This post is long overdue. I lost my motivation after reading and engaged in a bunch of bickering posts. My apology.

Quoting Eberhard Sengpiel (RIP): “Each sound source has a far field and a near field, which is only source dependent.” We can infer from Sengpiel's statement that the near field and far field characteristics, and how and where the transition takes place, are different for different types of sources.

(Source:

https://sengpielaudio.com/DirectFieldAndReverberantField.pdf)

We'll start by looking at, AFAIK, probably the most elementary type of sound wave — a steady, single frequency sinusoidal plane wave. (Note: The purpose of the colors is to help keeping track of the air "particles". The color of each particle remains the same and is independent of its position.)

**TL/DR — Plane Wave:**
- In this discussion, a plane wave is an idealized pressure wave (in a uniform and stagnant medium) that propagates in a straight line in one direction. The pressure and other properties are uniform across any plane that is perpendicular to the direction of the wave propagation.
- It may be generated by an infinitely large flat wall with an oscillating motion. The plane wave propagates in the direction of the wall movement (and perpendicular to the wall).
- In a lossless medium — a reasonable assumption when dealing with distances typical for domestic listening rooms, the pressure of the plane wave does not vary with distance. The pressure at position
*r + δ *is the same as the pressure at position *r* delayed by the amount of time it takes for the pressure wave to travel the distance of *δ* (delay *τ = δ / c*, where *c* is the speed of sound).
- Pressure and particle velocity are in-phase for a plane wave — pressure peaks and troughs are aligned to the velocity peaks and troughs. This alignment is typically a characteristic of being in the far field. There is no near-field / far-field transition for plane waves.

**The More Elaborate Version:**
The mathematical expression that describes the acoustic pressure field of this plane wave is shown below. This expression is a solution to the 1-D wave equation for pressure. (This solution shown is for a wave traveling to the right, i.e. in the positive

*r* direction. A left-going wave is also a solution to the wave equation.)

Let's spend some time to examine this expression. In the second expression of

*p'(t, r)*, it can be seen that

*p'(t, r)* is the product of 2 sinusoidal functions, with frequencies

* ω* and

*k*. When working with electrical signals, we often use the radial frequency

*ω = 2 **π f = 2 **π / T*, where

*T* is the period. Here we additionally have the acoustic wavenumber

*k*, where

* k = ω / c = 2 π f / c = 2 π / λ*, and

*λ* is the wavelength. Comparing the two,

*k* can be seen as the counterpart of

*ω*, where

*ω* is the radial frequency in the time domain, and

*k* is the radial frequency in the spatial domain. At any given fixed time (constant

*t*),

*p'* gives a sinusoidal function in space. At any given fixed position (constant

*r*),

*p'* gives a sinusoidal function in time.

The relationship between pressure and “acoustic particle” velocity is given by the momentum balance equation, which basically is Newton's second law,

*F = m a*. Plugging in the expression for the pressure and after a few mathematical operations, we have the expression for the particle velocity.

For the plane wave, the velocity at a given position is the pressure at the location divided by the product

*ρc*, a quantity we call the characteristic impedance of the fluid. In electricity, from Ohm's law, we have: current = voltage / electrical impedance. The analog in acoustics is: velocity = pressure / acoustic impedance.

Since the characteristic impedance

*ρc* is a scalar quantity (i.e. a real, not complex, number), and multiplication by a scalar has no impact on phase, pressure and velocity for plane waves are therefore in-phase.

**The Standing Wave**
Where pressure and velocity can get out-of-phase is in a standing wave. We have standing waves when we hit a room mode (resonance). As an example, we synthesize a standing wave by summing a plane wave and a copy of it but going in the opposite direction.

The particle velocity again can be calculated using the momentum balance. The mathematical manipulations of the last step of the derivation are not shown. Those unconvinced can take a look this from Wolfram Alpha (scroll down to the “Alternate forms”).

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

Particle velocity is pressure multiplied by a purely imaginary number, which means it is 90° out-of-phase with pressure for this standing wave.