Summary of Separation of Variables — Question 7

Question 7 – Solution

Strategy. The same initial Fourier coefficients can lead to a smooth heat field and a discontinuous wave field; convergence alone does not guarantee finite energy.

Step 1: Compute and evolve the coefficients. The coefficients are $b_n=2(1-(-1)^n)/(n\pi)$, so $$H=\frac 4\pi\sum_{n\ {\mathrm{odd}}}\frac{e^{-n^2t}\sin(nx)}n,\qquad W=\frac 4\pi\sum_{n\ {\mathrm{odd}}}\frac{\cos(nt)\sin(nx)}n.$$ Both recover $f$ in $L^2$ as $t\downarrow 0$. Heat derivatives converge uniformly for every $t\ge\tau>0$. The wave series represents the reflected function in $L^2$ and takes average values at jumps; its zero initial time derivative is understood in distributions.

Step 2: Track the two reflected fronts. The extension is $F(s)=\operatorname{sgn}(\sin s)$ away from multiples of $\pi$, with value zero at those multiples. For $0<t<\pi/2$, $$\boxed{W(x,t)= \begin{cases} 0,&0<x<t\text{ or }\pi-t<x<\pi,\\ 1,&t<x<\pi-t,\\ 1/2,&x=t\text{ or }x=\pi-t. \end{cases}}$$ Each endpoint trace is zero. Reflection has produced two moving jumps, not a gradually smoothing profile.

Step 3: Compare smoothing and initial convergence. For $t>0$, the heat field is smooth and $0<H<1$ in the interior, by diffusion and the maximum principle. The wave instead has the step profile above. For any positive time approaching zero, the wave’s zero strips give sup-norm error one; the heat’s continuous zero endpoint trace makes its supremum error also one over the open interval. Thus $L^2$ initial convergence does not imply uniform convergence across the incompatible endpoint layer.

Step 4: Test energy rather than assume it. The partial wave energy is $$E_N=\frac\pi 4\sum_{n\le N}n^2b_n^2 =\boxed{\frac 4\pi\,\#\{n\le N:n\text{ odd}\}\longrightarrow\infty.}$$ No finite-energy Dirichlet wave has this initial profile; it is not in the zero-trace energy space. The reflected bounded field still solves the wave equation distributionally, with the stated initial displacement and zero initial velocity in the corresponding weak sense. At a moving jump, its derivatives need not be ordinary square-integrable functions.