Introduction to Electrodynamics - ch09.pdf

文档介绍：Chapter 9
ic Waves
Problem
a:: = -2Ab(z - vt)e-b(z-vt)2; ~~1 = -2Ab [e-b(Z-vt)2- 2b(z - vt)2e-b(Z-vt)2] ;
8ft = 2Abv(z - vt)e-b(z-vt)2, 82ft = 24bv -ve-b(Z-vt)2 + 2bv (z - vt)2e-b(Z-vt)2 = V282ft .;
m '~ . [ ] M'
M ~h 2.
8z = Abcos[b(z - vt)]; 8z2 = -Ab sm[b(z - vt)];
8h 82h 2 2 . 282 h
8t = -Abvcos[b(z- vt)]; 8t2 = -Ab v sm[b(z:- vt)] = v 8z2";
-813 - -2Ab(z - vt) . --82h -2Ab + BAb2(z- vt)2 .
8z - [b(z- vt)2 + 1]2' 8z2 - [b(z- vt)2 + 1]2 [b(z- vt)2 + 1]3'
813 = 2Abv(z-vt); 82h = -2Abv2 + BAb2v2(Z-vt)2 =v282h..;
8t [b(z- vt)2 + 1]2 8t2 [b(z- vt)2 + 1]2 [b(z- vt)2 + 1]3 8z2
8i4 = -2Ab2ze-b(bz2+vt). 8214 -2Ab2 e-b(bZ2+vt) - 2b2z2e-b(bZ2+vt) .
= ]
8z I 8Z2 [ '
8i4 - - Ab -b(bz2+vt). 8214 - Ab2 2 -b(bz2+vt) -J. 2 82 i4
8t - ve , 8t2 - v e -r v 8Z2'
8 is 3 82 is 2 . 3 8 is 3 3 2 . . 3
8z = Abcos(bz) cos(bvt) ; 8z2 = -Ab sm(bz) cos(bvt) ; 8t = -3Ab v t sm(bz) sm(bvt) ;
8;~5 = -6Ab3v3t sin(bz) sin(bvt)3 - 9Ab6v6t4 sin(bz) coS(bvt)3 =I V2~~5 .
Problem
M ~i
8z = Akcos(kz)cos(kvt); 8z2 = -Ak2 sin(kz) cos(kvt);
~~ = -Akvsin(kz) sin(kvt); ~:; = -Ak2v2 sin(kz) cos(kvt) =v2 ~:{. ./
Usethe trig identity sin a cos 13= ~[sin(a + t3) + sin(a - 13)]to write
i = I ~ {sin[k(z + vt)] + sin[k(z - vt)]},
157
158 CHAPTER 9. IC WAVES
which is of the form , with 9 = (Aj2) sin[k(z - vt)] and h = (Aj2) sin[k(z + vt)].
Problem
(A3)2 = (A3eiO3) (A3e-iO3) = (A1eiOl + A2eio2) (A1e-iol + A2e-io2)
= (Ad2 + (A2)2 + A1A2 (eiOle-io2 + e-iOleio2) = (Ad2 + (A2)2+ A1A22cos(61- 62);
A3 = I v(Ad2 + (A2)2 + 2A1A2 cos(61 - 62).1
A3eio3 = A3 (cos 63 + i sin 63) = Al (cos 61 + i sin 61) + A2 (cos 62 + i sin 62)
... A3 sin 63 Al sin 61 + A2 sin 62
- (AI COS61 + A2 cos (h) + z(AI sm 61 + A2 sm 62). tan 63 = A3COS 63 = A1cos61 +A2coS62 ;
-1 AI sin 61 + A2 sin 82
83 - tan .
(Al COS81 + A2 cos 82)
Problem
82j 1 82j
The wave equation (Eq. ) says 2 = -:1 2. Look for solutions of the form j(z, t) = Z(z)T(t). Plug