Resumen
The coordinate-free one-way wave equation is transferred in spherical coordinates. Therefore it is necessary to achieve consistency between ????????????????
g
r
a
d
i
e
n
t
, ????????????????????
d
i
v
e
r
g
e
n
c
e
and ??????????????
L
a
p
l
a
c
e
operators and to establish, beside the conventional radial Nabla operator ?F/???
?
F
/
?
r
, a new variant ???F/?????
?
r
F
/
r
?
r
. The two Nabla operator variants differ in the near field term F/??
F
/
r
whereas in the far field ??»0
r
»
0
there is asymptotic approximation. Surprisingly, the more complicated gradient ???F/?????
?
r
F
/
r
?
r
results in unexpected simplifications for ? and only for ? spherical waves with the 1/??
1
/
r
amplitude decrease. Thus the calculation always remains elementary without the wattless imaginary near fields, and the spherical Bessel functions are obsolete.