Chapter 20 Electromagnetic Waves a The Waves The Wave equations Electromagnetic spectrum Poynting vector s
Chapter 20 Electromagnetic Waves ◼ The Waves ◼ The Wave Equations ◼ Electromagnetic Spectrum ◼ Poynting Vector S
20.1 Waves Wave on a stretched string See fig 20-1 Figure 20-1 Wave on a stretched string
Use y to denote the hight of the string Th en gen erally it is a function of both the time t and the coordinate z (2,x) Moreover, as a wave. it is a function as where v is the speed of the wave. So, for instance the hight of z at the time t is equal to the hight of x=0 at an earlier time t-a/v 0(-x/v)=y(t-z/)-0/v)
Use y to denote the hight of the string. Then gener- ally it is a function y(t, a)of both the time t and the coordinate 2. Moreover, as a wave, it is a function as where v is the speed of the wave So, for instance the hight of z at the time t is equal to the hight of z=0 at an earlier time t-2/v 0(-x/v)=y(t-x/v)-0/) Remarks Waves can be in air water. vacuum or other media The quantity propagated can be either a scalar, a vector. and a tensor
An oscillating quantity a can be written as a=ao cos(wt), where ao is the amplitude, w is the angular fre- quency, f=w/2T is the frequency, and T=1/f is the period a plane wave a=a(t-a/v) travelling along the z-axis can be written as a= ao coslw(t The quantity w(t-v in the bracket is the phase The wave is also unattenuated since ao is a constant
The quantity v is called the phase velocity of the wave, since it is the velocity with which the wave front propagates in space For instance, in 1-dimensional case, take C= 0 The wave front is given by that is at times t the position of the front is at z=ut The wave length is the distance travelled over a pe riod T=1/f 入=0T=/f=0/(/2x)=27/u
a plane wave can be also written as a complex func- tion In the 1-dim case a=a0el(a(t-2/)-=a0ect+-272/) ao cow(t-2/u)+iao sinw(t-2/v) a0 COS(ut-2x/入)+isin(ut-2丌2/入)
20.2 The Wave Equation We take the 2nd derivative of a- goei(w(t-2/v) w.r.t. the time t We take the 2nd derivative of a= coei(w(t-2/u) w.r.t. the coordinate 2. 02a(2) From these two equations we get 02a(2m)2a2a1a2 a22a2入2at2 O This is the partial differential equation of waves in 1-dim case
In 3-dimensional case the plane wave equation is 1 a2c V-a at a general wave is a more complicated than this one
2rU Figure 20-2 The quantity a= o cos o[t-(z/u) as a function of z and as a function of t