= Solution
Each supplied <eigenvector> gives relative <displacements> at $x=1,2,3$ in a <standing wave>. Insert zero at $x=0,4$ to describe the whole sampled string. A real <normal mode> is $u_j(t)=C v_j\cos(\omega t+\phi)$: its shape stays fixed while its <amplitude> oscillates.
* $(0,1/2,-1/\sqrt2,1/2,0)$ is the highest-<frequency> resolved <normal mode>. Adjacent interior <displacements> alternate in sign: the middle point moves opposite to both outer interior points. Its interpolated shape has three lobes, corresponding to the third string <normal mode>.
* $(0,1/\sqrt2,0,-1/\sqrt2,0)$ is the middle-<frequency> <normal mode>. The midpoint is a <node of a standing wave>, and the two sides move in opposite directions. It corresponds to the second string <normal mode>.
* $(0,1/2,1/\sqrt2,1/2,0)$ is the lowest-<frequency> <normal mode>, corresponding to the <fundamental frequency>. All interior points move in the same direction, with greatest <displacement> at the centre; the only <nodes of a standing wave> are the endpoints in the smooth shape through these samples.
These <semidiscrete fixed-string normal modes> are samples of $\sin(k\pi x/4)$, with $k=3,2,1$ in the order listed. More precisely, their interior <vectors> are $v_j^{(k)}=2^{-1/2}\sin(k\pi j/4)$, which have unit <Euclidean norm> and form an <orthonormal basis> of $\mathbb R^3$. Their <eigenvalues> and <angular frequencies> obey
$$
\lambda_k=-4\sin^2\!\left(\frac{k\pi}{8}\right),\qquad
\omega_k=2\sin\!\left(\frac{k\pi}{8}\right),\qquad k=1,2,3.
$$
Thus larger <absolute value> of a negative <eigenvalue> means faster <oscillation>, rather than faster decay. The numerical <angular frequencies> approximate the continuous fixed-string values $k\pi c/L$; the <finite difference> spatial approximation changes them. Only three independent interior <normal modes> exist on this grid, and any interior <displacement> or initial velocity <vector> can be expanded in this <orthonormal basis>.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3-q5-string-modes-and-four-box-transport.png]
{title=Three fixed-string <normal modes> and conservative four-box transport}
{description=The left panels show the supplied unit <eigenvectors> with zero endpoints; smooth sine curves guide the eye through the samples. The right panel is an original directed graph of the numerical transport <matrix> in Question 6: every arrow from j to i is labelled by Aij. It is not a reproduction of source Figure 6.}
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