Mathematical Methods for Physics and Engineering - Matematica.NET

21.2 SUPERPOSITION OF SEPARATED SOLUTIONSybu =0u = f(y)u → 00 u =0xA semi-infinite metal plate whose edges are kept at fixed tem-Figure 21.1peratures.exponentials rather than of hyperbolic functions. We therefore write the separable solutionin the form (21.15) asu(x, y) =[A exp λx + B exp(−λx)](C cos λy + D sin λy).Applying the boundary conditions, we see firstly that u(∞,y) = 0 implies A =0ifwetake λ>0. Secondly, since u(x, 0) = 0 we may set C = 0, which, if we absorb the constantD into B, leaves us withu(x, y) =B exp(−λx)sinλy.But, using the condition u(x, b) = 0, we require sin λb =0andsoλ must be equal to nπ/b,where n is any positive integer.Using the principle of superposition (21.16), the general solution satisfying the givenboundary conditions can therefore be written∞∑u(x, y) = B n exp(−nπx/b) sin(nπy/b), (21.17)n=1for some constants B n . Notice that in the sum in (21.17) we have omitted negative values ofn since they would lead to exponential terms that diverge as x →∞.Then = 0 term is alsoomitted since it is identically zero. Using the remaining boundary condition u(0,y)=f(y)we see that the constants B n must satisfy∞∑f(y) = B n sin(nπy/b). (21.18)n=1This is clearly a Fourier sine series expansion of f(y) (see chapter 12). For (21.18) tohold, however, the continuation of f(y) outside the region 0 ≤ y ≤ b must be an oddperiodic function with period 2b (see figure 21.2). We also see from figure 21.2 that ifthe original function f(y) does not equal zero at either of y =0andy = b then itscontinuation has a discontinuity at the corresponding point(s); nevertheless, as discussedin chapter 12, the Fourier series will converge to the mid-points of these jumps and hencetend to zero in this case. If, however, the top and bottom edges of the plate were held notat 0 ◦ C but at some other non-zero temperature, then, in general, the final solution wouldpossess discontinuities at the corners x =0,y =0andx =0,y = b.Bearing in mind these technicalities, the coefficients B n in (21.18) are given byB n = 2 ∫ b ( nπy)f(y)sin dy. (21.19)bb0719