Derive the relation between the frequencies of a wave in two reference frmames (the Doppler effect,) and the relation between the value of the consines of the angles the ray makes with the direction of the source's motion (in both reference frames).

Explanation

Equation (59.22) follows from the phase invariance. Applying the Lorentz transformation, we obtain
`omega((t_(0)+vx_(0)//c^(2))/(sqrt(1-v^(2)//c^(2)))-(costheta)/(c).(x_(0)+vt_(0))/(sqrt(1-v^(2)//c^(2)))-(z_(0)sintheta)/(c))=omega_(0)(t_(0)-(x_(0)costheta_(0))/(c)-(z_(0)sintheta_(0))/(c))`
Removing brackets and regrouping the terma, we have
`(omegat_(0))/(sqrt(1-v^(2)//c^(2)))(1-(vcostheta))/(c)+(omegax_(0))/(csqrt(1-v^(2)//c^(2)))((v)/(c)-costheta)-(omegaz_(0)sintheta)/(c)-omega_(a)t_(0)-(omega_(0)x_(0)costheta_(0))/(c)-(omega_(0)z_(0)sintheta_(0))/(c)`
Nothing that `x_(0), z_(0) and t_(0)` are independent variables we see that equality obtained is possible only if the factors precending these vari = albes are equal. Hence putting `beta = v//c` we obtain
`(omega(1-betacostheta))/(sqrt(1-beta^(2)))=omega_(0).(omega(costheta-beta))/(sqrt(1-beta^(2)))=omega_(0)costheta_(0),omegasintheta=omega_(0)sin0_(theta)`
The first equality is the expression for the Doppler effect, Dividing the second equation by the first, we obtain the relation for the cosines
`(costheta-beta)/(1-betacostheta)=costheta_(0)`