Wave equation boundary problem

[JI567]

There are not two different solutions to the problem. There is only one solution to the problem, but it is described in two different spatial regions by two separate equations. There is a discontinuity in the solution at x = ct. The region x < ct is described by one equation. The region x > ct is described by the other equation.The boundary condition does not apply to this region of the solution.

It doesn't satisfy the initial conditions because we must have made a mistake in the solution.

Chet

Okay but where do you think the mistake is? I did exactly as it was done in the document you linked here...

 

[JI567]

There are not two different solutions to the problem. There is only one solution to the problem, but it is described in two different spatial regions by two separate equations. There is a discontinuity in the solution at x = ct. The region x < ct is described by one equation. The region x > ct is described by the other equation.The boundary condition does not apply to this region of the solution.

There is no mistake in our solution. The initial condition is at t = 0. What is the region covered by 0<x<ct at t = 0?

Chet

The region covered is

For 0≤x<ct, $Y=\frac{3}{8}(e^{-(x+2t)^2}-e^{-(x-2t)^2})$

so now at t = 0 the whole equation becomes 0...but it should be $\ e^{-x^2} \$

 

[Chestermiller]

Okay but where do you think the mistake is? I did exactly as it was done in the document you linked here...

I've corrected my previous response. Our solution actually is correct. See the change that I made in my response.

Chet

The region covered is

For 0≤x<ct, $Y=\frac{3}{8}(e^{-(x+2t)^2}-e^{-(x-2t)^2})$

so now at t = 0 the whole equation becomes 0...but it should be $\ e^{-x^2} \$

But the region has no spatial extent at t = 0: 0<x<0.

Chet

 

[JI567]

I've corrected my previous response. Our solution actually is correct. See the change that I made in my response.

Chet

But the region has no spatial extent at t = 0: 0<x<0.

Chet

Okay so we just ignore this region as its range is just 0. Ct < x < infinity results in 0<x<infinity when t is 0 which is the semi infinite domain and the question only states initial conditions satisfies the semi infinite domain so we just consider that region for satisfying the initial conditions.

And as for the boundary condition as it states y(0,t) = 0 so now we can't take Ct<x<infinity as t is not 0 so x in this region can't be 0 as a result we ignore this region. Now 0 ≤ x < ct for this range x can have a value of 0 even when t is not 0 so boundary condition applies only to this region.

Is my understanding of this thing correct?

 

[Chestermiller]

Okay so we just ignore this region as its range is just 0. Ct < x < infinity results in 0<x<infinity when t is 0 which is the semi infinite domain and the question only states initial conditions satisfies the semi infinite domain so we just consider that region for satisfying the initial conditions.

And as for the boundary condition as it states y(0,t) = 0 so now we can't take Ct<x<infinity as t is not 0 so x in this region can't be 0 as a result we ignore this region. Now 0 ≤ x < ct for this range x can have a value of 0 even when t is not 0 so boundary condition applies only to this region.

Is my understanding of this thing correct?

Yes.