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Unformatted text preview: Phys 218 11 February, 2008 - The displacement along the direction of propogation on wave in the x direction. Restoring force due to compression with -density, S - Surface Area, 2 2 Y and the Y - Young's Modulus. F = SY d so since x = v12 2 with v 2 = S . 2 ds dt Waves Through a Rigid Rod: y(x, t) = A exp [i (t - kx + )] = A exp [i (kx - t)] (since cos x = cos -x) Example: Because the forces at x = 0 and x = L are zero, the (x, t) = aei(t-kx) + bei(t-kx) . Because at x = 0: F (o, t) = SY |x=0 = 0. |x=0 = -ik (-a + b) eit = 0 . a = bv . With this we then see that (x, t) = dx dx aeit e-ikv + eikx = 2aeit cos(kx). Now, because at x = L the F = 0, |x=L = 0, then sin(kL) = 0. So we then dx see that kL = n for n N.k = n so that = 2 = 2L . If (x, t) = cn ein t cos (kn x): (for example) L k n So, What is Happening at the lowest mode? (n = 1) kn = n , so n = v = L L d Force Impedance of the Rod: Z = Velocity so since F = SY dx and v = , Z = S Y. Y S L. So, v1 = 1 2 = Y 1 S 2L . Sound Waves: Pressure Waves in Gases: Positive Feedback Mechanism. 1 1. Gas Moves, Density changes 2. Density Changes Pressure Changes 3. Pressure Changes Gas Then Moves, so that the cycle proceeds With P =Pressure, we see that these are true: t=0 P V 0 t=t P +p V +v 0 + So, if x + x + (x + x, t ) is the ending poition of the cylinder, then the mass m = 0 S x = 0 V so that m = (x, t) V (t)= (x, t) S x 1 + x and nally, m = 0 + S x 1 + x so that 0 S x = 0 + S (x, t) 1 + x so, d (x, t) -0 x . 2 ...
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