Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 5

With the stated Fourier transform convention, the exponential transform is a shifted Dirac delta distribution:Writing the sine as $(e^{i{\omega }_{0}t}-e^{-i{\omega }_{0}t})/(2i)$ then givesThese are tempered distributions, since the time-domain functions are not integrable on the full real line. The second formula also correctly vanishes when ${\omega }_0=0$.

By the definition of convolution and Fourier inversion,The second line uses $\nu =\omega -\eta$. The factor $2\pi$ comes from the inverse-transform normalization. This is the product form of the convolution theorem, initially justified for sufficiently decaying functions and extended when the indicated products and convolutions exist.

The reciprocal in this transform must mean the principal-value reciprocal distribution. Its inverse transform isusing the given Dirichlet integral and the sign of $t$. Also ${\mathcal{F}}^{-1}[\delta ](t)=1/(2\pi )$. The inverse of the proposed expression is consequently $(iA/2)\mathrm{sgn}t+B/(2\pi )$. Comparison with $H(t)=[1+\mathrm{sgn}t]/2$ for $t\ne 0$ givesThe value assigned at the single point $t=0$ does not affect the distribution. This is the Fourier transform of the Heaviside step function; treating $1/\omega$ as an ordinary integral would miss the required prescription.

Multiply $H$ by the exponential form of the sine and use the modulation property of the Fourier transform:Inserting part (iii) givesThe reciprocal terms are real principal-value distributions and the delta terms are imaginary. Equivalently, exponential damping $H(t)e^{-ϵt}$ gives ${\omega }_0/[(ϵ+i\omega {)}^2+{\omega }_0^2]$, whose distributional limit is the boxed expression. The shifted-difference formula remains valid at ${\omega }_0=0$, where all terms cancel.

Transform only the time variable. Two integrations by parts give ${\mathcal{F}}_t[y_{tt}]=(i\omega {)}^2\tilde{y}=-{\omega }^2\tilde{y}$, when the ordinary transform boundary terms vanish; the same derivative rule holds for tempered distributions. Spatial derivatives and the fixed spatial delta commute with the transform. ThusorThe spatial Dirichlet boundary conditions transform to $\tilde{y}(\omega ,0)=\tilde{y}(\omega ,L)=0$.

Multiply the transformed equation by $\sin (k_{n}x)$ and integrate from zero to $L$. Two integrations by parts, using the zero endpoint values of both $\tilde{y}$ and the sine, giveThe delta source contributes $\tilde{F}(\omega )\sin (k_n{x}_0)$. Multiplying by $2/L$ and using the coefficient definition therefore yieldsAway from the resonant frequencies this givesAt $\omega =\pm k_n$ the quotient must be interpreted with a pole prescription if the source excites that mode. In distributional form, arbitrary delta terms at those frequencies may also be added; they are the free homogeneous oscillations. This is the frequency-space issue addressed explicitly in part (iii).

Expand in the complete Fourier sine series $\tilde{y}(\omega ,x)={\sum }_{n\ge 1}{b}_n(\omega )\sin (k_{n}x)$ and apply Fourier inversion. Substitution of part (ii) givesTo check the source normalization, apply ${\partial }_t^2-{\partial }_x^2$ mode by mode. It cancels the denominator; the remaining inverse transform gives $2\pi F(t)$, and completeness givesThe endpoints vanish termwise. These identities establish the equation in the distributional sense, with classical validity wherever the series and derivatives converge suitably.

The displayed real-axis integrals in the paper are not ordinary convergent integrals at their poles. One definite choice is the Cauchy principal value at each $\omega =\pm k_n$. If a causal solution is wanted, use $[k_n^2-(\omega -i0{)}^2{]}^{-1}$ instead. Its inverse transform is $H(t)\sin (k_{n}t)/k_n$, so the causal solution iswhen the past integral exists. The principal-value choice instead uses the time-symmetric kernel $\sin (k_n|t|)/(2k_n)$. No temporal initial data are given, so homogeneous sine and cosine modes remain permissible additions. This is the causal Fourier resolvent of a point-driven fixed string.