Phase is the AC representation of a waveform.  We are all familiar with Alternating Current every time we plug a cord into a wall socket, we know that current first flows one direction, then reverses and flows the reverse direction- alternating current.  The number of times that it alternates direction in one second is what defines Frequency.  

Acoustic energy also has an analogy to this.  All vibration is a compression followed by a rarefaction, so in this sense alternating like AC current, with a positive half cycle (compression) followed by a negative half cycle (rarefaction) then repeating (see Fig 1).  Think of a 15” woofer playing a low E on a bass guitar (40Hz), the woofer pushes outward compressing the air thus making it denser, and then it sucks 

Acoustic waveforms also have an impedance, just like AC electricity (impedance is AC resistance).  Acoustic impedance is the ratio of acoustic pressure (voltage if you will) and flow (analogous to current).   Think of this as an acoustic Ohms law…acoustic pressure/acoustic flow= acoustic impedance (R=V/I).  

If you are buying into the alternating current aspect of acoustic waveforms, then lets now extend that analogy to how alternating current is almost always represented, by an alternating (oscillating) wave, such as the voltage sine wave show below in Fig 2:

Mathematically, the way that voltage starts at zero, rises to its maximum level, drops back to zero, changes direction, repeating the previous half a cycle, is likened to a radius of a circle rotating 360 degrees (OK, vector rotation- see Fig. 3), so this is where the degree thing in phase measurements comes from.  It describes where the waveform is in this repeating duty cycle. Again, think of the positive part of the cycle as acoustic compression (high pressure), and the negative part as acoustic rarefaction (low pressure).

So why does phase matter?  Well, by itself from a single sound source, it really doesn’t matter much at all, but more of that later.  However, if we talk about two sound sources combining, now it’s a completely different story.  Using the sine wave analogy, if we sum together two waveforms (sine wave or acoustic) and they start off at the same time, called “in phase” and are like the example below identical waves, the result is the same waveform, but with greater amplitude (Fig. 4).  

If on the other hand, one sine wave starts going the positive direction and the other starts going in the negative directions (which is called “out of phase”), then they just cancel each other out (Fig. 5).

Obviously, they can be just partially in phase as seen in Fig. 6, or just partially out of phase and all points in between, and you get a summation that depends on where they start their duty cycle.

Two waveforms, in this case two sine waves, will sum together to produce different composite waveforms depending on the relative phase of each wave.

So what does an acoustic (loudspeaker) phase curve look like?  Below is a typical phase curve, commonly referred to as a “wrapped” phase curve.  Why “wrapped”?  Mostly just to make it conveniently fit on a graph with the kind of aspect ratio (width to height) that fits comfortably on a graph with the same aspect ratio as a loudspeaker frequency response graph.  Virtually all “wrapped” phase curves start with a positive degree value at the lowest frequency and slope downward to the maximum negative -180° on the X axis of the graph, but in reality phase curve don’t stop at -180°, but continues as a negative slope.  To “force” phase curves fit on the graph, the curve starts over again at +180°, and again slopes downward to -180° as seen in magnitude and phase graph depicted in Fig. 7 (the phase curve is grey, the magnitude frequency response curve is black), repeating this wrapping convenience up to the most upper frequency (usually 20 kHz) of the graph.  However, in a reality the phase curve is just one long sloping line.  This can be seen in the Fig. 8, a phase only graph where the phase is commonly referred to as “unwrapped”

Why is the unwrapped phase curve just one long downward sloping curve that keeps increasing in negative phase degrees as frequency increases from low to high?  In a word, it’s a reflection of the relationship between Phase, frequency and wavelength. If you look at the depiction of sine waves as frequency increases (see Fig. 9), the time period for 1 cycle gets progressively shorter, such that compared to a single 360° time period at low frequency, a high frequency time period may include 2 or more cycles in the same time period as a lower frequency.

Since wavelength gets shorter as frequency increases (ok, the time period is frequency dependent), higher frequencies are represented, on the same graph, with increasing higher numerical degrees of phase, hence the downward sloping phase curve as frequency increases. 

But what about the “wiggles” in a phase curve?  As the magnitude of a frequency response rises and falls, and since phase is an AC representation of frequency response, relative changes in phase also rise and fall.  In other words, phase changes with the slope of the magnitude curve.  If the slope of the acoustic magnitude curve (loudspeaker frequency response) is zero (flat curve shape), and the because the reference that the frequency magnitude is being compared to (the electronic signal) is always a flat curve, there is no net phase change, or 0°.  If the magnitude slope is positive (like a high-pass filter), then compared to the flat reference, phase there will be a positive change in phase.  Likewise, if the magnitude slope is negative (like a low-pass filter), then compared to the flat reference signal, phase change of a negative magnitude slope will be a negative change in phase (see Fig. 10). 

So we ask the question again, is acoustic phase important? Does it matter subjectively when listening to loudspeakers?  Yes, but it depends.  If we are talking phase from a single source, empirical evidence suggest that you really can’t hear phase, just mostly the magnitude of the waveform.  It is also suggested that once a loudspeaker is in a listening room, the anechoic phase of the speaker becomes scattered due to all the reflections in the room.  However, if there are two or more sources, then they can sum together in ways that cause amplitude cancelations and summations that are definitely hearable.

If you follow the latest technology in the studio monitor market, you may have heard or read the claims being made about “zero phase”.  Zero phase is accomplished artificially using DSP to render a phase curve that instead of going from +120° to -720° hovers around 0° (at least above 50-100Hz) as shown in Fig. 11.  Also remember that we are talking about the anechoic (no reflections) magnitude (frequency response) and phase response of the speaker. 

The claim made for this type of DSP processing using FIR filters to achieve Zero Phase is that it produces a tighter low end, wider sound stage, and just plain sounds better.  However, as I previously commented, acoustic phase is not very hearable, so rendering the phase response close to zero is probably not going to change much, despite the all the claims being made.  

References:

1. Loudspeaker Measurements and Their Relationship to Listener Preferences: Part 2- by Dr. Floyd E. Toole; J. Audio Eng. Soc., Vol. 34, No. 5, 1986 May.

2. The Loudspeaker Design Cookbook, by Vance Dickason, 7th Edition, 2006, Library of Congress Cat. # 87-060653