# I p4 r i1 i2 wednesday october 9 2013 2 22 r2 r1 eg

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: day, October 9, 2013 √ 2 τ µω A 1 = Pmax 2 2 and since τ2 √ 2 τ k ω = ω = τ µω v Pave 1√ 22 = τ µω A 2 1 ￿sin θ￿ = ￿cos θ￿ = 2 2 2 3d wave, intensity Intensity = I = energy ﬂux, i.e. energy ﬂow per area, per time. In 3d, I drops as distance r-squared, since energy is conserved and area grows as r-squared. I = P/4π r I1 /I2 = Wednesday, October 9, 2013 2 22 r2 /r1 E.g. 60W bulb emits P=60W, intensity of the light drops as 1/r^2, since light spreads out. Play / tune your guitar Fingers shorten string length, shorter length = higher frequency. Bass: fatter + longer strings = lower frequency. Tune: tighten the knobs (increase tension) to get higher frequency. Wednesday, October 9, 2013 String waves, modes 1 ∂2 ∂2 ( 2 2− )y (t, x) = 0. 2 v ∂t ∂x Ends, y (t, 0) = y (t, L) = 0 BCs: 1 L=n λ 2 The wave eqn + B.C. solution: n=3 Ends, and nodes, have y(t)=0. n = 1, 2, 3, 4, . . . Fundamental mode higher harmonics nπ x y (t, x) = A sin( ) cos(ωn t) L ￿ τπ ω1 = ωn = vkn = nω1 µL Wednesday, October 9, 2013 =“mode number” ￿ 2π nπ kn ≡ = λn L ￿ Tune your guitar! More bass (lower freq) from fatter or longer strings. Higher freq. from more tension. Harmonics n=1 “fundamental” n=2, (1 node) n=3 (n-1 nodes) n=4 λn = 2L/n ωn = vkn = nω1 Wednesday, October 9, 2013 ￿ 2π nπ kn ≡ = λn L ω1 = ￿ τπ µL ￿ ω ≡ 2π f ≡ 2πν Wave B.C.s at the ends: Two common choices: Fixed (Dirichlet): y (t, xend ) = 0 Free (Neumann): ∂y (t, xend ) = 0 ∂x Fixed (Dirichlet) Wednesday, October 9, 2013 Free (Neumann) Free ends case, sol’n: Suppose free ends at x=0, and x=L ∂y ∂y (t, 0) = (t, L) = 0 ∂x ∂x nπ x nπ vt y = A cos( ) cos( ) L L Fixed ends (few slides ago): this cos was instead sin. The harmonics are similar in the two cases. Wednesday, October 9, 2013 Free ends harmonics Slope =0 at ends λ = 2L λ=L λ = 2L/3 ... λn = 2L/n − fn = Tn 1 = v/λn = nv/2L Wednesday, October 9, 2013 Just for fun: string theory! Similar wave equation. Different harmonics are different particles. Known particles are the fundamental harmonic, others would be new particles. Wednesday, October 9, 2013...
View Full Document

## This note was uploaded on 02/11/2014 for the course PHYSICS 2C taught by Professor Hicks during the Fall '09 term at UCSD.

Ask a homework question - tutors are online