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6. Assuming that the wave field does indeed take the form n(x,t) = R(x – Cot)+...
3. The last question is about physical interpretation of PDES. Assume throughout that u(r, t) describes...
3. The last question is about physical interpretation of PDES. Assume throughout that u(r, t) describes the temperature in a thin rod, with r [0, L] andt2 0. Note: when I ask for physical interpretation, I mean things like "u(0,t) = 0 means that fixing one of the ends at constant temperature" (a) Consider the IBVP we are O (3 points) at Or2 u(0, t) Ou (L, t) u(x,0) =0 Give a physical interpretation of the boundary conditions and of...
8.3 Consider the periodic wave equation for t E R and x e T. Suppose the...
8.3 Consider the periodic wave equation for t E R and x e T. Suppose the initial conditions are or (0, x) h(x), for g є Cm +1 (T) and h є Cm(T), for m E N. 152 8 Fourier Series (a) Assuming that u(t, x) can be represented as a Fourier series u(t, x)- av(l)ekx, kEZ (8.48) find an expression for ak() in terms of the Fourier coefficients of g and h.
The general solution y(t, p. 6) to the wave equation on a disc of radius R...
The general solution y(t, p. 6) to the wave equation on a disc of radius R with boundary condition v(t, R, 1) = 0 is given by vlt,0,0) = EE - ( ) [cos (ES) (Am.cos(nb) + Bu sin(no)) + (ME) (C..cos(no) + Dm (no) n=0 s=0 sin sin(ne)) 728 where Jn (2) is a Bessel function and x is the s'th root of In(x). (i) Derive the expressions for y and Oy/at at t = 0. (ii) Find all...
4. (4.5 pts) Consider a 3D electromagnetic plane wave in vacuum, described in usual complex form...
4. (4.5 pts) Consider a 3D electromagnetic plane wave in vacuum, described in usual complex form by, (r, t) = Ēelkr-wt) in which ło = Epein/2y. Where k = -kx is the wave vector (assume k > 0) and w > 0 is the angular frequency. As usual, the real field is Er, t) = ReLEr, t)] (a) In which direction is the wave propagating? In terms of the given values k and w, what is the speed, wavelength, and...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m,...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3} a. List all numbers of S. b. List all ordered pairs in R. c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)? d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges. e. Give each edge in...
Torsional vibration of a shaft is governed by the wave equation = 16- where e(r,t) is...
Torsional vibration of a shaft is governed by the wave equation = 16- where e(r,t) is the angular displacement (angle of twist) along the shaft, is the distance from the end of the shaft and t is time. For a shaft of length 3T that is supported by frictionless bearings at each end, the boundary conditions are 0 r (0,t) = 0x(3mT, t) = 0, t > 0. Suppose that the initial angular displacement and angular velocity are e(xr,0)= 4cos(4x),...
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t)...
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t) (2,3)? (c) Using only characteristic lines and your knowledge of d'Alembert's solution, find the following values of u(r,t) c) u9,9)- (b) u2,5)
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave...
kr ky (4) A plane EM wave with frequency o propagates in vacuum and approaches a...
kr ky (4) A plane EM wave with frequency o propagates in vacuum and approaches a dielectric interface (permittivity &, permeability u and refractive index n) at an angle of incidence Oi with its electric field parallel to the plane of incidence. If the incident E and B fields are of the form E(r, t) = Eel(k.r-wt) and B(r,t) = -K XE 'By (a) Write the boundary conditions satisfied by the fields at the interface (b) Derive the reflection and...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the particle, E ita energy, (z) the potential (a) Consider a particle moving in a constant pote E> Vo, show that the following wave function is a solution of the TISE and determine the relationahip betwoen E an zero inside the well, ie. V(2)a 0foros L, and is infinite ou , ie, V(x)-w (4) Assuming (b) Consider an infinite square well with walls at 1-0...
(6%) Problem 3: The magnetic field of an electromagnetic wave is described by By = Bocos(kor...
(6%) Problem 3: The magnetic field of an electromagnetic wave is described by By = Bocos(kor - wt), where Bo = 2.5 x 10-6 T and w=2.5 107 rad/s. Randomized Variables Bo = 2.5 x 10-61 w=2.5 * 107 rad/s A 20% Part (a) What is the amplitude of the corresponding electric field oscillations, Eo, in terms of Bo? Eo=1 G D PO a 00 HOME B 0 d 7 TA 4 8 9 6 St A Во 5 с...
3. The last question is about physical interpretation of PDES. Assume throughout that u(r, t) describes the temperature in a thin rod, with r [0, L] andt2 0. Note: when I ask for physical interpretation, I mean things like "u(0,t) = 0 means that fixing one of the ends at constant temperature" (a) Consider the IBVP we are O (3 points) at Or2 u(0, t) Ou (L, t) u(x,0) =0 Give a physical interpretation of the boundary conditions and of...
8.3 Consider the periodic wave equation for t E R and x e T. Suppose the initial conditions are or (0, x) h(x), for g є Cm +1 (T) and h є Cm(T), for m E N. 152 8 Fourier Series (a) Assuming that u(t, x) can be represented as a Fourier series u(t, x)- av(l)ekx, kEZ (8.48) find an expression for ak() in terms of the Fourier coefficients of g and h.
The general solution y(t, p. 6) to the wave equation on a disc of radius R with boundary condition v(t, R, 1) = 0 is given by vlt,0,0) = EE - ( ) [cos (ES) (Am.cos(nb) + Bu sin(no)) + (ME) (C..cos(no) + Dm (no) n=0 s=0 sin sin(ne)) 728 where Jn (2) is a Bessel function and x is the s'th root of In(x). (i) Derive the expressions for y and Oy/at at t = 0. (ii) Find all...
4. (4.5 pts) Consider a 3D electromagnetic plane wave in vacuum, described in usual complex form by, (r, t) = Ēelkr-wt) in which ło = Epein/2y. Where k = -kx is the wave vector (assume k > 0) and w > 0 is the angular frequency. As usual, the real field is Er, t) = ReLEr, t)] (a) In which direction is the wave propagating? In terms of the given values k and w, what is the speed, wavelength, and...
Torsional vibration of a shaft is governed by the wave equation = 16- where e(r,t) is the angular displacement (angle of twist) along the shaft, is the distance from the end of the shaft and t is time. For a shaft of length 3T that is supported by frictionless bearings at each end, the boundary conditions are 0 r (0,t) = 0x(3mT, t) = 0, t > 0. Suppose that the initial angular displacement and angular velocity are e(xr,0)= 4cos(4x),...
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave speed? c (b) What is the domain of dependence for the point (r, t) (2,3)? (c) Using only characteristic lines and your knowledge of d'Alembert's solution, find the following values of u(r,t) c) u9,9)- (b) u2,5)
Consider the inlínitm ar ls sby subject to the initial conditions u(x,0)-f() and u(,0)-9(x) where 0, otherwise (a) What is the wave...
kr ky (4) A plane EM wave with frequency o propagates in vacuum and approaches a dielectric interface (permittivity &, permeability u and refractive index n) at an angle of incidence Oi with its electric field parallel to the plane of incidence. If the incident E and B fields are of the form E(r, t) = Eel(k.r-wt) and B(r,t) = -K XE 'By (a) Write the boundary conditions satisfied by the fields at the interface (b) Derive the reflection and...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the particle, E ita energy, (z) the potential (a) Consider a particle moving in a constant pote E> Vo, show that the following wave function is a solution of the TISE and determine the relationahip betwoen E an zero inside the well, ie. V(2)a 0foros L, and is infinite ou , ie, V(x)-w (4) Assuming (b) Consider an infinite square well with walls at 1-0...
(6%) Problem 3: The magnetic field of an electromagnetic wave is described by By = Bocos(kor - wt), where Bo = 2.5 x 10-6 T and w=2.5 107 rad/s. Randomized Variables Bo = 2.5 x 10-61 w=2.5 * 107 rad/s A 20% Part (a) What is the amplitude of the corresponding electric field oscillations, Eo, in terms of Bo? Eo=1 G D PO a 00 HOME B 0 d 7 TA 4 8 9 6 St A Во 5 с...
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