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

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

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