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tions predict an acoustic pressure that is proportional to the square root of the driving frequency (of a biased system). More recent models of CNT thermophones in free space were published by Xiao et al. (2008), Vesterinen et al. (2010),
ations and Aliev et al. (2013). Each of these models differ slightly, but under the same basic assumption of a negligible active- element heat capacity, the far-field acoustic pressure is lin- early proportional to its frequency and reduces to
appears to be limited by the properties of air rather than the active element.
To complicate things further, it is difficult to provide a com-
parison of thermophone performance to that of convention-
al transducers. Most transducers operate within a region of
essentially constant efficiency no matter what power is pro-
vided to the device. Thermoacoustic devices are completely
different, and the conversion process is similar to that of
a car engine or power plant that has a behavior limited by
Carnot’s cycle in which the efficiency is dependent on the temperature difference between the “hot” fluid and “cold” reservoir. Therefore, so long as the background temperature of the “cold” reservoir surrounding a thermophone is main- tained, an increase in input power provides a proportional increase in efficiency. For acousticians, this translates to a 6 dB increase in sound pressure level (SPL) for each doubling of input power as opposed to the 3 dB increase seen in con- ventional devices.
As discussed, thermophone efficiency may be greatly in- creased by operating at a higher frequency and higher pow- er and by creating resonant devices. However, efficiency still remains orders of magnitude lower than in conventional devices, and each of these requirements limits application potential. It is for this reason that commercialization of ther- mophones at this point has been stymied.
Thermophone transducers generate acoustic signals by modulating the temperature of an active element via Joule heating. Heat transfer to gas adjacent to the element causes thermal rarefactions and compressions producing an acous- tic wave. The historical origins of these transducers are en- tangled with the invention of Bell’s telephone and scientific observations that followed, leading to the development of photoacoustic spectroscopy, thermoacoustic engines, and thermoacoustic refrigeration. Arnold and Crandall (1917) developed the theoretical foundation for sound projection by thermophones that enabled their use as a precision source of sound for microphone calibration. Decades went by with relatively few developments in thermophone technology un- til highly porous nanoscopic materials such as porous doped silicon and carbon nanotubes were utilized in the late 1990s and 2000s, respectively. The discovery that such materials significantly improve thermophone efficiency has led to a resurgence in interest as well as new theoretical models and potential use cases.
2√2𝑟𝑟𝑐𝑐 𝑇𝑇
, ./0
𝑓𝑓𝑃𝑃 (1)
 where p is the acoustic pressure at a distance r from the mono- pole source that is driven with an electrical input power Pel producing an acoustic signal at frequency f in an ambient gas- eous environment at temperature Tamb with gas-specific heat capacity at constant pressure cp. Therefore, in the “absence” of the thermophone active element, the thermoacoustic trans- duction process is only determined by the drive power, fre- quency, and properties of the surrounding gas.
These formulations are limited in scope to sources small with respect to the acoustic wavelength and the ability of the thermophone to maintain its background ambient tempera- ture. At high power, the acoustic pressure will thermally sat- urate as the background temperature approaches the surface temperature on the heater, eventually causing the active ele- ment to degrade, either burning or melting in extreme cases (Aliev et al., 2014a). Both Xiao et al. (2008) and Aliev et al. (2015) have proposed that the difference in the frequency dependence between various thermophones is due to a more substantial heat accumulation of the active elements used in the early 20th century compared with the CNTs often uti- lized in thermophones today.
Models for the far-field acoustic pressure of encapsulated thermophones are studied less and, although few exist, none appear robust enough to predict the performance of devices that vary significantly in dimension or housing composition. Even fewer models of underwater thermophone projectors exist, an area much in need of development for any practical implementation of thermophones in naval applications.
The largest criticism of thermophones by far is their low efficiency. People often think, “can you just make a better nanomaterial that converts heat to sound more efficiently,” but often it isn’t the element itself that is the problem. At the current stage of development, the efficiency of a ther- mophone open to its environment (i.e., not encapsulated)
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