## How to solve a system of Hyperbolic PDE of the second order using the function "hyperbolic" ?

### Michele (view profile)

on 17 Feb 2013

Dear all,

I would like to solve the problem for the propagation of shear sound wave through two media using the hyperbolic function.

The governing hyperbolic equation is in the form u_tt=c^2*u_xx where u=f(x,t) and u_tt and u_xx are the second order PD with respect to time and space. Let's suppose that 0<x<2 and that for 0<x<1 c=2000 %m/s (for example) while for 1<x<2 c= Re+ i*Im (in the case of shear waves the propagation speed of sound in fluids is a complex quantity). Let's suppose the following boundary conditions: u(x,0)=f(x); u(0,t)=0 ; u(2,t)=0; u_t(x,0)=g(x) and continuity boundary condition for displacement at x=1 so that u_medium1=u_medium2.

Known the value of the coefficients c in the first and second medium how can I solve a system of hyperbolic pdes using the function hyperbolic in this form: U1=HYPERBOLIC(U0,UT0,TLIST,B,P,E,T,C,A,F,D)

U2=HYPERBOLIC(U0,UT0,TLIST,B,P,E,T,C,A,F,D)?

how do I implement the boundary conditions stated above (these are just an example anyway)?

Is there a better/more precise way to implement the stated problem?

how can I plot the results in an animated 3d contour plot?

Is it possible to solve the inverse problem with respect to c [given enough boundary conditions, is it possible to obtain the value of c in the second or in the first medium]?

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### Bill Greene (view profile)

on 19 Feb 2013

Hi,

I suggest you start by looking at this example:

http://www.mathworks.com/help/pde/examples/wave-equation-on-a-square-domain.html

The simplest way to define the c coefficient when it varies by region is to define it as a string with the value for each region separated by a !-character. For example, '5.4!2.3' for a two-region model with c=5.4 in the first region and 2.3 in the second.

Bill

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