Rarefaction Waves

PDEs · Axiom Academy

Understanding expansion waves in hyperbolic conservation laws 1 When Characteristics Spread Apart For hyperbolic PDEs like Burgers' equation, characteristics carry information about the solution. When the initial data decreases, characteristics spread apart, creating a rarefaction wave. Key Concept: Characteristic Spreading Consider Burgers' equation with initial condition where u decreases. Characteristics have speed u , so: When u L > u R , faster characteristics (left) move away from slower ones (right). In fluid dynamics, spreading characteristics represent expansion of the medium. The solution must "fill in" the gap between diverging characteristics. 2 Rarefaction Fan: Continuous Expansion Waves The rarefaction fan (or expansion fan) is a continuous family of characteristics emanating from a single point, connecting two constant states. Definition: Rarefaction Fan Solution For the Riemann problem with u L > u R , the solution is: This creates a smooth transition region where u varies continuously from u L to u R . The rarefaction fan is a self-similar solution : it depends only on the ratio x/t . The fan spreads linearly in time but maintains the same profile shape. 3 Smooth Connection Between States Watch how the rarefaction wave creates a smooth, continuous transition between the left state u L and right state u R as time evolves. The rarefaction solution divides space into three distinct regions:

This is the written version of the interactive lesson above. See the full PDEs course.