Change in Reverberation Due to Acoustic Damping

Beljeanne Forester and Jacquelyn Hoseth

Methods of Experimental Physics

Introduction

The reverberation of an aperiodic transient sound in a closed non-rectangular box was characterized, with and without acoustic dampening material, by examining the frequency response and reverberation time. Work by previous MXP students was tested on a smaller scale, and on the same or larger scale. Results are compared to relevant theory and related empirical data. The acoustic dampening material yielded a steeper exponential curve fit in the output waveform’s decay as a function of time, and hence, a shorter reverberation time. We predicted that lower frequencies will yield longer reverberation times when we examine our results in the frequency domain.

II. Theory

The standard reverberation time is the time it takes for the echoes to drop in amplitude by 60 dB, or to one-millionth of the initial sound. To estimate the reverberation time, the echoes are fitted by an exponential. We can denote the relationship between the voltage received V and source voltage V_0 with the exponential in the equation below:

Where T is the reverberation time, a is a scaling constant, and t is the time interval. This equation is then evaluated for which corresponds to the 60 dB drop below the original sound level, which then gives the scaling constant a as such:

Having solved for a, equation 2 can now be solved for reverberation time T:

The reverberation time T fits the data with an approximate exponential.

In the frequency domain, the output waveform generated after a single pulse of sound travels through the system would be expected to be constant over all frequencies. However, in practice, some frequencies will be attenuated. In particular, we predict that lower frequencies will have longer reverberation times than higher frequencies.

This is due to the fact that materials like air and wood absorb higher frequencies more efficiently than lower frequencies. Also, longer wavelengths in lower frequencies allow for diffraction around obstructions in the wave’s path.

III. Apparatus & Data Analysis

[Figure 1]: Image of closed wooden box with non-parallel sides

The transient pulse is generated using a pulse generator and function generator in succession. The pulse generator controls the length of the pulse and the time between pulses. The function generator is set in burst mode so that it only emits waves in pulses; this is then fed through a speaker into the box. That wave is reflected off the sides of the box and received at the microphone, where it is sent through a pre-amplifier and received by a National Instruments DAQ driver and sent into the LabView program. The LabView program created for this experiment displays both the output waveform received by the microphone and the input waveform and transfers the data to a comma separated variable file to be imported in Matlab. Matlab was used to complete the exponential fit and Fourier transform required to examine our results in the frequency domain in addition to fitting the exponential decay of the output waveforms in the time domain.

[Figure 2]: schematic of our experimental apparatus

In order to fit the decay of sound, the natural log of the square of the decay was taken and any points that were claimed at ‘infinity’ were eliminated in order to ensure a good fit. That result was fit with a linear polynomial, and that gave us the exponential decay of the reverberation. The error for our fits is coming mostly from the time interval, which is what the error will be based significantly on.

IV. Results & Analysis

In our experiment the decay was measured and fit as an exponential to study exactly how the sound decays inside a closed system. From this we estimate the reverberation time. Below is an example of how the decay changes after adding padding (black: no padding; red: all sides padded)

[Figure 3]: V(t) example plot of output waveforms with and without padding (1kHz)

The plot to the below shows the reverberation time as a function of the input frequency.

We observed that when no padding is used in the box, lower input frequencies tend to have a longer reverberation time than higher frequencies.

However, we also observed that when padding is used in the box, reverberation time still decreases, but the input frequency dependence is not as prevalent. Longer wavelengths of low-frequency waves allow the wave to diffract more easily around obstructions in its path

[Figure 4]: T(f) plot of reverberation time as a function of frequency with and without padding

Without padding, the output waveform has a strong frequency component near 2 kHz. However, with padding, the output waveform has stronger components of low frequencies in comparison to the 2 kHz component.

The plots below exemplify how waves of higher frequencies are absorbed more efficiently than that of lower frequencies.

[Figure 5]: P(f) example plot of Fourier transform for 2 kHz pulse with no padding

[Figure 6]: P(f) example plot of Fourier transform for 2 kHz pulse with no padding

V. Conclusion

We confirmed that reverberation time decreases as input frequency increases. We also found that without padding, the output waveform has a strong frequency component near its input frequency. However, with padding, the output waveform has stronger components of lower frequencies in comparison to the input frequency. This leads us to conclude that waves of higher frequencies are absorbed more efficiently than that of lower frequencies, which was expected because the longer wavelength of lower frequency waves can diffract more easily around obstructions in their paths.

For future experimenters it might be interesting to investigate paths and reflections of waves within the box to analyze and identify echoes. They could also experiment with different padding materials and calculate absorption coefficients. Also, as a note to future experimenters, make sure that your pulse width is significantly smaller than the time it will take sound to travel across the space that your measuring. If you don't, the echoes will become complicated. Also, this project is great for an MXP II project, because the data taking time is very short and the data analysis is reletively simple compared to other groups' data analysis.

VI. References

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