Boundary Value Problem from Laplace's eq (Thermal)

[lanan]

Homework Statement

A rectangular plate extends to infinity along the y-axis and has a width of 20 cm. At all faces except y=0, T= 0°C. Solve the semi-infinite plate problem if the bottom edge is held at
T = {0°C when, 0 < x < 10,
T = {100°C when, 10 < x < 20.

Homework Equations

∇²T=0

The Attempt at a Solution

I generalized Laplace's equation for two dimensions, then used separation of variables to find solutions of the form T(x,y)=X(x)Y(x) such that
Y = {e^ky , e^-ky }
X = {sinkx, coskx }

Then, by applying two of the initial conditions required to satisfy T (y→∞ T=0 and x=0 T=0) I was able to eliminate one of the possible choices for X and Y each, leaving a T of:

T=e^-kysinkx.

Here is where I'm having trouble. In other problems from this section I could continue to apply the boundary conditions and solve for k then use a Fourier series approximation to solve the remaining condition where T=100°C. However, with x broken into several more cases I am unsure how to solve for a k that works for all of the conditions (for which there are a confusing six, rather than the previous 3 I solved). Any guidance would be much appreciated.

 

[Chestermiller]

Hi lanan. Welcome to Physics Forums.

I don't consider myself an expert on this type of problem, but the first thing I would do would be to shift the origin to x = 10, so that the region of interest is from x = -10 to x = + 10. The base temperature would then be 0 from x = -10 to x = 0, and 100 from x = 0 to x = +10. I would then represent the solution as the linear superposition of two solutions. In solution 1, T = 50 over the entire base from x = -10 to x = +10. This could be described by a cosine series. In solution 2, the base temperature would be T = -50 from x = -10 to x = 0, and T =+50 from x = 0 to x = +10. This could be described by a sine series. This approach should simplify things considerably (I think).

Chet

 

[lanan]

In solution 1, T = 50 over the entire base from x = -10 to x = +10. This could be described by a cosine series. In solution 2, the base temperature would be T = -50 from x = -10 to x = 0, and T =+50 from x = 0 to x = +10. This could be described by a sine series.

Chet

Thank you, this is helping a little. However, this is my first time actually solving a physical problem with difeq, so I'm not sure how to apply multiple boundary conditions over x for these two circumstances (do I just use 20?). Also, I'm not sure why you used T=50 over the entire base for solution one. Also, do I construct the same T for each solution? I'm really quite lost.

 

[vela]

There are only certain values of k that are allowed. Use the condition for x=20 cm to find these. So you have a solution of the form
$$T(x,y) = \sum_n c_nT_n(x,y) = \sum_n c_ne^{-k_n y}\sin k_n x.$$ You now apply the boundary condition at y=0 to solve for the $c_n$'s.

 

[Chestermiller]

This is follow up to vela's post. You need to determine the kn's and cn's. To determine the kn's, you choose the sequence of values for the kn's that automatically satisfy the boundary conditions at both x boundaries. The solution vela wrote down already satisfies the boundary condition at x = 0.