Past exam of the physics A course of the University of Cambridge / 2016 / Paper 2 / Section B / B7

For a uniform taut sheet in the small-displacement approximation, the two-dimensional wave equation isSeek a harmonic solution $Y(y)e^{i\omega t-i{k}_{x}x}$. The clamped edges impose Dirichlet boundary conditions $Y(0)=Y(b)=0$. Hence $Y^{\prime \prime }+k_y^{2}Y=0$ has nontrivial solutions only forTaking the real part gives the stated sinusoidal transverse profile. There is no nonzero $n=0$ mode satisfying both clamped boundaries.

Substitution gives the dispersion relationThe three branches have cutoffs ${\omega }_{c,n}=n\pi v/b$ at $k_x=0$ and approach $\omega =v|k_x|$ at large $|k_x|$. Only frequencies above a branch's cutoff give traveling waves along an infinitely long guide; below it, the axial dependence is evanescent. For a forward propagating mode,The phase velocity exceeds $v$, while the group velocity is smaller: the carrier phase speed is not the signal speed.

Here the first two cutoff frequencies are $50$ and $100\mathrm{Hz}$, so both modes propagate at $200\mathrm{Hz}$. With $\omega =400\pi {\mathrm{s}}^{-1}$,Each transverse mode independently acquires its axial phase. The second mode first changes sign relative to the first when $(k_1-k_2)x_1=\pi$, givingAt that distance a common temporal phase is $\varphi =-k_1{x}_1$ modulo $2\pi$; the second term is negative relative to the first for every $y$ and every time, not just at one selected instant.

In an optical fibre, different guided modes have different group velocities, so an input pulse becomes a sum of differently delayed pulses. This intermodal dispersion broadens pulses and can make adjacent digital symbols overlap. A single-mode optical fiber removes this particular mode-delay mechanism; a graded-index optical fiber can reduce it in a multimode design. Single-mode propagation still permits wavelength-dependent group delay, so it does not remove every kind of optical dispersion.