Principles of Electromagnetic Waves and Materials, 2nd Edition

7Miscellaneous Topics on Waves*

Most of the problems we have dealt with till now are monochromatic waves unbounded in space. Practical signals involve pulse waves or beam waves. The simple solutions obtained for monochromatic waves can be used to construct solutions for more difficult problems. In this connection, we first explore the concept of group velocity.

7.1Group Velocity v_g

Group velocity is the velocity with which a group of waves travel. Let us consider that at z = 0, we have two sinusoidal oscillations:

(7.1)$$E\left(0\text{,}t\right)=sin\left({\mathrm{\omega }}_0-\mathrm{\Delta }\mathrm{\omega }\right)t+sin\left({\mathrm{\omega }}_0+\mathrm{\Delta }\mathrm{\omega }\right)t.$$

The two oscillations differ in frequency by 2∆ω.

Equation 7.1 may be also written as

(7.2)$$E\left(0\text{,}t\right)=2sin{\mathrm{\omega }}_{0}tcos\mathrm{\Delta }\mathrm{\omega }t.$$

A sketch of Equation 7.2 is shown in Figure 7.1.

FIGURE 7.1Two oscillations with ...