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Physical Model for Sound Speed and Attenuation in Gas Mixtures
In a gaseous medium, the molecules exchange energy via collisions. A complex process where the molecules change state upon collision can be described theoretically by quantum-mechanical inelastic scattering theory. The interaction is governed by an intermolecular potential typically characterized by a long-range attractive part and a short-range repulsive core." The attraction forces arise from the electric fields induced in each molecule by their rapidly fluctuating electric dipoles. At very small separations, the electronic clouds of the collision pair overlap, leading to strong, short-ranged repulsive forces, which are presently not fully understood. The usual potential energy function for non-polar molecules is the Lennard-Jones or 12-6 potential characterized by two force constants, namely the depth of the potential well, ε, and the zero-potential point (or collision diameter), σ. The probability of transition between molecular vibrational levels has a strongly temperature-dependent exponential part describing the inelastic exchange of energy with translation, and a pre-exponential part, virtually insensitive to temperature, that characterizes the vibrational transitions. We have co-developed and refined a first-principles model, where we consider bimolecular collisions.
The key feature of the refined model is an algorithm that calculates the force constants by iteratively fitting of the Lennard-Jones potential for each energy exchange process, whether interspecies or intraspecies.
Subsequently, a set of relaxation equations involving the vibrational and translational temperature fluctuations, coupled with the equations of linear acoustics yield an effective wave number, which fully characterizes the excitable gas mixture. This frequency-dependent wave number also depends on the static sound speed, the translational specific heat, and the specific heat of each vibration mode. The sound speed and attenuation are then determined from the real and imaginary parts of this effective wave number, respectively. The model can be applied to any number of polyatomic gases, provided the molecular thermodynamic and spectroscopic properties are known. (A more complete description should include molecular rotational degrees of freedom via V-R and R-R relaxation processes; I will address this in the future.)
We have found very good agreement between our model predictions and experimental data for various mixtures of N2, O2, CO2, CH4, C2H4, and air. (The measurements were done in conjunction with Commercial Electronics in Broken Arrow, OK.) We published the results here.
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Acoustic Absorption and Sound Speed in Alien Atmospheres
Planetary atmospheres are intriguing dynamic fluid systems in continuous interaction with the solar radiation and the surface. The sensitivity of sound wave propagation to pressure, temperature, and composition variation in a planet's atmosphere can be fully exploited in descent-phase atmospheric sensing. Moreover, since acoustic perturbations are sustained by the gaseous medium itself, they can be used to probe the structure and dynamics of an atmosphere both passively (listen mode) and actively (transmit-receive mode). In the 1980s, two Soviet Venera spacecraft carried microphones in the hope of detecting thunder signatures on Venus. The recorded data was inconclusive, most probably generated by air flowing past the lander. It wasn't until the late 1990s that another attempt to launch a microphone on board a planetary probe was made, on board the unfortunate Mars Polar Lander, lost during descent in the Martian atmosphere in September 1999. On January 14, 2004, as part of the Cassini-Huygens mission to Saturn, the Huygens lander made a historical splash-down on Titan, during which it broadcast the first sounds of an alien world in over two decades. The Huygens Atmospheric Structure Instrument carried microphones for recording ambient sounds and potential lightning events, while the Surface Science Package incorporated active acoustic sensors for measuring surface topography, sound speed, altitude, wind speed, as well as the surface acoustic impedance of the landing zone. The Huygens data obtained is currently being analyzed, and more work will be needed to understand Titans environment. The successes of recent planetary exploration missions such as Cassini-Huygens, Venus Express, and the Mars Exploration Rovers herald a new era in the exploration of our solar system. As the data relayed back to Earth by Huygens has shown, acoustics can play an ever-increasing role in gathering information about alien environments.
I am using the model described above to predict the frequency-dependent sound speed and attenuation in the surface atmospheres of Titan, Venus, Mars, and Earth, as well as the characteristic acoustic impedance at each planet's surface. These quantities are shown in the figures below.
The characteristic acoustic impedance
Zacof a medium is a very important parameter to consider in the development of acoustic transducers for various fluid media: it dictates the acoustic insertion losses occurring at the emitter-fluid and fluid-receiver interfaces. Assuming plane waves, Zac = &rho0c, where &rho0is the ambient density and cis the speed of sound. The acoustic impedance depends on frequency via the frequency-depenent sound speed.
Using available data for the vertical atmospheric profiles of temperature, density, and pressure, the model can be used to calculate the dependence of sound speed and attenuation with altitude for the four planetary bodies. The sound speed and attenuation profiles at 15 kHz for the four planets are shown below.
More details about the application of the molecular acoustics model to predict the acoustic properties of Mars, Venus, and Titan's atmospheres can be found in this paper.
Divertimento: Music on Mars, Venus, Titan, and Earth
The attenuation-frequency plots show that the different environments of Mars, Venus, Titan, and Earth "filter" sounds differently. This is shown in this audio file. The file repeats a soundbite three times, spaced a few seconds apart. The three parts correspond to passing the soundbite through 1) the "Earth filter", 2) the "Titan filter", and 3) the "Venus filter". The "Mars filter" is much too lossy due to the tenuousBach's Toccata and Fugue in D Minor:
CO2-based atmosphere, so I didn't include it. (The Venus part is relatively weak but also the most interesting, so you might want to increase the volume.)
Below are samples of a fragment of Bach's masterpiece, processed with the molecular acoustics model and propagated to a distance of 50 meters on Mars, Venus, Titan, and Earth. Before running the processor, I adjusted the pitch based on the different sound speeds (resonance frequencies of organ pipes being proportional to the sound speed).
Bach on Earth
Bach on Titan
Bach on Venus
Bach on Mars
Reality-Based Immersive Soundscapes
Immersive sound is a relatively new area of computer simulation, devoted to creating a 3D acoustic environment. Computer science and audio engineers are continuously developing the hardware and software needed to implement a 3D listening experience. The current state-of-the-art in 3D audio perception is obtained by a mix of computer and audio engineering with psychological acoustics - the branch of acoustics that studies how the brain perceives and processes sounds. The common thread passing through the current technologies is the focus on multi-channel audio rather than the realism of the soundscape. On the educational front, documentaries about other planets abound yet they lack realism when it comes to sounds. For instance, Mars is often shown as a dusty and quiet place. However, low-frequency sound can propagate quite a long way on the Red Planet and their sources abound: meteorites, electric discharges, dust devils, dust storms, etc. Saturn's moon Titan may be awash in sound - from the gurgling of liquid hydrocarbon-spewing volcanoes, to methane-falls, to lightning, to winds, to off-world landers splashing down in hydrocarbon melt... But space science missions have so far been effectively deaf: we practically have no idea what other worlds sound like!
Beyond the lighter, fun side of seeing what a soundbite sounds like on other planets lies, of course, the realistic layer, where the listeners would be able to experience (close-to-)real alien soundscapes.
By mixing several channels of audio (each channel corresponding to an acoustic "event" accurately propagated to the listener's location), the model described above would be able to realistically simulate previously unknown soundscapes, enabling the listener to immerse herself/himself in a realistic 3D acoustic world, whether in the thick atmosphere of Venus, the tenuous environment of Mars, or the cold, liquid-hydrocarbon world of Titan. These characteristics may help bring acoustic realism in documentaries, sci-fi films, games, as well as 3D-immersion movies developed for planetaria.
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Acoustic Gas Sensing
Toward quantitative acoustic relaxation spectroscopy
Detailed studies of molecular interactions have benefitted greatly from the development of laser spectroscopy techniques which can provide unrivaled information on the structure and dynamics of molecules. However, critical applications such as gas monitoring in life-support systems often require sensors that are not only fast and sufficiently accurate but also rugged and not needing extensive maintenance and calibration. Gas-coupled ultrasonic devices are commonly used in gas sensing, relying on changes in the sound speed. Measuring sound speed alone, however, does not confer the ability to detect and identify contaminant molecules - it only shows changes in the mean molecular weight. For reasons unknown, researchers have habitually not considered using acoustic attenuation in addition to sound speed. Beside the classical sound loss mechanisms in fluids due to viscosity, heat conduction, and diffusion, the relaxational contribution to attenuation arises from the inability of internal molecular degrees of freedom to follow the acoustic temperature fluctuations. This connects acoustic attenuation to the molecular relaxation times. In the laboratory, acoustic studies of molecular relaxation in fluids require that the ambient pressure be varied over a wide range at a given frequency in order to cover the relaxation processes. Needless to say, this would be highly impractical in a fast-sensing device.
We have developed a novel algorithm that "synthesizes" the full frequency dependence of the effective specific heat based on knowing the sound speed and attenuation at a set pressure and two frequencies only.
As we show in the paper, this would enable a new generation of "smart" acoustic sensors to infer the concentration, molecular weight, and geometry of contaminant molecules in a base gas.
(Could this be the end of the mantra that acoustics cannot provide quantitative gas analysis?)
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Pattern Formation in Colloidal Suspensions
Milk Patterns in a Coffee Cup
I took a time-lapse movie showing the formation of a central milk pattern in a cup of coffee, compressing ~6 hours into ~6 min (the sound is annoying and irrelevant so you may want to turn it down). I have also posted a high-resolution (but slow-loading!) version. The coffee volume is tapered from D= 4 cm (bottom) to 6cm (top), with a height of ~ 4 cm. The colloidal suspension (milk) first seems to aggregate more or less uniformly (at least at the coffee surface). After about 1 hour, thin "tendrils," probably of colloidal milk fat, appear on the surface and a white patch is slowly being "woven." The patch then evolves toward the center of the cup, being continuously "anchored" (fed?) by the tendrils.
Is this predominantly a mixing-segregation problem?
What happens beneath the surface? Does convection play a center role throughout the experiment, given the long evolution time scales observed?
Do the milkfat particles (fat globules) tend toward a jammed state? What about the coffee particles?
Are casein micelles aggregating as well? (Maybe this could be tested using skim milk.)
Electrostatic interactions? (What happens if one adds NaCl?)>
Dependence on container geometry?
For more discussions and various views on pattern formation in a coffee cup, see also Thomas Witten's "Coffee Caustics" website.
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Various data files (Warning: Due to their sizes, certain files' life here may be fleeting. I can post data upon request.)
Acoustic sound speed and attenuation predictions on Earth, Mars, Venus, Titan.
(Warning: Large Files ~65 MB each !!)
There are four fields in each text file, as follows: frequency (Hz), speed (m/s), alpha*lambda, alpha (1/m)
earth mars venus titan
Audio Files for Earth, Mars, Venus, Titan:
AT: Boosted Audio Files - 3 dB Gain
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