x(t) = R cos (ω 0 t + θ) + x s (t)

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Lucas Ramsey
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1 Formula Sheet Final Exam Springs and masses: dt x(t + b d x(t + kx(t = F (t dt More general differential equation with harmonic driving force: m d Steady state solutions: where d dt x(t + Γ d dt x(t + ω 0x(t = F 0 m cos (ω dt x s (t = A cos (ω d t δ A = F 0 m (ω 0 ω d + ω d Γ and General solutions: For Γ = 0 (undamped system: tan δ = Γω d ω 0 ω d where R and θ are unknown coefficients. For Γ < ω 0 (under damped system: ( x(t = R cos (ω 0 t + θ + x s (t x(t = Re Γ t cos where R and θ are unknown coefficients. For Γ = ω 0 (critically damped system: where R and R are unknown coefficients. For Γ > ω 0 (over damped system: ω 0 Γ 4 t + θ x(t = (R + R te Γ t + x s (t + x s (t ( ( x(t = R e Γ + Γ 4 ω 0 t + R e Γ Γ 4 ω 0 t + xs (t where R and R are unknown coefficients.

2 Coupled oscillators Examples for n = F j = n K jk x k k= x (t X (t = x (t K K K = K K m 0 M = 0 m Matrix equation of motion, matrices M, K, I are n n, vectors X, Z are n. d dt X (t = M KX (t Z(t = Ae iωt (M K ω IA = 0 To obtain the frequencies of normal modes solve: det(m K ω I = 0 For n = M M det = M M M M M M If the system is driven by force one can find the response amplitudes C(ω d F(t = F 0 e iω dt W(t = C(ω d e iω dt C(ω d = c (ω d c (ω d (M K ω dic(ω d = F 0

3 solving the equation above one can find the response amplitudies for the first (c (ω d and second (c (ω d objects in the system. Reflection symmetry matrix: 0 S = 0 Eigenvalues (β and eigenvectors (A of this S matrix: ( β =, A = [ ] ( β =, A = D infinite coupled system which satisfy space translation symmetry: Given a eigenvalue β, the corresponding eigenvector is where A j = β j A 0 A j (A 0 is the normal amplitude of jth(0th object in the system. Consider an one dimentional system which consists infinite number of masses coupled by springs, β can be written as β = e ika where k is the wave number and a is the distance between the masses. Kirchoff s aws (be careful about the signs! Node : i I i = 0 oop : i V i = 0 Capacitors : V = Q C Trigonometric equalities: Inductors : V = di dt Current : I = dq dt sin(a ± b = sin(a cos(b ± cos(a sin(b cos(a ± b = cos(a cos(b sin(a sin(b ( ( a + b a b sin(a + sin(b = sin cos ( ( a + b a b sin(a sin(b = cos sin ( ( a + b a b cos(a + cos(b = cos cos 3

4 ( ( a + b a b cos(a cos(b = sin sin Some useful integrals involving sin and cos: 0 0 sin cos ( nπx sin ( nπx cos e iθ = cos θ + i sin θ ( mπx ( mπx {, if n = m. dx = 0, otherwise. dx = {, if n = m. 0, otherwise. ( nπx ( mπx cos sin dx = 0 0 x sin(xdx = sin(x x cos(x + C x cos(xdx = cos(x + x sin(x + C 4

5 Maxwell Equations in vacuum E y x E x y = B z ; E z y E y z = B x ; E x z E z x = B y B y x B x y = µ E z 0ɛ 0 ; B z y B y z = µ 0ɛ 0 E x ; B x z B z x = µ E y 0ɛ 0 E x x + E y y + E z z = 0 ; B x x + B y y + B z z = 0 orentz force Wave equation for EM fields in vacuum F = q( E + v B E i x + E i y + E i z = c E i where i = x, y, z B i x + B i y + B i z = c B i where i = x, y, z For EM plane waves in vacuum: B( r, t = c ˆk E( r, t E( r, t = c B( r, t ˆk inear energy density in a string with tension T and mass density ρ dk dx = ρ ( y du dx = T EM energy per unit volume and Poynting vector: Transmission and reflection ( y x U E = ɛ 0 E U B = µ 0 B S = µ 0 E B Phase velocity and impedance: R = Z Z Z + Z, T = Z Z + Z 5

6 T v =, Z = T ρ ρ (string Snell s law v = C, Z = C (transmission line Fourier transform n sin θ = n sin θ Delta function f(t = C(ω = π π dωc(ωe iωt dtf(te iωt e i(ω ω t dt = δ(ω ω δ(xdx = δ(x af(xdx = f(a Electric and magnetic field from an accelerated charge: E( r, t = q a (t r /c 4πɛ 0 rc E( r, t = ˆr B c Total power emitted by the accelerated charge: P (t = q a (t r/c 6πɛ 0 c 3 Interference of two sources with amplitudes A and A with a relative phase difference δ: (A + A + A A cos δ Interference of N fields of equal amplitude with phases δ m+ δ m = δ: 6

7 = sin(nδ/ sin(δ/ Single slit diffraction where β is the phase difference between rays coming from edges and the center of the slit: [ sin(β = β Rayleigh s criterion for resolution: Diffraction peak of one image falls on te first minimum of the diffraction pattern of the second image. Electric field transmission and reflection ratios, magnitude and sign, for radiation incident normally on an interface between lossless dielectrics with indices of refraction n and n. Schrodinger s Equation E t E i = n n + n i [ ψ(x, t = m ] E r E i = n n n + n ] + V (x, t ψ(x, t x where V is the potential energy, m is the mass of the particle and ψ is the wave function. 7

8 MIT OpenCourseWare SC Physics III: Vibrations and Waves Fall 06 For information about citing these materials or our Terms of Use, visit:

Formula Sheet. ( γ. 0 : X(t) = (A 1 + A 2 t) e 2 )t. + X p (t) (3) 2 γ Γ +t Γ 0 : X(t) = A 1 e + A 2 e + X p (t) (4) 2

Formula Sheet. ( γ. 0 : X(t) = (A 1 + A 2 t) e 2 )t. + X p (t) (3) 2 γ Γ +t Γ 0 : X(t) = A 1 e + A 2 e + X p (t) (4) 2 Formula Sheet The differential equation Has the general solutions; with ẍ + γẋ + ω 0 x = f cos(ωt + φ) (1) γ ( γ )t < ω 0 : X(t) = A 1 e cos(ω 0 t + β) + X p (t) () γ = ω ( γ 0 : X(t) = (A 1 + A t) e )t