On Acoustc Field

The instantaneous acoustic intensity correspond to the energy flux
density per unit of time, it is the instantaneous rate per unit of
area at which work is done by one element of fluid on an adjacent
element. The acoustic intensity is measured in W/m² and is
determined by the product of acoustic pressure and the complex
conjugate of the particle velocity (if we're treating those
acoustics variables as complex exponentials).

I = P U*

Let's suppose that both acoustic pressure and particle velocity
are harmonic functions of time (which is quite a suitable
assumption). So they might be expressed as:
P = P₀ e^{- j ω t}
U = U₀ e^{- j ω t + θ}
so the instantaneous intensity is
I = P₀ U₀ e^{- j θ}

If we're dealing with real entities we have:
p = P₀ cos(ω t)
u = U₀ cos(ω t + θ)
i = P₀ U₀ cos(ω t) cos(ω t + θ)
i = P₀ U₀ cos²(ω t)cos(θ) - (P₀ U₀/2) sen(2 ω t)sen(θ)
i = (P₀ U₀/2) cos(θ) + (P₀ U₀/2) cos(θ) cos(2 ω
t) - (P₀ U₀/2) sen(θ) sen(2 ω t)
i = P + P cos(2 ω t) - Q sen(2 ω t)
where
P = (P₀ U₀/2) cos(θ)
and
Q = (P₀ U₀/2) sen(θ)

P is called the mean intensity, or the real intensity, and Q the
reactive intensity. The real intensity describes an energy
transfer conveyed by the sound wave. The reactive intensity
corresponds to an oscillation of energy around a fixed point for
which the mean value in time is zero. When pressure and speed are
90^o out of phase, the average intensity is zero, the phenomena
is pure reactive and no energy transfer produced by the sound wave
is observed.

The space surround the acoustic source is usually subdivided into
three regions: (a) reactive near-field, (b) radiating near-field
(Fresnel) and (c) far-field (Fraunhofer) regions. These regions
are so designated to identify the field structure in each.
Although no abrupt changes in the field configurations are noted
as the boundaries are crossed, there are distinct differences
among them. The boundaries separating these regions are not
unique, although various criteria have been established and are
commonly used to identify the regions.

Reactive near field is defined as that portion of the near-field
region immediately surrounding the acoustic source wherein the
reactive field predominates. The radiating near-field (Fresnel)
region is defined as that region of the field of the acoustic
source between the reactive near-field and the far-field region
wherein radiation fields predominate and wherein the angular field
distribution is dependent upon the distance from the source. The
far-field (Fraunhofer) region is defined as that region of the
field of the acoustic source where the angular field distribution
is essentially independent of the distance from the source.