APPENDIX D
OBTAINING EQUATION (3.79)
In this Appendix we show the algebra involved in obtaining equation (3.79) from equa
tion (3.51) using the assumption that the dispersion relationship is written as
u a + k cos OU
We start by using the following form of <¡>
- -{g i4(I,y) C*'(/ CO. i dz)
and then make a phase shift by using the substitution
cof z)z + (pv)v + k2p(¡> + [w2 a1 + i'w(Vfc U)] + iaw4>
-(U24>z)z-{v24>v)v-{uv4>x)v-{uviv)z + 2i = o (d.5)
Writing out all the components of equation (D.5) and rearranging terms gives
{p u2)Jx + {p- u2)4>xx + [(p V2)4>v]v + k2p4> + (oj2 o2)~4>
+ojUx4> + iuVv4> + icrw (UVx)y [UV*v)x + 2iuU**x + 2iuV^>y = 0 (D.6)
The derivatives of defined by equation D.2 are as follows
4>z = ig [(^) + ik cos 9^ e*(fkeo,tdz)
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