University:
Massachusetts Institute of Technology
Course:
8.03SC | Physics III: Vibrations and Waves
Academic year:

2020
Views:

255

Pages:

11
Author:

folclord7m

We have discussed the motion of a massive string extensively. This time: more examples which can be described by the wave equation. "Longitudinal Waves" x,t+Δx Equilibrium Position Po : room pressure ρo : air density p(x) = Po + ψp(x) p(x+Δx) = Po + ψp(x+Δx) (2) The compression is actually adiabatic meaning that there is almost NO HEAT flowing IN and OUT of the volume. The time scale of the heat flow is larger than the time scale of oscillation ⇒ PV^γ = Const PV^γ = C Consider : small vibration ψp << Po ΔV << Vo Before : Po Vo^γ = C After : (Po+Δp) (Vo+ΔV)^γ = C (iv) Drive the organ → D cos(ωdt) L ψ(x,0) = 0. ψ(L,t) = D cos(ωdt) kd = ωd/v (decided by the dispersion relation w=vk) ψ(x) = A₀ sin(kdx + α) cos(ωdt) ① ψ(x,0) = 0 ⇒ α=0 ② ψ(L,t) = D cos(ωdt) ⇒ A₀ sin(kdL) = D ⇒ A₀ = D/sin(kdL) ⇒ ψ(x) = D/sin(kdL) sin(kdx) cos(ωdt) when kd = (n+½)π/L ⇒ Resonance ! (Huge amplitude)