Two independent derivations

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter

Modes, Packets and Media, Ch Dispersion and the propagation of wave packets

Problem

Two independent derivations, each on a medium already held. ((a)) Capillary ripples on a water surface obey ω2=(σ/ρ)k3\omega^2 = (\sigma/\rho)k^3, where σ\sigma is the surface tension and ρ\rho the density; gravity is negligible at these wavelengths. Take σ=0.072 N m−1\sigma = 0.072\,\mathrm{N\,m^{-1}} and ρ=1.00×103 kg m−3\rho = 1.00\times10^3\,\mathrm{kg\,m^{-3}} for water. For a packet centred on a wavelength of 2.0 mm2.0\,\mathrm{mm}, find ω′′\omega'' numerically and write the resulting envelope equation. ((b)) A monatomic chain of masses mm joined by springs of stiffness κ\kappa and spacing aa has dispersion relation ω(k)=2ω0∣sin⁡(ka/2)∣\omega(k) = 2\omega_0|\sin(ka/2)| with ω0=κ/m\omega_0 = \sqrt{\kappa/m}. Define an energy density per site ene_n (kinetic plus half of each neighbouring bond's potential energy) and an energy flux Sn=−κ(xn+1−xn)x˙nS_n = -\kappa(x_{n+1}-x_n)\dot x_n between neighbouring sites. Cycle-average both for a travelling wave xn=Acos⁡(ωt−kna)x_n = A\cos(\omega t - kna), and show that the energy velocity vE≡a⟨Sn⟩/⟨en⟩v_E \equiv a\langle S_n\rangle/\langle e_n\rangle equals dω/dk\mathrm{d}\omega/\mathrm{d}k. Evaluate vEv_E at ka=π/2ka = \pi/2 for ω0=50 rad s−1\omega_0 = 50\,\mathrm{rad\,s^{-1}}, a=2.0 mma = 2.0\,\mathrm{mm}, and compare it with the phase velocity at the same point.

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