Three-scale asymptotics for a diffusion problem coupled with the wave equation

Abstract

In this article, we present an asymptotic analysis of waves of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load. The problem under consideration couples the wave equation and the heat equation, and the asymptotic approximation of the solution requires three-scaled variables. The asymptotic approximation is supplied with a rigorous remainder estimate and is illustrated numerically.

Keywords:

• Thermo-elastic waves
• Asymptotic analysis
• Rapid thermal load
• AMS Subject Classifications: 41A60
• 74F05
• 74J40

Notes

This is especially pronounced for the materials with relatively high thermal diffusivity, for example, germanium (κ = 0.31 cm²s⁻¹), gallium nitride (κ = 0.43 cm²s⁻¹) or gallium arsenide (κ = 0.31 cm²s⁻¹) widely used in modern electronics applications (see, for example, http://www.carondelet.pvt.kl2.ca.us/Family/Science/GroupIVA/germanium.htm). The speed a of dilatational waves in germanium is around 5400 m/s. Hence, one can deduce that even if the duration of thermal loading is of the order 10⁻¹² of a second, such a time interval should not be treated as a negligibly small quantity if represented in terms of the normalized time-variable τ = a ² t/κ.