Help with Ray-tracing and Extensions

[rexasaurus]

1. I am looking for some examples on how to perform ray-tracing and extensions for a function.

u_{tt}=u_{xx} for 0<x<1 with homogeneous Dirichlet conditions (i.e., the boundary conditions are that u(t,0) and u(t,1) are 0 for all t). As initial data we assume that u(0,x)= x for 0 < x < 1/3 and = (1-x)/2 for 1/3 < x < 1 and that u_t(0,x)=0 for every x.

Extend the initial data as odd across x=0 and across x=1 and use the d'Alembert formula to compute u(3,1/2).

Use `ray tracing' starting with your answer to (a) (with reflection at the boundaries) to compute u(3,1/2).


2. D'Alambert Formula u(t,x)=f(x+ct)+g(x-ct)



3. I used D'alambert formula to solve for u(t,x) for the two ranges of "x". I created a graph plotting "u" not sure where to go from there.

 

[lippyka]

From here, you can use the ray tracing technique to trace the solution of u(t,x) as it propagates through the domain. This involves creating a graph with the initial data plotted, and then following the solution as it reflects off the two boundaries, eventually arriving at the desired point (in this case u(3,1/2)).