## 5

## Optical Applications of Lorentz Transformation

##### 5.1 ELEMENTS OF WAVE PROPAGATION

Let us consider a function *f*_{1}(*x*, *t*) defined by the relation

*f*_{1}(*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 *δ**x* and *t* by *δt* then

*f*_{1}(*x* + *δx*, *t* + *δt*) = *a* sin [*b* (*x* + *δx*) – *g* (*t* + *δt*)]

= *a* sin [(*bx *– *gt*) + (*bδx* – *gδt*)]

Now,

*f*_{1}(*x* + *δx*, *t* + *δt*) = *f*_{1}(*x, t*)

provided

*bδx* – *gδt* = 0

or

where, *v*_{ω} is called the phase velocity. The function *f*_{1}(*x*, ...