To show that frequency of a Helmholtz resonator varies inversely as the square root of its volume and to estimate the neck correction

4. Theory

A Helmholtz resonator consists of a cavity (volume V) connected to the outside air through a narrow neck of cross-sectional area A and length L. When air is forced into the cavity through the neck, the pressure inside increases, pushing the air back out. Due to inertia, the air overshoots and creates a pressure lower than atmospheric pressure, drawing air back in. This oscillatory motion creates a resonance at a specific frequency.

The resonator can be modeled as a mass-spring system where:

• The air in the neck acts as a mass (m)
• The air in the cavity acts as a spring (with stiffness k)

The resonance frequency (f) of a Helmholtz resonator is given by:

Where:

• c is the speed of sound in air
• A is the cross-sectional area of the neck
• V is the volume of the cavity
• L' is the effective length of the neck (L + ΔL)
• ΔL is the end correction

The end correction (ΔL) accounts for the fact that the oscillating air extends slightly beyond the physical ends of the neck. For a flanged opening, ΔL ≈ 0.85r, where r is the radius of the neck.

Rearranging the equation:

This shows that $f^2 \propto \frac{1}{V}$, meaning $f \propto \frac{1}{\sqrt{V}}$, which is what we aim to verify.

5. Formula

The primary formula used:

Where:

• f is the resonance frequency (Hz)
• c is the speed of sound in air (≈ 343 m/s at 20°C)
• A is the cross-sectional area of the neck (m²)
• V is the volume of the cavity (m³)
• L' is the effective length of the neck (m) = L + ΔL
• ΔL is the end correction (m)

Speed of sound in air at temperature T (in °C):

For plotting:

From the slope, we can calculate L' and then determine ΔL = L' - L.

6. Procedure

Part A: Verification of f ∝ 1/√V

1. Measure the dimensions of each Helmholtz resonator:
   • Neck diameter (d) using vernier calipers
   • Neck length (L) using measuring scale
   • Calculate the neck area A = πd²/4
   • Measure the internal volume (V) of each resonator
2. For each resonator:
   • Hold a vibrating tuning fork near the mouth of the resonator
   • Try different tuning forks until maximum resonance is observed
   • Record the frequency of the tuning fork that produces maximum resonance
   • Alternatively, use a frequency generator and speaker, varying the frequency until maximum resonance is detected
3. Repeat the procedure for all resonators with different volumes.
4. Record the room temperature to calculate the speed of sound.

Part B: Determining the End Correction

1. Use a single resonator with adjustable volume (e.g., by adding water to change the air volume)
2. Measure the initial volume V₁ and find the resonance frequency f₁
3. Change the volume to V₂ by adding a measured amount of water and find the new resonance frequency f₂
4. Repeat for 5-6 different volumes
5. Plot f² vs 1/V and determine the slope
6. From the slope, calculate L' and then ΔL = L' - L

7. Observation Table

Table 1: Resonator Dimensions

Table 2: Resonance Frequency Measurements

Table 3: Variable Volume Measurements (For End Correction)

8. Calculations

1. For each resonator, calculate:

   • 1/V (m⁻³)
   • f² (Hz²)

2. Plot a graph of f² vs 1/V

   • The graph should be a straight line passing through the origin
   • Calculate the slope (m) of the best-fit line

3. From the slope, calculate L':

   $$ m = \frac{c^2A}{4\pi^2L'} $$ $$ L' = \frac{c^2A}{4\pi^2m} $$

4. Calculate the end correction:

   $$ \Delta L = L' - L $$

5. Verify the theoretical end correction:

   • For a flanged opening: ΔL ≈ 0.85r (where r = d/2)
   • Compare your experimental value with this theoretical value

9. Result

1. The graph of f² vs 1/V shows a linear relationship, confirming that f ∝ 1/√V.

2. The experimental value of the end correction (ΔL) = _____ m

3. The theoretical value of the end correction = _____ m

4. Percentage error = [(|Experimental - Theoretical|)/Theoretical] × 100 = _____ %

5. The relationship between frequency and volume of a Helmholtz resonator has been verified, and the end correction of the neck has been experimentally determined.

10. Precautions

11. Viva Voice Questions