A detailed analysis about penumbra caustics
Résumé
A penumbra caustic is an interrupted fold caustic. It looks like a fold caustic but with a finite size. This kind of caustic results from the focusing of a semi-infinite concave wavefront. The pressure around the extremity of the penumbra caustic can be expressed analytically in terms of incomplete Airy function for linear monochromatic waves. Using asymptotic expansions in the vicinity of the extremity, that classical result is rederived. It can be matched with the classical Fresnel diffraction before the extremity and with the classical diffraction catastrophe theory after the extremity. Nevertheless the linear modeling is not valid for incoming shock waves. A theoretical description of the phenomenon of focusing of shock waves at a penumbra caustic is given. It relies on the Zabolotskaya-Khokhlov equation. Numerical simulations are used to compute the behavior of this phenomenon. In particular, the numerical simulations show the presence of a triple point inside the pressure field. Finally, the theory and the numerical simulations are applied to explain the apparent paradox of non-causality around fold caustic.