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 [2],[3].
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

and

respectively.

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.

Fri Aug 30 16:31:47 EDT 1996