5

Optical Applications of Lorentz Transformation

5.1  ELEMENTS OF WAVE PROPAGATION

Let us consider a function f1(x, t) defined by the relation

 

f1(x, t) ≡ a sin (bxgt)         (5.1)

 

where a, b, and g are all positive. The function is periodic both in x and in t and varies between ±a. It can be shown to represent a wave of amplitude a, propagating in the direction of +x with velocity g/b. If x is increased by δx and t by δt then

 

f1(x + δx, t + δt) = a sin [b (x + δx) – g (t + δt)]
= a sin [(bx gt) + (bδxgδt)]

Now,

f1(x + δx, t + δt) = f1(x, t)

provided

bδxgδt = 0

or

image

where, vω is called the phase velocity. The function f1(x, ...

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