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