Consider now a rotation of coordinates to the "asynoptic" s,r axes, 
inclined an angle -a relative to the A,t axes (Fig. 3). It follows then that 
|c,|à-t à +|c,lt 
gum ee (7,1) ris, —— 1 (7,2) 
a 1.4 UT 
Similarly in the transform plane, the wave vector can be expressed in terms 
of its asynoptic components, kir (k. .k,.) (Fig. 3), with 
   
  
  
  
  
  
  
(5.2) 
{ EN (8,33 kit Fol (8.2) 
y. 0077 y 2 
] Co 1 + Co 
Da 
(6.3) 
time 
It 
ctangle 
(b) 
Fig. 4 Synoptic rectangle, D, and asynoptic strip, D': domains of trans- 
formation. The asynoptic strip is defined by the descending nodes 
Located at r = -R/2 and their periodic images at r = +R/2. Ascending 
nodes are not in general equidistant, from the two descending loci. 
Note (7) represent hybrids of space and time, and (8) an amalgam of wavenumber 
and frequency. It can be shown (Salby, 1982a) that the space-time spectrum is 
1 equivalently expressible as the double Fourier transform over the asynoptic 
Y strip D' - [-R/2,R/2]X[-S/2,S/2] oriented parallel to the s,r axes (Fig. 4). 
Thus 
S 1 /29. 5/2 -ikex 
: w(K) = I< f pole 7 7 dx (9.1) 
=R/2 *S/2 
where 
‚nd time. lc. | 
egration C 
J R=—2— (9.2) S. T. Zl +c? . (9.3) 
1 + Co 
159