Acoustic Stop Bands in a Periodic Waveguide

Andrew White and Yinchuan Yu

Methods of Experimental Physics Spring 2013

University of Minnesota

Abstract

The concept of a periodic acoustic waveguide is defined. A theoretical description of Bloch waves and dispersion is provided. In an analogous method to the Floquet theory, it is argued that the propagation of an acoustic wave in a periodic waveguide can be described by these Bloch waves. An experiment is described in which this hypothesis is tested, and the procedure for analyzing the data from the experiment is set forth. The experiment and analysis can be realistically performed in the time available.

Introduction

Hypothesis: When generated in a waveguide with periodic scattering centers, certain frequencies of acoustic waves will not propagate; there will be stop bands.

Goals: Identify the frequency ranges in which the acoustic waves do not propagate and find the imaginary part of the Bloch wavenumber. Demonstrate that stop bands (like those found in semiconductors) are a natural consequence of wave dynamics in a periodic structure.

Theory

A wave propagating in a periodic structure is described as a Bloch function;

The Bloch wave number, q, describes the wave as it propagates from cell to cell and obeys the dispersion relation (where kwg and ksb are the wave numbers of simple waves in the waveguide and scattering centers, respectively);

When there is an imaginary part of q, the amplitude of the Bloch wave experiences exponential decay and there will be a stop band. The imaginary part of q at each frequency can be found by plotting the natural logarithm of the dimensionless gain of an coustic wave at that frequency against the distance from the mouth of the waveguide; the slope of this line is Im[q].

Im[q_f ]=(∆ ln(mic2/mic1))/∆x

Experimental Setup

The experimental setup is shown as Figure 1.

Figure 1: Microphones are powered by 2.5V power supply and send the signal to SRS810 Lock–in Amplifiers. Speaker is powered by a MPA-200 Power Amplifier. Signal generated by HP 33210A Function Generator. GPIB connected the Function Generator and Lock-In Amplifiers to the computer. The waveguide is shown as Figure 2.

Figure 2: Shown is one period of wave guide, where a=38.1mm, b=25.4mm, d=38.1mm, l=9.5mm and h=100mm.

Data taking was done in LabView; the reference microphone was placed near to the mouth of the waveguide, before the first scattering center. The second microphone was placed from the 2nd port to the 23rd port; for each port, data was taken from 500Hz to 5000Hz at intervals of 5Hz; the output was the ratio of the response of the second microphone over the first microphone.

Results

The dimensionless gain as a function of frequency is shown in Figure 3. Stop bands are obviously seen at seen at approximately 1400-1700Hz, 2000-2500Hz, and 3500-3700Hz.

Figure 3: Dimensionless gain fo mic1/mic2, line in blue is data from prot 6 and line in red is data from port 6.

The prediction and measured value of Im[q] as function of frequency is shown on Figure 4 below.

Figure 4: The above graph compares the plots of the experimental values for Im[q] (blue line) and its theoretical values (red line). Although there are three stop bands in both the experimental and theoretical plots, they are quantitatively dissimilar.

Conclusion

The data clearly shows stop bands with qualitative similarity to the predicted values. However, the values at which the bands begin and end as well as the magnitudes of Im[q] are quantitatively different. Attempts to explain the discrepancy were extensive, but not exhaustive. Future work can be done in exploring the original boundary value problem to verify the theory, and use a longer waveguide to guarantee that the periodic structure dominates the wave propagation.

Acknowledgements

Thanks to Lee Wienkes, Kurt Wick, Dr. Clem Pryke, and Dr. Gregory Pawloski for their support on this project.