Using rays to describe spinning sound

Chirag Gokani – chiragokani@gmail.com

University of Texas at Austin, Applied Research Laboratories and Walker Department of Mechanical Engineering, Austin, Texas, 78766-9767, United States

Michael R. Haberman; Mark F. Hamilton (both at Applied Research Laboratories and Walker Department of Mechanical Engineering)

Popular version of 5pPA13 – Effects of increasing orbital number on the field transformation in focused vortex beams
Presented at the 186th ASA Meeting
Read the abstract at https://eppro02.ativ.me/web/index.php?page=IntHtml&project=ASASPRING24&id=3657527

–The research described in this Acoustics Lay Language Paper may not have yet been peer reviewed–

When a chef tosses pizza dough, the spinning motion stretches the dough into a circular disk. The more rapidly the dough is spun, the wider the disk becomes.

Fig 1. Pizza dough gets stretched out into a circular disk when it is spun. Source

 

A similar phenomenon occurs when sound waves are subjected to spinning motion: the beam spreads out more rapidly with increased spinning. One can use the theory of diffraction—the study of how waves constructively and destructively interfere to form a field pattern that evolves with distance—to explain this unique sound field, known as a vortex beam.

Fig 2. The wavefronts of vortex beams are helical in shape, like the threads on a screw. Adapted from Jiang et al., Phys. Rev. Lett. 117, 034301 (2016).

 

In addition to exhibiting a helical field structure, vortex beams can be focused, the same way sunlight passing through a magnifying glass can be focused to a bright spot. When sound is simultaneously spun and focused, something unexpected happens. Rather than converging to a point, the combination of spinning and focusing can cause the sound field to create a region of zero acoustic pressure, analogous to a shadow in optics, between the source and focal point, the shape of which resembles a rugby ball.

While the theory of diffraction predicts this effect, it does not provide insight into what creates the shadow region when the acoustic field is simultaneously spun and focused. To understand why this happens, one can resort to a simpler concept that approximates sound as a collection of rays. This simpler description, known as ray theory, is based on the assumption that waves do not interfere with one another, and that the sound field can be described by straight arrows emerging from a source, just like sun rays emerging from behind a cloud. According to this description, the pressure is proportional to the number of rays present in a given region in space.

Fig 3. Rays in a vortex beam. Adapted from Richard et al., New J. Phys. 22, 063021 (2020).

 

Analysis of the paths of individual sound rays permits one to unravel how the overall shape and intensity of the beam are affected by spinning and focusing. One key finding is the formation of an annular channel, resembling a tunnel, within the beam’s structure. This channel is created by a multitude of individual sound rays that are converging due to focusing but are skewed away from the beam axis due to spinning.

By studying this channel, one can calculate the amplitude of the sound field according to ray theory, offering perspectives that the theory of diffraction does not readily reveal. Specifically, the annular channels reveal that the sound field is greatest on the surface of a spheroid, coinciding with the feature shaped like a rugby ball predicted by the theory of diffraction.

In the figure below from the work of Gokani et al., the annular channels and spheroidal shadow zone predicted by ray theory are overlaid as white lines on the upper half of the field predicted by the theory of diffraction, represented by colors corresponding to intensity increasing from blue to red. The amount by which the sound is spun is characterized by ℓ, the orbital number, which increases from left to right in the figure.

Fig 4. Annular channels (thin white lines) and spheroidal shadow zones (thick white lines) overlaid on the diffraction pattern (colors). From Gokani et al., J. Acoust. Soc. Am. 155, 2707-2723 (2024).

 

As can be seen from Fig. 4, ray theory distills the intricate dynamics of sound that is spun and focused to a tractable geometry problem. Insights gained from this theory not only expand one’s fundamental knowledge of sound and waves but also have practical applications related to particle manipulation, biomedical ultrasonics, and acoustic communications.