Phase velocity and group velocity (c) Zhengqing Yun,

Transcription

1 Phase velocity and grou velocity (c) Zhengqing Yun, Objective: Observe the difference between hase and grou velocity; understand that the grou velocity can be less than, equal to, and greater than the hase velocity and can exceed the seed of light; the grou velocity can even be negative! 1. Phase velocity and grou velocity For a harmonic lane wave, e.g., E( x, Acos( x, the hase velocity is defined as v. This is obtained by assuming the hase is constant, i.e., x t C differentiation of this equation, obtaining dx dt 0. Thus we have hase velocity. dx dt, and we take which is the When two harmonic lane waves with slightly different frequencies are added, a resultant wave can be reresented by, assuming they have the same amlitude, E 0, E( x, E ( x, E2( x, 2E0 cos( x cos( x ). (1) 1 t where 1 ( 1 2) 2 and 1 ( ). The two cosine functions in (1) are both traveling waves and we can define their hase velocities as v and v. (2) Fig. 1 shows the two original lane waves (green and ink), the two traveling wave in (1) (blue and red), and the resultant wave (black), i.e., E(x, in (1). Note that v is the hase velocity corresonding to the enveloe (modulating) curve in Fig. 1 (blue); while v is the hase velocity corresonding to the carrier wave (red). Since the enveloe curve encloses a grou of wavelets, v is called the grou velocity which may travel at a different velocity from velocity can be defined as v. More generally, the grou 1

2 d v g. (3) d cos( x cos( x E = E 1 +E 2 E 2 E 1 Fig. 1. Two lane waves and the resultant wave (E 0 =1/2) It is known that the hase velocity can be greater than the seed of light in free sace without violating the rule of Einstein s theory of relativity: nothing can travel at a seed greater than light. The reason is that the hase velocity does not reresent the signal or energy velocity. The grou velocity, on the other hand, was used to reresent the energy velocity which should not exceed the seed of light. Recently, lab exeriments have shown that the grou velocity can be less than, equal to, and greater than the seed of light, and grou velocity can even be negative! Reasons to these strange observations have been discussed and it is found that the energy is still traveling at a seed less than the light seed. So Einstein s theory is still valid. This lab does not try to exlain these advanced roblems in hysics. Instead, we lan to observe the hase and grou velocity based on the addition of two harmonic waves and draw some conclusions about the conditions which make the grou velocity less than, equal to, greater than, and oosite to the hase velocity. 2. Lab software In this lab, we exlore under what conditions the grou velocity can be less than, equal to, and greater than the seed of light, and even be negative! We only observe the grou velocity vs. hase velocity defined in (2) and (3). Fig. 2 shows the starting screen shot of the software. The lower art shows the two waves (urle and green curves; at beginning only the urle curve can be seen since these two curves are overlaed). The uer art (the white curve) is the sum of the two waves in the lower art. There are five controllers you can use. The first one is the red shere 2

3 which can be dragged on the slider to change the relative amlitudes of the two harmonic lane waves; the second is the yellow shere which is used to change the relative frequencies of the two lane waves; the third is the blue shere used to change the relative hase velocities; the fourth is the white shere which, when right clicked, will turn on/off the enveloe; the last one is the green one which controls the animation seed. You can click inside the window to start or sto the animation. Use right-click on the white shere to toggle the enveloe on/off. Fig. 2. The starting screen of the software. Fig. 3 is a screen shot when various arameters including the amlitudes, the frequencies, and the velocities are set, and the enveloe is turned on. 3

4 Resultant wave: E = E 1 +E 2 Enveloes E 2 E 1 Fig. 3. A screen shot when arameters are set and the enveloe is turned on. 3. Exercises In the following, the grou velocity is the velocity of the enveloe(s); the hase velocity is the velocity of the resultant wave (white curve); the arameters include amlitude, the frequency, and the velocity for the original two waves, E 1 and E 2. Note that these arameters are exressed in a relative manner, i.e., E 02 in terms of E 01, f 2 in terms of f 1, and v 2 in terms of v 1. Also when we say a velocity is ositive, we mean that the corresonding wave is moving from left to right; when a velocity is negative, the wave is moving from right to left. Exercise 3.1. Adjust the arameters E 02, f 2, v 2 so that the grou velocity is ositive and less than the hase velocity. Record these arameters. (Note: These arameters are not unique. Just list the arameters you find.) Exercise 3.2. Adjust the arameters E 02, f 2, v 2 so that the grou velocity is ositive and greater than the hase velocity. Record these arameters. (Note: These arameters are not unique. Just list the arameters you find.) 4

5 Exercise 3.3. Adjust the arameters E 02, f 2, v 2 so that the grou velocity is ositive and equal to the hase velocity. Record these arameters. (Note: These arameters are not unique. Just list the arameters you find.) Exercise 3.4. Adjust the arameters E 02, f 2, v 2 so that the grou velocity is zero. (Note: These arameters are not unique. Just list the arameters you find.) Exercise 3.5. Adjust the arameters E 02, f 2, v 2 so that the grou velocity is negative. (Note: These arameters are not unique. Just list the arameters you find.) Exercise 3.6. Based on the exeriences you ve obtained from Exercises 1~5, develo a theory that tells how to adjust arameters E 02, f 2, v 2 to get the grou velocity designated in Exercises 1~5. 5