Chapter 20 Electromagnetic Waves 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 ave on a stretched string See Fig 20- Figure 20-1 Wave on a stretched string
Usey to denote the hight of the string. Then gen erally it is a function of both the time t and the coordinate 2 Moreover, as a wave. it is a function as u(t-2/u), where v is the speed of the wave. SO, tor instance, the hight of z at the time t is equal to the hight of x=0 at an earlier time t-v 0(t-x/u)=y(t-21/)-0/0)
Usey 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 x=0 at an earlier time t-x/v (t-x/)=y(-2/)-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- q ueno cy, f=w/ 2T is the frequency, and T=1/f is the period a plane wave a =a(t-0)travelling along the x-axis can be written as a= an coslwlt The quantity w(t-i) in the bracket is the phase The wave is also unattenuated since an 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 b t-x/v=0, that is, at times t the position of the front is at z=. The wave length is the distance travelled over a pe- riod T=1/f 入=T=0/f=v/(0/2m)=270/
a plane wave can be also written as a complex func- tion In the 1-dim case loe 2((t-2/) ane(wt-2T2 o cow(t-x/v)+ ico sin w(t-2/u) co cos(t-2T2/A)+isin(wt-2TZ/A
20.2 The Wave Equation We take the 2nd derivative of onei((t-x/u )) w.r.t. the time t a2a at2 We take the 2nd derivative of a- cnei(w(t-2/v) w.r.t. the coordinate 2 02a(2r 0 From these two equations we get 02a(2m)202a12a 2at2220t This is the partial differential equation of waves in 1-dim case
In 3-dimensional case the plane wave equation is 1a2 at a general wave is a more complicated than this one
Figure 20-2 The quantity a o cos o[t -(z/0)] as a function of and as a function of t
