helmholtz resonance?
2 Answers
Helmholtz resonance describes the phenomenon in which air inside a cavity resonates when excited by a sound wave. A classic example is blowing over the mouth of a bottle to produce a tone. The frequency of this resonance depends on the geometry of the cavity and the neck, as well as the speed of sound in air.
### Formula for Helmholtz Resonance Frequency
The Helmholtz resonance frequency (\( f_0 \)) is given by:
\[
f_0 = \frac{v}{2\pi} \sqrt{\frac{A}{V \cdot L_{\text{eff}}}}
\]
where:
- \( f_0 \) = resonance frequency (in Hz)
- \( v \) = speed of sound in air (typically 343 m/s at room temperature)
- \( A \) = cross-sectional area of the neck of the cavity (in square meters)
- \( V \) = volume of the cavity (in cubic meters)
- \( L_{\text{eff}} \) = effective length of the neck, which includes the actual neck length and an end correction due to air inertia at the open end of the neck.
### Components in Detail:
1. **Speed of sound (v):**
The speed of sound in air is affected by temperature and is approximately 343 m/s at 20°C. You can use the following formula to adjust for temperature:
\[
v = 331.3 + 0.6 \cdot T
\]
where \( T \) is the temperature in °C.
2. **Cross-sectional area of the neck (A):**
For a circular neck, the area \( A \) is given by:
\[
A = \pi r^2
\]
where \( r \) is the radius of the neck.
3. **Volume of the cavity (V):**
The volume of the cavity can be calculated based on its geometry. For example, if the cavity is a cylinder, the volume is:
\[
V = \pi r_{\text{cavity}}^2 h
\]
where \( r_{\text{cavity}} \) is the radius of the cavity and \( h \) is the height.
4. **Effective length of the neck (L_{\text{eff}}):**
The effective length accounts for the actual length \( L \) of the neck plus an "end correction" \( 0.3r \) (for a neck open to free space):
\[
L_{\text{eff}} = L + 0.3r
\]
where \( r \) is the radius of the neck.
### Example Calculation:
Let’s say we have a bottle with:
- Neck length \( L = 0.05 \, \text{m} \)
- Neck radius \( r = 0.01 \, \text{m} \)
- Bottle cavity volume \( V = 0.001 \, \text{m}^3 \)
The speed of sound at room temperature is \( v = 343 \, \text{m/s} \).
1. First, calculate the cross-sectional area of the neck:
\[
A = \pi (0.01)^2 = 3.14 \times 10^{-4} \, \text{m}^2
\]
2. Calculate the effective length of the neck:
\[
L_{\text{eff}} = 0.05 + 0.3(0.01) = 0.053 \, \text{m}
\]
3. Finally, calculate the resonance frequency:
\[
f_0 = \frac{343}{2\pi} \sqrt{\frac{3.14 \times 10^{-4}}{0.001 \times 0.053}} \approx 146 \, \text{Hz}
\]
So, the Helmholtz resonance frequency is approximately **146 Hz**.
This method can be used for various applications, including designing speakers, understanding wind instrument behavior, and solving acoustic problems.
### Formula for Helmholtz Resonance Frequency
The Helmholtz resonance frequency (\( f_0 \)) is given by:
\[
f_0 = \frac{v}{2\pi} \sqrt{\frac{A}{V \cdot L_{\text{eff}}}}
\]
where:
- \( f_0 \) = resonance frequency (in Hz)
- \( v \) = speed of sound in air (typically 343 m/s at room temperature)
- \( A \) = cross-sectional area of the neck of the cavity (in square meters)
- \( V \) = volume of the cavity (in cubic meters)
- \( L_{\text{eff}} \) = effective length of the neck, which includes the actual neck length and an end correction due to air inertia at the open end of the neck.
### Components in Detail:
1. **Speed of sound (v):**
The speed of sound in air is affected by temperature and is approximately 343 m/s at 20°C. You can use the following formula to adjust for temperature:
\[
v = 331.3 + 0.6 \cdot T
\]
where \( T \) is the temperature in °C.
2. **Cross-sectional area of the neck (A):**
For a circular neck, the area \( A \) is given by:
\[
A = \pi r^2
\]
where \( r \) is the radius of the neck.
3. **Volume of the cavity (V):**
The volume of the cavity can be calculated based on its geometry. For example, if the cavity is a cylinder, the volume is:
\[
V = \pi r_{\text{cavity}}^2 h
\]
where \( r_{\text{cavity}} \) is the radius of the cavity and \( h \) is the height.
4. **Effective length of the neck (L_{\text{eff}}):**
The effective length accounts for the actual length \( L \) of the neck plus an "end correction" \( 0.3r \) (for a neck open to free space):
\[
L_{\text{eff}} = L + 0.3r
\]
where \( r \) is the radius of the neck.
### Example Calculation:
Let’s say we have a bottle with:
- Neck length \( L = 0.05 \, \text{m} \)
- Neck radius \( r = 0.01 \, \text{m} \)
- Bottle cavity volume \( V = 0.001 \, \text{m}^3 \)
The speed of sound at room temperature is \( v = 343 \, \text{m/s} \).
1. First, calculate the cross-sectional area of the neck:
\[
A = \pi (0.01)^2 = 3.14 \times 10^{-4} \, \text{m}^2
\]
2. Calculate the effective length of the neck:
\[
L_{\text{eff}} = 0.05 + 0.3(0.01) = 0.053 \, \text{m}
\]
3. Finally, calculate the resonance frequency:
\[
f_0 = \frac{343}{2\pi} \sqrt{\frac{3.14 \times 10^{-4}}{0.001 \times 0.053}} \approx 146 \, \text{Hz}
\]
So, the Helmholtz resonance frequency is approximately **146 Hz**.
This method can be used for various applications, including designing speakers, understanding wind instrument behavior, and solving acoustic problems.