Special Theory of Relativity

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 (bx – gt)         (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) + (bδx – gδt)]

Now,

f₁(x + δx, t + δt) = f₁(x, t)

provided

bδx – gδt = 0

or

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