Speed of Sound in Bubbly Liquids

By: Travis Graham

University of Minnesota (Methods of Experimental Physics, Spring 2015)

Abstract

In this experiment, the speed of sound was measured for bubbly mixtures of gas and water at various gas fractions; the gases used were air and helium. This was achieved by using a hydrophone to observe standing wave frequencies in a large cylinder (a one-dimensional acoustic waveguide) containing water that was injected with gas bubbles. The speed of sound in a bubbly liquid mixture exhibits interesting behavior, described by Wood’s Equation, and varies with the gas volume fraction at low frequencies. The results agreed with the general behavior described by the theory, but there was a noticeable discrepancy in both cases.

Introduction

For a gas-liquid mixture, the physical properties of the fluid cannot be easily explained as a combination of the two phases. If a property depends on multiple parameters of the liquid and gas, behavior that is not very intuitive can arise. For example, the addition of gas bubbles to a liquid greatly changes the mixture’s compressibility, but does not change the density much due to the dominance of the liquid mass. This affects one of the notable physical properties changed by the presence of gas bubbles: the propagation of sound through the fluid. That is, the speed of sound changes with the gas volume fraction of the mixture. One of the interesting attributes of this effect is the immediate drastic decline in the speed of sound with the addition of a small amount of gas bubbles. For instance, if gas bubbles make up just 0.1% of the volume of a column of water, the speed of sound decreases to about 20%-25% of the normal speed of sound in water. The relationship between the speed of sound and the gas volume fraction at low frequencies was described by A.B. Wood in 1930, and this experiment aimed to confirm said relationship.

Theory

The speed of sound in a bubbly liquid is dependent on two factors: the percentage of gas in the mixture, and the size of the bubbles. However, the effects of the size of the gas bubbles can be considered negligible under two conditions. First, the bubbles must be on the order of a few millimeters in radius in order to reduce the effects that surface tension have on the mixture's compressibility. Second, the frequencies of the standing waves, which are key to this experiment, must be lower than the resonance frequency of the bubbles. This is so that the resonance does not interfere with the standing waves, and also so that the mixture will continue to behave isothermally (as it does only at low frequencies; it behaves adiabatically for higher frequencies). For bubbles of radii on the order of millimeters, the resonance frequencies are in the kHz range, meaning standing waves lower than this frequency range will behave as desired. With these two conditions met, the speed of sound depends only on the gas fraction.

The speed of sound in any medium is generally given by the density and compressibility, ρ and κ:

Where ρ and κ are given by their fractional components (with X being the gas fraction, which is given in terms of the change in height and total height of a column of fluid, including a correction term '0.03 (meters)' for the volume taken up by the hydrophone):

Which, with a bit of algebra, gives Wood's Equation(s):

A graphical representation is given below in Figure 1.

Sound speed vs. gas fraction for air (blue/bottom) and helium (red/top). Cut off at 200m/s for ease of viewing. Original figure.

For this model, the known values of the densities and speeds of sound for each component at room temperature and 1 atm were used. It is worth noting that, as can be seen by the miniscule difference between the water-air and water-helium curves, this model is not easily changed, and parameters need to be altered to an extreme extent in order to see tiny changes in the curvature.

As for measuring the speed of sound, standing waves within the water column were studied. When the gas bubbles formed and snapped off of the injector needles, they oscillated and created ambient noise. This noise excited standing waves in the water column, which had frequencies given by:

This displays a linear relationship between mode number and frequency, with the slope given by the speed of sound and the height of the column.

Experimental Setup

The apparatus consisted of a large, water-filled tube standing vertically on a base-plate, which was connected with tubing to a gas source. On the base-plate, at the bottom of the cylinder, were multiple injector needles that released air bubbles into the water. A hydrophone (essentially an underwater microphone) was placed within the mixture, which sent the signal from the pressure waves it detected into a pre-amp, which behaved as both an amplifier and a band-pass filter. The new signal was then sent into a DAQ card and into a computer, where a spectrum analyzer displayed the signal’s frequency domain. A diagram of the full setup is shown in Figure 2 below.

Figure 2: Full experimental setup. The hydrophone was actually quite large in the tube, having a radius of 1.4cm, where the tube's radius was 2.2cm. Original figure.

Using a ping-pong ball to measure the height of the column with and without gas being injected, H and X could be calculated. The spectrum analyzer was used at each gas fraction to look at the frequency domain, which contained linearly-spaced peaks representing the standing wave frequency modes. See Figure 3 below.

Figure 3: frequency domain for 7.69% air. Original Figure.

A few things are worth noting here. First, if the hydrophone was placed around the middle of the column, additional standing waves accumulated between it and the bottom/top of the column, resulting in poor spectra due to overlapping peaks. These additional standing waves disappeared when the hydrophone was placed near the bottom of the column (near the top worked too, but additional noise due to the snapping bubbles on the surface created unwanted background effects). Additionally, the first one or two frequency modes would often be hidden under a large, oddly-shaped peak and very low frequencies (as seen in the figure), which were the result of flow noise. However, since the calculation of the speed of sound only depended on the slope of the frequencies versus mode numbers, any peak could be called "mode 0" so long as the following peaks were sequential and linear.

Results

In order to obtain the speed of sound, the frequencies and mode numbers needed to be linear, giving a slope of c/2H. Plots of frequency versus mode number for each gas fraction for water-air and water-helium are given below in Figures 4 and 5.

Figures 4 and 5: Frequency vs. mode number for air at sixteen gas fractions and helium at six gas fractions. The higher, steeper slopes are the lower gas fractions. Error bars too small to see. Original figure.

The frequencies were then all multiplied by 2H given their respective H values, and a least-squares fit of these values and the mode numbers gave slopes of c. Then, speed of sound was plotted against gas fraction, giving Figures 6 and 7.

Figure 6: Speed of sound vs. gas fraction for air, including the model from Wood's Equations. Most of the error bars too small to see. Original Figure.

Figure 7: Speed of sound vs. gas fraction for helium, including the model from Wood's Equations. Some error bars too small to see. Original figure.

While both plots show that the data follows the same general behavior as the models, there is definitely a source, if not sources, of error causing some discrepancy. For water-air, all of the data (having an absurdly high χ² of 2086) lies above the model, and the shape of the curve seems to be altered a bit, looking “flatter”. For water-helium, the data fits a bit better and the shape is more similar, but the trend is still below the curve (disregarding the “bad” data point, the χ² is 26; better, but not great). In both cases, the error bars very rarely encompass the model. The fact that the error bars are so small and that the trends follow some consistent behaviors point to the possibility that there was some constant source, or sources, of error. That is, the discrepancies are likely due to some unaccounted-for factor that caused consistent offsets and shape changing. This could be a number of things, including: the pressure gradient of the water column affecting the bubble sizes and densities along the column height, the largeness of the hydrophone causing a number of undesirable effects, inhomogeneous bubble spacing due to poor injector setup, and faulty equipment altering the signal fed into the spectrum analyzer.

Conclusion

While the speed of sound versus gas fraction plots of the water-air and water-helium curves followed the expected behavior of the model given by Wood’s Equation, there were noticeable discrepancies in the data that suggest deviance from the theory. Whether this deviance was a parameter that was unaccounted for or a problem caused by the equipment is unknown, though it is likely a combination of effects. Regardless, the data still shows the strange effects on the speed of sound in bubbly liquid, and there are multiple steps future experimenters could take in order to find better agreement with the theory.

Moving Forward

This experiment was performed only within the span of a few weeks, and could be expanded on greatly. First and foremost, the discrepancies encountered could be tested and better data could possibly be obtained if the hydrophone was smaller and if the bubble injectors were more effective. The hydrophone was very large, and not only could it have changed the effective column height by allowing waves to propagate through it, but its mere presence and swaying motions could have easily effected the spacing of the frequency peaks. So, a smaller (preferably rigid) hydrophone would hopefully eliminate these effects. The bubble injectors, on the other hand, were fairly corroded and were questionably spaced, making the bubble injection process very inhomogeneous near the bottom of the column. Making sure these needles are new/clean, adding a few more of them, and potentially having them not protrude so far upwards would help relieve the effects of inhomogeneous bubbles.

Lastly, in order to further prove the theoretical models, more liquids and gases could be used in a variety of combinations in order to prove the accuracy of Wood's Equation across many bubbly liquids. Additionally, for demonstration purposes, a speaker system could be set up to the output signal from the pre-amp, allowing the change in the speed of sound to actually be heard when adjusting the gas fraction.

References

Main references:

• 1. Wilson, Preston S., and Ronald A. Roy. "An Audible Demonstration of the Speed of Sound in Bubbly Liquids." American Journal of Physics. Vol. 76. No. 10. 2008

  2. !McWilliam, D., and R. K. Duggins. "Speed of Sound in Bubbly Liquids." Proceedings of the Institution of Mechanical Engineers. Vol. 184. No. 3. SAGE Publications, 1969.

  3. Wood, Albert Beaumont. “A Textbook of Sound.” G. Bell and Sons, 1964.

Additional references:

• 1. Carey, William M., and James W. Fitzgerald. “Low Frequency Noise from Breaking Waves.” Natural Physical Sources of Underwater Sound. 1993.

  2. Kieffer, Susan Werner. "Sound Speed in Liquid-Gas Mixtures: Water-Air and Water-Steam." Journal of Geophysical Research. Vol. 82. No. 20. 1977.

  3. Yang, Xinmai, Ronald A. Roy, and R. Glynn Holt. “Bubble Dynamics and Size Distributions during Focused Ultrasound Insonation." The Journal of the Acoustical Society of America. Vol. 116. No. 6. 2004.

  4. Minnaert, M. "XVI. On musical air-bubbles and the sounds of running water." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 16.104 (1933): 235-248.

  5. Haddle, G. P., and E. J. Skudrzyk. "The physics of flow noise." The Journal of the Acoustical Society of America 46.1B (1969): 130-157.

Thanks to:

Professor Clement Pryke and Professor Elias Puchner for their instruction and assistance throughout the experiment.