Introduction: Design and Characterization of a Handmade Helmholtz Resonator
This project details the acoustic characterization of a Helmholtz Resonator built from common, low-cost materials. Using pink noise and FFT analysis, the system was confirmed to function as a highly effective narrowband passive bass trap. This performance highlights the significant energy dissipation caused by the resonator's flexible cardboard structure.
Supplies
Cardboard panel measuring 1 x 0.8[m]
Cardboard tube 8 cm long and 4 cm in diameter
Hot glue gun
Set of ruler, set square, and protractor
Pencil
Box cutter
Adhesive tape
Speaker
Microphone
Laptop
Step 1: Design of the Box Dimensions
For the box dimensions, I opted for measurements that were large enough to allow for easy assembly, but without surpassing certain limits that would result in a resonant frequency too low (sub-bass) and outside the audible spectrum. This estimation is clear when observing the effect of volume on the resonant frequency formula [a larger volume yields a lower resonant frequency].Similarly, the formula confirms that the larger the area of the tube (neck), the higher the resonant frequency will be.
Based on the previous estimations, I chose to fabricate a cardboard box with dimensions of 30 x 20 x 26 {cm}. I paired this with a cardboard tube, sourced from a standard toilet paper roll, having a diameter of 4 {cm} and a length of 8 {cm}, as this material is easily accessible at home.
Step 2: Theoretical Calculation of the System's Resonant Frequency
The resonant frequency is calculated using the formula attached in the image above; its parameters are the following ones:
- c (Speed of Sound): The value 343 m/s is used, which is the speed of sound in air at 20ºC
- A (Area of the Neck's Cross-Section): This is the cross-sectional area of the resonator's neck, measured in square meters.
- V (Volume of the Cavity/Box): This is the volume of the main body or cavity of the resonator, measured in cubic meters.
- Leq (Effective Length of the Neck): This is the physical length of the neck plus a correction factor for the air vibrating outside the neck (end correction), measured in meters.
The resonant frequency is determined by the box's dimensions, knowing these values makes it possible to easily calculate all parameters, as attached in the image, resulting in a theoretical resonant frequency of 47 Hz.
Step 3: Assembly
The image details the assembly steps I followed, which are explained below:
1. I opted for a single-piece design to minimize the number of seams between walls, which could otherwise lead to air leaks . For this purpose, I used a cardboard sheet measuring 1 x0.8 {m}, on which I traced the required dimensions using a ruler, set square, and protractor.
2. Once the box's silhouette was drawn onto the sheet, it was time to cut it out using a utility knife. I then made the necessary folds and secured the end walls using hot glue, leaving the box open only on its top face.
3.To reinforce the proper sealing of the box, I applied a layer of adhesive tape over the segments that were glued with silicone.
4. This step involves creating a circular opening in the center of one of the smaller faces of the box to facilitate the sound's directionality . This opening must have the same diameter as the tube we intend to insert. Again, after fixing the tube with silicone, we will use adhesive tape. The reason for leaving the top face open was to subsequently seal the inner edges of the tube section from the inside as well. Once this step is complete, we can close the top face of the box in the same manner as the others.
Result: The interior of our box is now completely isolated from the exterior, except for the neck area, meaning the sole path for air exchange with the outside occurs through that section.
Step 4: Experimental Measurements: Pink Noise Analysis
In this section, we will obtain the frequency response of the system using pink noise.
To achieve this, we will position the box so that the mouth of the neck (tube) is facing the microphone, at a distance of approximately 8–10 cm. Simultaneously, we will apply the pink noise directly over the neck using a speaker with a good low-frequency response, given that the theoretically calculated resonant frequency of our box is around 50 Hz.
We will also measure the pink noise without the box system present and compare both signals collected by the microphone. This comparison will be done by performing a Fast Fourier Transform (FFT) on both signals to determine their amplitude at each frequency. The Matlab code, the two captured sound files (with and without the box), and a screenshot of both FFT-processed signals are attached in this section.
Observing the results, we can state that the largest attenuation produced is 35.27 dB, and it occurs at 54.6 Hz. Therefore, this is the measured resonant frequency of the system.
Step 5: Comparison and Explanation of Results
The practical result obtained (54.6 Hz) differs by 7.6 Hz from the theoretically calculated frequency (47 Hz). This difference demonstrates that the system functions as a resonator and that the characterization process was carried out correctly.
However, the significant amplitude attenuation provided by the system—over 30 dB—can largely be attributed to the construction material:
- Material Rigidity and Energy Dissipation: Although the material choice (cardboard) was made for accessibility and easy shaping, cardboard is not sufficiently rigid. This lack of rigidity causes a large part of the resonant energy to be dissipated through the box walls as they slightly flex, which reduces the peak performance.
- Imperfect Air Exchange (Wavelength Interaction): Furthermore, the long wavelength of the internal resonant frequency does not entirely exit back through the neck. This phenomenon also contributes to the observed attenuation of the signal at the resonant frequency.
For these reasons, the box is effectively functioning as a Bass Trap.
