Solve Laplace Equation Inside Rectangular Pipe

[leonne]

Homework Statement

Find potential inside rectangular pipe.

Homework Equations

Laplace equation

The Attempt at a Solution

This is an example in the book I am confused about one part.They get the general solution as v(xy)=(Ae^kx+ Be^-kx)(Csin(ky)+Dcos(ky)
Now the part i don't understand is getting from the formula to v(xy)=Ccosh(npiex/a)sin(npiey/a)
like they say " we cannot set A=0 ;region in question does not extend to x=infinity On the other hand situation is symmetric with x so A=B than D=0 k=npie/a.I am guessing this has to do with the boundary conditions, but don't fully understand how they get this.

 

[Mindscrape]

Yeah, it comes from the boundary conditions. I'm guessing you have V=0 at y=0 and Y = a. If you use those boundary conditions you'll see that one solution is that C=0 and D=0, but that's the trivial solution, so another solution is to make the sin term itself zero. Exponentials can be written as cosh and sinh.

List your BCs and we can help you a bit more.

 

[leonne]

o ok I think i get it now ill try it out later thxs