2.2 Cauchy Problem for the Wave Equation MAT 3724 Applied Analysis I 2.2 Cauchy Problem for the Wave Equation http://myhome.spu.edu/lauw

Wave Equation: Set Up Assumptions: 1. 2. 3.

Cauchy Problem

Thought Experiment What would you expect to happen?

Thought Experiment

Thought Experiment

Modeling Wave Propagation in a String

Modeling Wave Propagation in a String

Compare with Advection What are the differences? What are the similarities? What thing do you think we can mimic from the advection equation when solving the wave equation?

Solve the Wave Equation… …without the initial conditions

Solve the Wave Equation… …without the initial conditions

Solution: d’Alembert’s Formula

Classwork Verify d’Alembert’s Formula Show that it satisfy the wave equation (2.12) and the initial conditions (2.13)