# Fourier Series, Temperature Distribution.

Cuthbert Nyack
Fourier Series can also be used to help find solutions of Laplace equation shown below. Using separation of variables the following can be obtained. The general solution of this equation involves sin, cos, sinh and cosh.
The solution for this case can be derived by using the boundary conditions. Assuming a rectangle of length b along the y direction and a along the x direction. and initial boundary conditions given by
T(x,0) = T(x) for 0 <= x <= a
T(x,b) = T0 for 0 <= x <= a.
T(0,y) = T(a,y) = T0.
Then the solution is shown below. Assuming T(x) is a constant Tc from c to d, then the solution is shown in the equation below. The terms shown in red represent the coefficient of the Fourier Series representation of T(x). The applet below shows the temperature distribution on the plate. nsum determines how many terms are summed in the Fourier Series. Its effect is most noticeable near the transition points where the temperature changes. The shape of the red portion of the plot is also affected by nsum. Unfortunately as nsum increases the applet's response time increases.

When activated the following gif images show how the applet should appear.  The following 2 images show what can happen if enough terms in the Fourier Series are not summed.  Using superposition the temperaure distribution can be found for any temperature along the 4 edges. The identical result is obtained for any system described by Laplace equation eg eloctrostatic fields.