Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 19

For $c>0$ and $l>0$, the zero-endpoint Dirichlet boundary conditions select the Fourier sine basis functions $\sin (n\pi x/l)$. Separation of variables gives temporal factors with angular frequencies ${\omega }_n=n\pi c/l$. Zero initial displacement removes the temporal cosine terms, givingDifferentiating at $t=0$ and using Fourier orthogonality givesThe evaluation property of the given initial velocity means $v(x)=\delta (x-l/2)$ in the sense of a Dirac delta, an idealized impulsively struck fixed-end string. Thus $a_n=2\sin (n\pi /2)/(n\pi c)$: every even mode vanishes, while the lowest two excited modes have $n=1,3$. Their ordinary frequencies and amplitude ratio areThe signed coefficient ratio is $a_3/a_1=-1/3$. The impulse solution is interpreted as a distributional weak solution, rather than one with a smooth initial velocity.

For a separated harmonic function $V=F(x)G(y)$, Laplace's equation gives $F^{\prime \prime }/F=-G^{\prime \prime }/G=-k^2$. Bounded periodic dependence in $x$ has $F=A\cos (kx)+B\sin (kx)$; decay on each half-plane selects $G=e^{-k|y|}$ for $k>0$, when the potential is matched continuously at the sheet. These are the separated decaying modes. General decaying solutions can be superpositions of them; decay alone does not force a single value of $k$.

The prescribed cosine density selects $k=1$ and $V=C\cos x{e}^{-|y|}$. Its normal derivative jump is $-2C\cos x$. Matching the stated jump $2\pi {\Sigma }_0\cos x$ givesThis is the gravitational potential of a sinusoidal sheet in the normalization of the question. It is continuous at the sheet and decays on both sides. Since ${\Sigma }_0\cos x$ changes sign, it can physically describe a Fourier component of a density perturbation rather than an everywhere positive isolated mass sheet.