Question

A wave defined by the expression E (z,t) = A cos(ot-kz) fills all space directions perpendicular to the z-axis in 3-dimensional space. A wave confined in the directions transverse to the direction of propagation can be constructed by superimposing such waves propagating in different directions. This problem is an illustration on how it can be done (a) Consider the space defined by the Cartesian coordinates x-y-z. Rotate the coordinates about the y-axis by an angle A to form a new system x'-y-z' A wave propagating along the z'-direction can be written as E, (z',t) = A cos(ot-kz) Write z' in terms of x and z in the expression for E'(z,t) (b) Construct a similar system x"y-z by rotating about the y-axis by an angle -A instead, and write down the similar expression E"(z",t) for a wave propagating along the z" direction, and then write z in terms of x and z. (c) Add the two waves E' and E", and show that the resultant wave propagates along the z- direction, with an increased velocity over that of either component waves, and that the amplitude is sinusoidally modulated in the x-direction. A ave confined in the x-direction can be constructed by superimposing many waves propagating at different directions

Answer 1

"A wave defined by the expression E(z, t) = Acos(omega t - kz) a) let the space defined by the Cartesian co-ordinates x - y - z A wave propagating along the z^' direction can be written as E^'(z^', t) = A cos(omega t - kz^') cos theta = x/z^' z^' = z/cos theta = x/cosA Therefore but tan A = x/z A = tan^-1(x/z Therefore X^' = x/cos[tan^-1(x/z)] rightarrow (1) \r\n E^'(x^', t) = A cos[omega t - z^' k] E^' (x, t) = A cos[omega t - z/cos[tan^-1(x/z)]k] rightarrow (2) b) cos(-A) = x/z^n z^n = x/cos(-A) = -x/z tanA = +x/z A = tan^-1x/z Therefore E^""(x^"", t) = a cos(omega t - kx^"") Therefore x^n = x/cos[tan^-1(x/z)] rightarrow (3) E^""(x^:, t) = A cos[omega t - kx/cos(tan^-1(x/z)] rightarrow (4)"