Eigenvalues and Eigenfunctions — Question 3

Question 3 – Solution

Strategy. Use equally spaced frequencies, not equally spaced eigenvalues, to interpret the measurements.

Step 1: Classify the mixed spectrum. The endpoint term vanishes in integration by parts, so $\lambda\int y^2=\int(y')^2>0$ for a nonzero admissible function; equality would force a constant and then $y=0$. With $\lambda=k^2>0$, the left condition gives $y=A\sin(kx)$ and the right requires $\cos(kL)=0$. Hence $$\boxed{\lambda_n=((n+1/2)\pi/L)^2,\qquad n=0,1,\ldots.}$$ Each eigenspace consists of multiples of the corresponding sine.

Step 2: Recover the consecutive indices. Consecutive positive frequencies differ by $\pi/L$. The measured difference is $5\pi/4-3\pi/4=\pi/2$, giving $\boxed{L=2}$. The first measured frequency satisfies $(n+1/2)\pi/2=3\pi/4$, so its index is $n=1$ and the next is $n=2$. They were not the first two eigenvalues.

Step 3: Recover the missing lowest mode. The index-zero eigenvalue is $\boxed{\lambda_0=\pi^2/16}$. Since $\int_0^2\sin^2(\pi x/4)dx=1$ and its initial slope is positive, the requested eigenfunction is $\boxed{\phi_0=\sin(\pi x/4)}$.

Step 4: Identify the remaining inverse ambiguity. Let the measured indices be $r<s$. Their frequency ratio implies $5(2r+1)=3(2s+1)$. Since three and five are coprime, all solutions are $$2r+1=3d,\quad 2s+1=5d,\quad \boxed{L=2d,\quad r=(3d-1)/2,\quad s=(5d-1)/2,}$$ where $d$ is any positive odd integer. Conversely, substitution verifies every such choice. The index gap is $s-r=d$; consecutiveness forces $d=1$ and removes this infinite ambiguity.