exam2formula

exam2formula - Useful Formulae General dierential equation...

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± ± ² ³² ³ Useful Formulae General differential equation for oscillators x + γx ˙ + ω 2 ¨ 0 x = f 0 cos ( ωt ) has solutions where x ( t )= A e γt 2 cos ´ µ ω 2 0 γ 2 4 t + α + x ss ( t ) x ( t ( A + Bt )e 2 + x ss ( t ) x ( t A e Γ 1 t + B e Γ 2 t + x ss ( t ) Γ 1 2 = γ 2 ± µ γ 2 4 ω 2 0 ω 0 > γ 2 ω 0 = γ 2 ω 0 < γ 2 and the steady-state solution is x ss ( t A ( ω ) cos ( δ ( ω )) γω A ( ω ( ω 2 ω 2 ) 2 f 0 + γ 2 ω 2 · 1 / 2 tan δ ( ω ω 2 ω 2 0 0 Non-dispersive wave equation 2 1 2 ∂x 2 y ( x, t ∂t 2 y ( x, t ) 2 v where v = T/µ for a string or v = κ/ρ for a gas. Kinetic, potential energy and power dK 1 ² ∂y ³ 2 dU = 1 ² ³ 2 P ( t T = dx 2 µ dx 2 T Reflection and transmission coefficients v 2 v 1 2 v 2 R = T = v 2 + v 1 v 2 + v 1 1
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± ± ± ± Fourier series for a function f ( θ )= f ( θ +2 π ) ² ³ f ( θ ´ A 0 + A m cos ( )+ B m sin ( ) 2 m =1 µ π 1 A m = f ( θ ) cos ( ) m =0 , 1 , 2 ,... π π µ π 1 B m = f ( θ )s in ( ) m =1 , 2 , 3 π π Dispersion ω d ω v phase = and v group = k d k Maxwell’s equations
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This note was uploaded on 10/12/2010 for the course BC BC367 taught by Professor Millard during the Spring '10 term at Colby.

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exam2formula - Useful Formulae General dierential equation...

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