A transport equation for the evolution of shock amplitudes along rays

• Main Authors: Giovanni Russo, John Hunter
• Format: Article
• Language: English
• Published: Università degli Studi di Catania 1991-05-01
• Series: Le Matematiche
• Online Access: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/633
• ISSN: 0373-3505; 2037-5298

&lt;!-- @page { size: 21cm 29.7cm; margin: 2cm } --&gt; <p style="font-style: normal;"><span style="font-family: DejaVu Sans,sans-serif;">A new asymptotic method is derived for the study of the evolution of weak shocks in several dimension. The method is based on the <em>Generalized Wavefront Expansion</em> derived in [1]. In that paper the propagation of a shock into a known background was studied under the assumption that shock is weak, i.e. Mach Number <em>=1+O(&epsilon;)</em>, <em>&epsilon; ≪ 1</em>, and that the perturbation of the field varies over a length scale <em>O(&epsilon;).</em> To the lowest order, the shock surface evolves along the rays associated with the unperturbed state.</span></p> <p style="font-style: normal;"><span style="font-family: DejaVu Sans,sans-serif;">An infinite system of compatibility relations was derived for the jump in the field and its normal derivatives along the shock, but no valid criterion was found for a truncation of the system.</span></p> <p style="font-style: normal;"><span style="font-family: DejaVu Sans,sans-serif;">Here we show that the infinite hierarchy is equivalent to a single equation that describes the evolution of the shock along the rays. We show that this method gives equivalent results to those obtained by Weakly Nonlinear Geometrical Optics [2].</span></p>