The acoustic particle velocity in the r direction, U, is in phase with the acoustic pressure, p, everywhere in the far field of a simple acoustic source radiating spherical waves. The quantities p and U are related by the characteristic impedance as
using the notations of Ref. 1 wherein a far field assumption was assumed but not stated.
In general p and U are related by the r component of Euler's equation, = 0 ,. Suppose that in the near field of a time harmonic source one has the pressure
Then the corresponding particle velocity component in the r direction is
One can see that as kr becomes very large in the far field, the simple plane wave relation Eq. (1) results. The real parts of Eqs. (2) and (3) are
In Ref. 1 the additional nearfield term of in Eq. (5) was omitted. Thus the corresponding equation for the instantaneous intensity should be
The time averaged intensity is then
which is unchanged from Ref. 1.