Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section D / D12

One instrument is a diffraction grating spectrometer. Its principal maxima satisfy $d\sin \theta =m\lambda$ at normal incidence. N coherently illuminated grooves give a narrow principal peak; applying the Rayleigh criterion to the next wavelength yields the resolving power of a diffraction gratingHigher order and more illuminated grooves improve resolution, but finite entrance-slit width, aberrations, detector sampling and overlap between spectral orders limit useful resolution. Angular dispersion $d\theta /d\lambda =m/(d\cos \theta )$ is distinct from the ability to resolve two finite-width peaks.

A second instrument is a Fabry–Pérot interferometer. For plate spacing L, refractive index n and internal angle theta, transmission resonances satisfy $2nL\cos \theta =m\lambda$. Multiple coherent reflections produce an Airy transmission profile. For identical lossless mirrors of intensity reflectivity $\mathcal{R}$, the high-finesse limit givesThe frequency free spectral range is approximately $c/(2nL\cos \theta )$ for nondispersive n. A larger optical spacing narrows a line but also shrinks the unambiguous frequency interval; nearby orders need sorting. Mirror flatness, absorption, alignment and aperture broadening lower the effective finesse. This resolving-power formula assumes separated narrow lines and a stated line-separation convention, rather than equating transmission peak spacing with peak width.

The transient is the homogeneous part of a damped harmonic oscillator, $m\ddot{x}+b\dot{x}+kx=0$. Define $\alpha =b/(2m)$ and ${\omega }_0=\sqrt{k/m}$. Its characteristic roots are $-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$.

Initial position and velocity determine A,B. For an imposed force or support motion, the full solution is a particular response plus this transient, which decays for b>0 and k>0. The mechanical energy obeys $d(K+U)/dt=-b{\dot{x}}^2$. In light damping its cycle-averaged envelope decays as $e^{-2\alpha t}$, with $Q\simeq {\omega }_0/(2\alpha )$. Initial conditions can still produce an overshoot in a nonoscillatory regime; the damping category describes the roots rather than every possible trajectory.

Small sound disturbances in a gas couple density, pressure and longitudinal velocity. Linearizing mass conservation and the pressure-gradient force about a uniform state givesEliminating velocity yields ${\partial }_t^2{\rho }^{\prime }=c_s^2{\nabla }^2{\rho }^{\prime }$. When local heat exchange is negligible over an oscillation, $c_s^2=(\partial p/\partial \rho {)}_s$. For an ideal gas,The adiabatic index is $\gamma =C_p/C_V$; an isothermal disturbance instead has $c_T^2=p_0/{\rho }_0$. Pressure and velocity in a progressive plane wave are in phase and obey $p^{\prime }={\rho }_0{c}_{s}u$, with average energy flux $⟨p^{\prime }u⟩=p_{\mathrm{peak}}^{\prime 2}/(2{\rho }_0{c}_s)$. Particle oscillation speed is not the sound propagation speed. Viscosity, heat conduction and molecular relaxation dissipate sound; changes in active internal degrees of freedom can cause dispersion. The continuum acoustic model requires wavelength large compared with the molecular mean free path and small fractional perturbations.