Prove that the two linearly independent solutions to the one-dimensional wave equation 1 d² 4 a² Əz² are: 2 v² Ət² 2 2 Y(z,t) = f (z − vt) and Y(z,t) = f (z+vt), where ƒ is an analytical function. -
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- A uniform plane wave in air with E;(z) = ax. 10. e¬(6z) (V/m) is incident normally on an interface at Z = 0 with lossy medium {ɛ, 2.5, tan 8 = 0. 5}. Find: e) The expressions for E,(z, t), H;(z,t), E,(z, t) & H7(z, t) a) The average Poynting vector P, in air and in the lossy dielectric media avShow explicitly that the wave function, y (x, t) A cos(kx - wt), = satisfies the wave equation, 8² dx 2 y (x, t) = 1 8² v² Ət2 y(x, t). Write the explicit value of v as a function of the parameters of the wave function, w, k and A.The periodic function f(x) of period 27 is given by f(x)%3D = {,! 0A uniform plane wave has the generic expression Φ(z,t) = A cos(ωt – kz + δ) with the following given parameter values: wave amplitude = 10, wave frequency in Hz f = 500 Hz, phase velocity vph = 10 m/s, and the phase angle δ = 60o. Find the values of the parameters A, ω, and k. ..A sinusoidal wave of Vmax = 10 V and T= 40 ms periodic time, Calculate the following: Veff=Vrms of this wave: The frequency f : Write the equation of this wave using V=Vmax sin [ (2πf)]tNormalize the wavefunction Y(x,t)= cos(2x)e where t represents time over the region, 0Find the Fourier series of the function -TGiven the following wave function: y (x, t) = (7.66 cm) cos (5.39x + -0.97t) Determine: -The amplitude-Angular frequency-The term-The frequencyThe inverse Fourier transform of the position probability amplitude y(x,t) is: a. w(p.t): O b. y (k,t) OC. d. w (x,t) = w(p,t) = 1 2π ħ 1 2r ħ 1 2r ħ 1 P + - Sto ÿ (x,1) e " - 00 2π ħ - S + 0 % ( (x,t) e - 00 S. - 00 P ·S+ a + (k.1) e ² = 2 * ak dk + ∞o - S y - 00 x (x,t) e dp P ħ₂. dx -²x dxdx /what is the energy for the particle described by the wave function(x) = A exp(±ikx) (p₁) And wave along the interval (0 ≤x≤ a)The differential equation for transverse vibrations of a string whose density increases linearly from one end to the other is y + (Ax + B)y = 0, where A and B are constants. Find the general solution of this equation in terms of Bessel functions. Hint: Make the change of variable Ax + B = Au.12.20 Show that the Fourier series for |sinθ| in the range –π < 0 < π is given by M8 Σ | sin 0| m=1 1 = 4m2 2 By setting 0 = 0 and 0 = π /2, deduce values for 1 π 4 π ∞0 and m=1 cos 2mθ 4m2 – 1 ∞o Σ m=1 1 16m2 – 1SEE MORE QUESTIONSRecommended textbooks for youCollege PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University PressPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio…PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSONCollege PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University PressPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio…PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON