Observing a focused weak shock wave perform a Hilbert transform
Miguel A Alonso1,2, Pedro A Quinto-Su3
1Institut Fresnel, Centrale Med, CNRS, Aix Marseille Univ, UMR 7249, 13397 Marseille Cedex 20, France.
Physical Review. E
|June 19, 2026
Summary
Researchers imaged focusing shock waves in liquids. Their profiles before and after focusing form a Hilbert transform pair, explained by the Gouy phase shift.
Area of Science:
- Fluid dynamics
- Wave phenomena
- Optics
Background:
- Focusing shock waves are crucial in various physical phenomena.
- Understanding their spatial profiles is key to predicting wave behavior.
- Previous studies explored wave focusing but lacked detailed spatial imaging in liquid media.
Purpose of the Study:
- To image the spatial profiles of focusing two-dimensional shock waves.
- To investigate the relationship between shock wave profiles before and after focusing.
- To explain the observed spatial profile transformation using optical principles.
Main Methods:
- Utilized shadowgraphy with strobe photography for imaging.
- Created focusing shock waves in a thin liquid gap.
- Analyzed shock wave profiles at different focusing conditions and angles.
Main Results:
- Observed that shock wave profiles before and after focusing approximate a Hilbert transform pair.
- This relationship holds irrespective of focusing geometry (single point or caustic).
- The Hilbert transform behavior is consistent across various directional angles.
Conclusions:
- The spatial profiles of focusing shock waves exhibit a Hilbert transform relationship.
- This phenomenon is attributable to the π/2 Gouy phase shift in two-dimensional focused waves.
- The findings offer insights into wave propagation and focusing in liquid environments.
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