VISUAL EFFECT PRODUCED BY A COMPLEX WAVE 41 



average of the energy density associated with the wave. The instan- 

 taneous state of vibration of a wave can also be represented by a complex 

 number z which includes the time / and the period T of a complete 

 vibration in accordance with the relation 



2 = ae"'(^'"''). (5.1) 



Alternately, 



z = ae~^e''^. (5.2) 



In these equations a is the amplitude of the wave and </> is the phase 

 angle or jihase retardation.. It follows from Eq. 4.6 that the time 

 factor e~'^'^'^''^ assumes the same value each time t is increased by the 

 period T. The vibration described by Eq. 5.1 or 5.2 is therefore 

 periodic. 



It will be noted that the phase retardation appears with the positive 

 sign in the phase factor e+^* of Eq. 5.2. We see from Eq. 5.1 that, if 

 the phase retardation or optical path is increased, / will have to be 

 increased in order to leave the state of vibration represented by z un- 

 affected. This means physically that, if we increase the optical path 

 of a medium, it will take the wave a longer time to present to the 

 observer a preassigned state of vibration. In other words, a given 

 portion of the wave train arrives at a later moment. Accordingly, 

 the phase retardation 4> will be considered henceforth as positive when 

 it corresponds to an increase in the optical path, and the phase retarda- 

 tion will be preceded by -\-i in the phase factor e"^**. 



The time average of the energy density produced by the wave is 

 proportional to \z\'^. It is often convenient in manipulating complex 

 numbers z from eciuations of the type of Eq. 5.2 to write \z\ in its ex- 

 panded form: 



Ul = a|e-^"'/n W^l (5-3) 



Since 



W 



1^ = [cos 6 + i sin df = cos^ 6 + sin^ 0=1, 



where 6 is equal to </> or —2irt/T, 



\z\' = a\ (5.4) 



The average energy density is therefore proportional to the square of 

 the amplitude a. With rapid vibrations the eye can detect only the 

 average energy density as given by Eq. 5.4. Since brightness is pro- 

 portional to the average energy density, we conclude that the brightness 

 is proportional to the square of the amplitude of the complex number 

 which represents the amplitude and phase of the image-producing light 

 wave. 



