Prove that E(x,t) = E0ei(kx-ωt) is a solution to the wave equation.
1. A common solution to the wave equation is E(x,t) = A Cos(kx-wt). On paper take the needed derivatives and show that it actually is a solution. 2. A common solution to the wave equation is E(x,t) = A ei(kx+wt). On paper take the needed derivatives and show that it actually is a solution. Note that i is the square-root of -1
Determine if the functionD = A sin kx cos ωt is a solution of the wave equation. (Must show your proof, no credit given without correct work.)
Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where k = 2π/λ and ω = 2πf. The aim of the following exercises is to show that this expression captures many of the intuitive features of waves. a) Consider a snapshot of the wave at t = 0. Use the expression to find the possible values of x at which the crests (maximum points) of the wave are located. By what distance are neighboring...
check whether the function E(x,t)= Asin(kx^2-wt^2) satisfies the wave equation. if so, find the wave speed. if not explain
The magnetic field of an electromagnetic wave is described by By = B0cos(kx - ωt), where B0 = 3.5 × 10-6 T and ω = 2.5 × 107 rad/s. What is the amplitude of the corresponding electric field oscillations, E0, in terms of B0? What is the frequency of the electromagnetic wave, f, in terms of ω? What is the wavelength of the electromagnetic wave, λ, in terms of ω and the speed of light c?
A free particle moving in one dimension has wave function Ψ(x,t)=A[ei(kx−ωt)−ei(2kx−4ωt)] where k and ω are positive real constants. At t = π/(6ω) what are the two smallest positive values of x for which the probability function |Ψ(x,t)|2 is a maximum? Express your answer in terms of k.
A sound wave of the form s = sm cos(kx - ωt + φ) travels at 343 m/s through air in a long horizontal tube. At one instant, air molecule A at x = 2.04 m is at its maximum positive displacement of 6.20 nm and air molecule B at x = 2.08 m is at a positive displacement of 2.00 nm. All the molecules between A and B are at intermediate displacements. What is the frequency of the wave?
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
Which of the following is/are solution to the wave equation, 1. ei(kx-wt) a. b. (cos kx) (sin ot) sin et C. e sin (kx - t) d. e COs t e. e 2. A building made with a steel structure is 650 m high on a winter day when the temperature is 0° F. How much taller (in cm) is the building when it is 100° F? (The linear expansion coefficient of steel is 11 x 10 (C) ) A...
The equation of a transverse wave traveling along a string is y = (0.11 m)sin[(0.78 rad/m)x - (14 rad/s)t] (a) What is the displacement y at x = 2.6 m, t = 0.27 s? A second wave is to be added to the first wave to produce standing waves on the string. If the wave equation for the second wave is of the form y(x,t) = ymsin(kx + ωt), what are (b) ym, (c) k, and (d) ω (e) the...