When one loudspeaker moves backwards while the other moves forwards, are they out of phase? That was the question asked by a friend who had got into a discussion whilst otherwise enjoying a drink between the soundcheck and gig. The article below is a modification of an email reply that I sent. He said that the answer gave him a headache, which is what happens when you mix alcohol and technical discussions.

The concept of phase with regards to sound reinforcement and audio systems seems to generate quite a considerable amount of shrugged shoulders and heated debate.

The dictionary tells us that phase is:

Date: 1812

It goes on to state that:

                - in phase : in a synchronized or correlated manner

                out of phase : in an unsynchronized manner not in correlation

In audio related topics number 3 is the most relevant definition, although the other definitions should remind us that as engineers and technicians, we have borrowed the term because we are describing a phenomenon that is similar to the cyclic patterns of phenomena that have been around for much longer than modern technology.

The most common use of phase in audio is to describe the reversal of signal voltage at a loudspeaker, usually, but not always caused by the reversal of polarity of the connecting cable. The result of this reversal is to cause a cancellation of the acoustic signal on the main axis between the loudspeakers when the output from left and right loudspeakers sum together. The above is a mouthful, but most people into hi-fi and or PA systems will have heard what I am talking about. Instead of a clear defined sonic image the sound is undistinguished and as defined above not in correlation. Despite its common usage in this way, there are those who consider this meaning to be abhorrent and with mission like veal undertake to convert the heathens to the proper terminology of "reversed polarity", as my friend found out.

The main other uses of the word phase are to describe inconsistent sound reproduction caused by comb filtering effects caused by the arrival of multiple identical signals with slight time delays to them, and the propagation delay through a filter. These later two give rise to the idea that phase is the same thing as time. To complete the round up, the term phase is used describe numerous mathematical expressions, but it is important to remember that when defining phase, just like any other term, in this way that it is just a convenient name to describe the expression. It could be called anything.

The old celestial analogies work well to show us the nature of phase. Consider the various seasons, spring, summer, autumn and winter. Each can be considered to be out of phase with the other. Winter and spring are 90 degrees out of phase and winter and summer 180 degrees. A 90 degree phase shift could be considered to introduce a three month time delay, so summer is always six months after winter. The reason for considering this as a phase difference is that if the earth rotated faster the time difference would alter so that maybe there was only 4 months between winter and summer. The phase relationship would, however, remain at 180 degrees. This is why it is more useful to consider a filter introducing a phase shift rather than a time delay; it is a lot simpler. If you know that a device introduces a 60 degrees phase shift you can then work out the time relationship at any frequency. If instead you used a time delay to describe the device, you would also need to specify the frequency at which the delay was measured. Before leaving the seasonal analogy, it is worth considering that it shows us another useful reason for using the term phase. Which comes first, winter or summer? It can't really be answered as there were seasonal changes millions of years before man arrived and no doubt they will continue for millions of years after we have gone. All that we know is that they continue to cycle round one after the other with a frequency of 12 months.

In PA systems there are, more often than not, multiple loudspeakers reproducing the same signal. As it is impossible for them all to occupy the same space there has to a difference in path length between the individual loudspeakers and the listener and/or measuring device. It is common to refer to the different arrival times of the signals as being out of phase. While this is a short simple phrase that indeed is understandable to most, in this case it is better, when analysing systems and performing calculations, to stick with describing the system in terms of time or distance (these two terms are related through the equation time = distance/velocity and so long as the velocity is constant one can be determined from the other). If you have two loudspeakers and describe the spatial separation by some convenient coordinate system, then that separation is fixed at all frequencies and all measuring positions. To describe the separation in terms of phase is impossible. For example, if we determine that two speakers are on the same axis that we are measuring and that the signal is in phase at 340Hz (assuming the speed of sound to be 340m/sec) it is possible for the loudspeakers to be any multiple of 1 metre apart. With the same frequency but moving off axis the phase relationship will change even though the system being measured hasn't.

You may have noticed that in the above discussions we have mentioned measuring phase at a particular frequency and that the phase will change when the frequency changes. In the case of the filter, which had a constant phase change, it must be remembered that we are comparing the phase of the input signal with that of the output so even in this case we are comparing one frequency at a time. Complex wave forms can be constructed from summing multiple sine waves of various frequencies and amplitudes, therefore although a square wave may appear to have the form of a single frequency it is actually composed of a fundamental frequency and many harmonically related frequencies. The only true single frequency wave form is the sine wave. I'll come back to more complex wave forms later, but let us now look a little closer at the sine wave. Most posh text books use the term sinusoidal where I am using the term sine. Some articles also use sin. Just like the term phase, if the intended meaning of the word is fully understood don't worry too much.

If you want to delve more into the maths and theory behind this article, there are many books available. It isn't really necessary for the purposes of this article to explain how the expressions were derived, but briefly, imagine the radius of a circle rotating at a constant angular speed. Now imagine that the circle is dragged forwards. The point at the end of the radius will describe a sine wave. The image below shows the classic textbook diagram of this and the animated image hopefully makes it even clearer.

The basic mathematical format for the sine wave is

Where Am = the peak value of the wave form

And

θ = an angular value between 0 and 360 degrees.

If θ was fixed we wouldn't actually have a sine wave therefore, because the value of θ is changing with time, the more common expression that is encountered is

Where ω = the angular velocity. If you are not familiar with radians and angular velocity ω = 2πf where f = frequency. The two terms describe the same thing.

Using the expression above, a sine wave will always have a value of 0 when t = 0.

To introduce a phase shift we simply introduce a fixed angular shift so that the expression becomes

Along with sine waves there are cosine waves. For our purposes we can consider then similar, but if you want to know more there are many text books that will go into great detail.

One interesting equation is

-sin θ  is simply sin θ multiplied by -1. In practical terms this is the same as reversing the polarity of a signal, so the above equation tells us that a 180 degrees phase shift is the same as inverting the signal.

The last equation would seem to indicate that polarity reversal and phase inversion are therefore the same thing. Others may point towards the earlier equation and emphasise the fact that time plays an important function in the expression. Both are right, but sort of missing the point. During the formal educational phase of my life (pun intended) I was taught that sine waves are a steady state phenomena and as such are considered not to start and stop (that would constitute a transient and other factors would apply) Setting a start point is just a convenience to us humans. As an analogy we set 0 degrees longitude as running through Greenwich, but it doesn't make Greenwich a special place as far as the earth's surface is concerned and saying that it is a start or end point on the earth's surface is crazy.

Coming back to audio signals, imagine drawing a graph of a 2KHz note from a flute which we will assume is a sine wave. The note lasts 2 seconds, which is not something unreasonable. To make the wave form clear on the drawing, we will scale it so that one wavelength is equal to 1cm. With 2000 cycles per second and at 2 seconds duration that is 4000 wavelengths that we have to draw. That makes the graph 4000cm long or 40m. Now imagine that you have two such graphs and that you are positioned somewhere half way along the graphs trying to align them. The start is 20 m one way and the end 20 m the other. Look at the graphs in front of you (I've drawn a section of the graphs below) and say which leads the other with regards to the start time. Unless you have very good eye sight it is not possible. All you can do is compare where each waveform is in its cycle relative to the other that is, the phase difference.

Moving on to some of the types of diagrams shown in text books we can consider the case of the current flowing through a capacitor when an alternating voltage is placed across it. By convention the current is said to lead the voltage by 90 degrees. The convention of leading and lagging again comes from the idea of the rotating radius. This can clearly be seem in the gif image below where blue leads red.

The next diagram below shows a classic text book illustration of phase shift and similar graphs can be found in any text book on the subject. It is easy to consider that this relationship between an alternating voltage across and an alternating current through a capacitor is one of time.

Consider the waveforms shown below.

The top graph is the voltage. The second is a similar graph moved along the axis. This would represent the graph of the current if a 90 degrees phase shift was simply one of time. The actual current is represented by the third graph. To realise why, we need to delve a little deeper into the relationship between voltage and current for a capacitor.

When a voltage is applied to a capacitor a charge builds up on it. If the voltage is changing, or alternating, the charge on the capacitor also alternates. The current is simply the flow of charge. The faster the movement of charge is, the higher the current is. Because the rate of charge movement is dependent on the rate of change of voltage, we can say that the value of the current flowing through the capacitor is proportional to the rate of change of voltage. Mathematically, this is referred to as a derivative and is written in the form below. This can apply this to any waveform.

If the concept of derivatives sounds complex, consider distance. Metres, kilometres or miles, any will do. If you now start to move you will have a speed. Speed is the first derivative of distance, and is simply the rate at which your position changes with time. The faster the rate of change, the higher the speed. Acceleration, which is the rate of change of speed, is the derivative of speed or the second derivative of distance.

For the above capacitor, the rate of change of voltage is represented by the slope of the graph. The steeper the graph the higher the current flow is. Where the graph is flat the current is zero. With a sine wave the graph is flat at the peak voltage, meaning the current is a minimum when the voltage is a maximum. The sine wave is the only wave form where the graph of the derivative is also a sine wave, hence the apparent time shift. For all other wave forms the graph of the derivative is completely different as shown above.

The above example is moving away from acoustics but can be considered relevant as reactive components (capacitors and inductors) are used extensively in filter and cross over circuits. Getting back to audio signals, it is very unlike that in everyday circumstances that sine waves will be reproduced. In the strictest sense it is therefore inaccurate to use the term phase at all. It is in such common usage though that it seems churlish to argue about how it should and shouldn't be used. If the intended use of the term is clear why be pedantic. If the use of the term is not so clear then further information should be given to clarify the intended meaning.

To give some idea about applying the concept of phase to signals with different frequencies, consider just two as shown in the diagram below. Both start with an amplitude of 0 when t = 0 seconds. As the wave progresses the signals become more out of phase.

If we represent the phase of the wave forms as

If wave (a) is represented by the dotted line and wave (b) by the solid line then the phase difference is the distance between them. If the difference is considered to be linear then the gap between the two straight lines represent this difference and  will continue to increase. The difference can be either positive or negative depending which wave is considered the reference. If phase is considered to be a fraction of a cycle, that is 360 degrees is the same as 0 degrees, then when the graph reaches 1 it will return to 0. This will happen for each cycle. This is represented by the saw tooth graphs. In this case each of the waves can both lead and lag and the phase will change between positive and negative as shown by the graphs crossing. If complex waveforms can be represented as the sum of individual sine waves of varying frequency then where and what does one use as a reference to measure or compute the phase difference from.

Some articles that I have read show diagrams of phase shifted pink noise. By its very definition, noise is a random signal so, as mentioned above, just what is used as a reference by which the phase is measured? The only way that I can think of, is that the pink noise has been generated digitally by a sequence of random numbers. If the sequence is 1000 numbers long then number 250 along the sequence might be considered 90 degrees out of phase. As each number in the sequence is random it would be impossible, just looking at the output, to distinguish what if any phase shift has occurred. Despite the fact that the concept of phase can't be applied to a complex waveform, if such a waveform is split and then one of the resultant waves is delayed before being summed with the other, interference will still occur and cause dips and peaks in the response. This interference may be referred to as phase cancellations but it must be remembered that any signals of any shape will add together in some way to produce a resultant signal, there doesn't have to be any phase relationship between them.

To wind up and come back to the original question When one loudspeaker moves backwards while the other moves forwards then I would say that they are out of phase but it is more than likely to be caused by the polarity of the signal being reversed somewhere. So phase inversion and polarity reversal are different things but when the signal is a sine wave the result is the same; this is not so with complex wave forms usually associated with music and speech. However, phase and time are not the same thing either and this is where most of the missionaries come unstuck. They are just as wrong to say that a delayed signal has been phase shifted as the person who they attack for using phase instead of polarity. Myself, I use polarity when referring to connectors, cables, plugs or anywhere there is something that can physically be swapped. For other stuff I use phase. The argument of just what phase is has been going on for a long time and everyone seems to have their own version. Hopefully the above will give a little insight to some of its properties and you can now make up your own mind. Don't expect to get too many converts though.