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Home | Blog | Waves, Distances, Phase, and Delay in Cars
December 18, 2023

Waves, Distances, Phase, and Delay in Cars

This is a basic explanation of how we can use delay in Cars to get great sound in cars. Some concepts are simplified, and mathematics is avoided.

Sound waves are pressure changes that travel through air at a given speed (the “speed of sound”). The traditional “squiggly line” is a graph of air pressure at one point in space, as the pressure changes move past a point in space, the air pressure at that point rises and falls. These air pressure cycles make up a wave.

Figure 1: Rising and Falling Air Pressure

Figure 1: Rising and Falling Air Pressure

The difference between the highest and the lowest peaks of air pressure is the amplitude of the wave. That’s basically how loud it is.

Figure 2: The height of the peaks on the graph indicates the Amplitude

Figure 2: The height of the peaks on the graph indicates the Amplitude

When we talk about sound, we talk about cycles of air pressure rising and falling. Our ears hear this repeated cycle of rising and falling air pressure.

Figure 3: The physical distance between identical pressure values on the wave is the wave length

Figure 3: The physical distance between identical pressure values on the wave is the wave length

There is a unit of measurement - the Hertz - which signifies how many “cycles per second” comprise the wave. This is called the frequency.

Each air-pressure cycle has a physical length. This is the distance between the beginning of a cycle, and the end of that cycle and the beginning of the next cycle. That is called a wavelength. The length of a wave varies with its frequency. A 20 Hz wave length is 675 inches, and a 20kHz wavelength is 0.675 inches. A 440 Hz wavelength is 31 inches. (This image is not to scale.)

Figure 5: Wave lengths in the audible range vary significantly

Figure 5: Wavelengths in the audible range vary significantly

We describe the cycles as if they are circles, since they circle from ambient air pressure, up, then down, and then back to ambient air pressure. We start at 0 degrees, we go through 90 degrees to our peak air pressure, we go through another 90 degrees back to ambient, then another 90 degrees to our lowest point of air pressure at 270 degrees, and then we finally climb back to 360 degrees and ambient air pressure again.

Figure 6: Phase describes a point in the cycle

Figure 6: Phase describes a point in the cycle

 

Figure 7: A cycle is a circle.

Figure 7: A cycle is a circle.

Regardless of whether the cycle takes place 20 times a second, or 20,000 times a second, this manner of measuring the cycle in degrees works the same way.  We call this phase. The concept of phase allows us to define where in the cycle we are at any point in time. In car audio, we often discuss phase and polarity interchangeably, but this confuses matters. Polarity is binary, but phase exists on a spectrum.

If we do something to change the phase - if we cause phase shift - we are changing where the pressure wave is at a given point in space, without changing the frequency. A wave can be phase shifted in many ways.

One way is to reverse the (+) and (-) wires - the polarity. This shifts phase 180 degrees (for steady-state signals).

Figure 8: Polarity is binary.

Figure 8: Polarity is binary.

When you have one sound wave, phase changes are really hard to hear. If you play music over a single speaker, and you reverse polarity (which forces a 180-degree change in phase at every frequency), no one can determine which connection is correct and which one is reversed. There is no absolute polarity.

Similarly, researchers have manipulated the phase of signals at specific frequencies, and humans have been unable to hear the difference when listening to music over a single speaker. This surprised the researchers, who expected to measure how phase distortion affected music playback, and could not provide evidence that it did! (It is true that with test tones over headphones, humans can hear phase manipulations - but not with music over speakers).

However, when we have multiple sound waves at the same frequency - such as when more than one speaker plays the same sound - things get more complex.  The two pressure waves add together at our ears. The intuitive expectation is that two speakers are louder than one speaker, but anyone who’s put two subwoofers in a box and inadvertently wired one incorrectly knows that this is not always the case.

Figure 9: Two subwoofers, wired out of polarity.

Figure 9: Two subwoofers, wired out of polarity.

When two speakers play the same note, and they are the same distance from us (such as the two subwoofers above), the waves arrive at the same time, AND they are aligned with each other. (When sound waves travel the same distance since they both travel at the same speed, it takes them the same time to arrive at their destination.) This means they are aligned in phase.

Figure 10: Two waves aligned in phase.

Figure 10: Two waves aligned in phase.

When the waves arrive aligned, they add together to a larger wave, with taller peaks of air pressure and deeper dips of air pressure (greater amplitude).

Figure 11: Two waves, aligned in phase, summed.

Figure 11: Two waves, aligned in phase, summed.

When the waves arrive at different times, the waves may be misaligned to some degree. This means they are “out of phase” to some degree (phase is a relative term, not an absolute - that’s why we measure phase in degrees).

Figure 12: Two waves, slightly misaligned in phase due to distance.

Figure 12: Two waves, slightly misaligned in phase due to distance.

So, let’s say that the signal is a 440 Hz tone. The wavelength for 440 Hz is about 31 inches.

If the two speakers are both 31 inches away - the same distance away from us - the sounds arrive at the same time, and the waves are aligned in phase. They add together to one louder sound (+6dB louder). This explains why we expect two speakers to be louder than a single speaker.

Figure 13: Two identical sound waves emanating from the same point, in phase.

Figure 13: Two identical sound waves emanating from the same point, in phase.

If one speaker is one wavelength farther from us than the other - if one speaker is 31 inches farther from us than the other - the two waves will still be aligned, and they will sum together to one louder sound. They will add to almost 6dB (the sound which travels farther will be ever-so-slightly attenuated by its 31-inch-longer path).

Figure 14: Two 440 Hz waves (Middle C) emanating from two different points in space, on 31 inches farther from the measurement point, arriving in phase.

Figure 14: Two 440 Hz waves (Middle C) emanating from two different points in space, on 31 inches farther from the measurement point, arriving in phase.

The difference in these distances is called a path-length difference, and we will sometimes use this term to describe the relative positions of two speakers.

If the second speaker is farther away than the first speaker, and that path-length difference is a whole multiple of the wavelength - if one wavelength is 31 inches, then two wavelengths is 62 inches, and four wavelengths is 124 inches - then the waves are also aligned in phase, and they still get 6dB louder when they add together.

But what if the path-length difference is not a whole multiple of the wavelength? What if it is less than one wavelength?

Let’s start with the path-length difference being one-half of a wavelength. In that situation, the second wave is only halfway through its cycle, halfway past the measurement point - so it’s 180 degrees out of phase with the first wave.

This is as misaligned as they can get, so when the two waves add together, we get destructive interference and near-complete cancellation.

There’s no resulting pressure change at all!

Figure 15: Two identical waves 180° out of phase, and the resulting sum being near-complete cancellation.

Figure 15: Two identical waves 180° out of phase, and the resulting sum being near-complete cancellation.

This accounts for how two subwoofers cancel each other out when we inadvertently wire one backward (remember, inverting polarity forces a 180-degree phase shift).

“Well, this is good to know in theory”, you might say, “but music consists of many, many waves, at many various frequencies, all at the same time. And most of the time the second speaker isn’t the same distance, or a multiple of a wavelength, or a multiple of a half wavelength. Most of the time it’s somewhere in between.”

Yes, it is usually somewhere in between - since every frequency has a different wavelength, every speaker pair with a path length difference is aligned at some frequencies, slightly misaligned at some others, and badly misaligned at a few! This means that some frequencies reinforce each other and get louder, but some cancel each other and get quieter. Here are three tables to show what happens.

Table 1: Path-length differences and degrees of phase misalignment

Identical distances  = 0 degrees

0.125 wavelength   = 45 degrees

0.187 wavelength   = 60 degrees

0.25 wavelength     = 90 degrees

0.5 wavelength       = 180 degrees

0.75 wavelength     = 270 degrees

One wavelength     = 360 degrees/0 degrees

Table 2: Summing two identical waves together

0 degrees phase shift     = +6dB

45 degrees phase shift   = +5.65dB

60 degrees phase shift   = +5.35 dB

90 degrees phase shift   = +3dB

120 degrees phase shift = 0dB

150 degrees phase shift = -3dB

180 degrees phase shift = -30dB

210 degrees phase shift = -3dB

240 degrees phase shift = 0dB

270 degrees phase shift =  +3dB

300 degrees phase shift = +5.35db

315 degrees phase shift = +5.65 dB

360 degrees phase shift = +6dB

Testing has shown that amplitude differences of less than 3dB are not apparent as “louder” or “quieter” to the human hearing system. Those less-than 3dB differences sound like tonal changes, but are not noticeably louder or softer. Changes of less than 1dB are very subtle and difficult to discern reliably.

Two identical waves summed together, and aligned, result in +6dB increase in amplitude. That is our baseline expectation for the gain we get when summing two identical waves - or when using two speakers instead of one.

When we sum two signals, it’s often by playing two speakers, and when we use two speakers, we expect that +6dB of gain. If we don’t get it, we are probably wasting money on speakers and on amplifier power (remember, +3dB requires 2x the amplifier power, and -3dB is what we get when we lose half our amplifier power - so a 3dB swing is a very important change in our acoustic result!)

Here is what we get when path-length differences are present.

Table 3: Misalignment measured in wavelengths and its effect on the sum

Same path length  = gain of +6dB (the expected result from summing)

1/8th wavelength  = gain of 5.65dB (failed to gain an expected 0.35dB)

3/8  wavelength     = gain of 5.35dB (failed to gain an expected  0.65dB)

1/4 wavelength      = gain of 3dB (failed to gain an expected 3dB)

1/2 wavelength      = gain of -30dB (failed to gain an expected 36dB)

3/4 wavelength      = gain of 3dB (failed to gain an expected  3dB)

5/8 wavelength      = gain of 5.35dB (failed to gain an expected .65dB)

7/8 wavelength      = gain of 5.65dB (failed to gain an expected  0.35dB)

1 wavelength          = gain of +6dB (we again get the expected result)

Here are some visual examples of these misaligned waves adding together:

0 degrees phase shift = sum to +6dB

Figure 16: These sum to +6dB

Figure 16: These sum to +6dB

Figure 17: These sum to slightly less than +6dB.

Figure 17: These sum to slightly less than +6dB.

 

Figure 18: A 60-degree misalignment still has less than a 1dB impact.

Figure 18: A 60-degree misalignment still has less than a 1dB impact.

Figure 19: A 90-degree misalignment loses us 3dB of the potential 6dB (which is half of our power!)

Figure 19: A 90-degree misalignment loses us 3dB of the potential 6dB (which is half of our power!

Figure 20: At 120° of misalignment, there is no increase at all.

 

Figure 20: At 120° of misalignment, there is no increase at all.

Figure 21: At 180° of misalignment, the signal is nearly completely cancelled.

Figure 21: At 180° of misalignment, the signal is nearly completely canceled.

 

 

Figure 22: A complete wave length - 360* of misalignment - sums to nearly as much as no misalignment at all!

Figure 22: A complete wave length - 360* of misalignment - sums to nearly as much as no misalignment at all!

Every time the same sound arrives to our ears from more than one location, we have a potential increase in amplitude - or a potential decrease.

And this means that whenever two speakers play a wide audio range, some notes will get louder and some will get quieter - unless the two speakers are the same distance away from us. (This problem is not usually noticeable with subwoofers in cars, since the path length differences for subwoofer drivers are a small fraction of the wavelengths the speakers are playing - but if we let our subwoofers play midrange notes, the problem would affect them too.)

Fortunately, only the worst misalignments result in near-complete cancellation. Slight misalignments have been shown to be inaudible, and moderate misalignments - while they should be avoided - are not disastrous. It’s the worst misalignments - those which result in significant loss of total output - which must be avoided at all costs.

As a good friend says, “It’s not important that we be perfectly in phase, but it is very important that we not be perfectly out of phase”.

Here’s an animation that illustrates this wonderfully:

https://www.acs.psu.edu/drussell/Demos/superposition/interference.gif at https://www.acs.psu.edu/drussell/Demos/superposition/superposition.html

In car audio, these multiple arrivals can result from four different causes:

  1. You have left and right speakers, and you play a stereo recording with content meant for the center of the stage. To accomplish this, the recording engineer puts the content in left and right channels equally, in phase.
  2. You have rear speakers, and they play the same sounds as the front speakers.
  3. You have a multiple-element speaker system with crossover filters, so the low-passed speaker and the high-passed speaker play the same sounds in the overlapping transition band of the crossover filter network.
  4. A reflected sound in the cabin arrives later than the direct sound.

Figure 23: Stereo Path Lengths, Rear-Speaker Path Lengths, Crossover-Transition Path Lengths, and ReflectionsFigure 23: Stereo Path Lengths, Rear-Speaker Path Lengths, Crossover-Transition Path Lengths, and Reflections

What does this do to our sound?

A great deal of damage, it turns out. Fortunately, it turns out that we cannot hear multiple arrivals as multiple arrivals - as echoes - until the path-length differences involved are much longer than can fit inside a car. This is what lets us install multiple speakers without hearing echoes, but it has other effects.

Some notes get louder, and some get a lot quieter. Multiple arrivals create a pattern of peaks and valleys in our frequency response - the peaks due to the constructive reinforcement of some waves adding together to higher amplitudes and getting louder, and the valleys due to the destructive cancellations from other waves combining to lower amplitudes. This pattern is called a comb filter.

Here is an example. We have measured two sources of full-range pink noise, in phase, arriving at the same time. The red and blue lines measure Channel 1 and Channel 2, and the green line is the sum of 1+2.

Figure 24: The red and blue traces are two identical-response channels, and the green is the sum of the two when the two are aligned in phase and time (reflections eliminated).Figure 24: The red and blue traces are two identical-response channels, and the green is the sum of the two when the two are aligned in phase and time (reflections eliminated).

In this example, the combination, or sum, is +6dB greater - at every frequency -  than either signal alone. This indicates that the two signals we summed together are in phase at all frequencies. This is a simulation - in real life, we never get this perfect a result.

And here is an example of the same two signals, summed together, after one travels 27 inches of distance relative to the other, and is delayed 2.01 mS by this path-length difference. As you can see, there is significant loss of amplitude, and the frequency response is significantly damaged.

Figure 25: The comb filter created when the same two traces are summed together, when one is delayed by 2.0mS (or, 27 inches).Figure 25: The comb filter created when the same two traces are summed together, when one is delayed by 2.0mS (or, 27 inches).

Below about 100 Hz, it seems the expected 6dB increase can be seen, as well as at 500 Hz, 1000 Hz, etc. (In reality, the phase is not perfectly matched and the increase is a fraction below +6dB.) However, there is massive signal loss at 250 Hz, at 750 Hz, at 1250 Hz, etc. Above 5000 Hz, the individual cancellations are no longer visible, even on this high-resolution measurement - but a close examination reveals that the average sum is still only half of the expected +6dB.

Knowing the phase by seeing the sum

If we have two signals which have the same amplitude, summing them together can tell us how aligned the two signals are at any given frequency!

Figure 26: The sum tells us the phase offset.Figure 26: The sum tells us the phase offset.

In the diagram above, we know the two signals are aligned in phase if the sum is +6dB, while if the sum is -30dB, we know the two signals are 180° out of phase - and so on. All these various phase shifts are the result of one 27” path-length difference. The path length difference affects different wavelengths differently. Refer to Table Two and Three above for more information.

What do we do about this “multiple-arrival” cancellation problem?

Do we have to calculate and predict the effects of path-length differences for every wavelength for every frequency from every speaker in the system? The audible range covers at least 20 Hz to 20,000 Hz!

No. There is actually a much simpler approach. We can use distance-based delay to address these first three causes of multiple arrivals at a given listening position (that is, left/right, front/rear, and high/low interactions).

The distance-driven delay process calculates the delay to apply to each channel based on the path-length differences, and the flight times for the sound from each speaker to arrive at the listening position. Once all the absolute speaker distances are entered, the differences are calculated, the flight times are calculated based on the speed of sound, and all the channels are delayed by an appropriate amount. The signal for the speaker farthest from the listener is not delayed at all - we delay the signals for the others to align them with the signal coming from the one farthest away. It’s very straightforward, and very effective.

This does require our signals to be aligned in phase and time when we start, of course.

It also works for one listening position, but not for multiple listening positions. If we want to have great sound in multiple listening positions, we must use other approaches to manage the problem of phase cancellations resulting from path-length differences.

How precise must we be when measuring the distances?

We need to be accurate, but in terms of precision, microscopic differences in delay don’t result in massive changes in fidelity.

We do not have to be overly precise when we apply delay in the low bass frequencies, because the wavelengths involved are very long, and as we saw above, small misalignments result in very small amplitude differences and are not audible. An 80 Hz wavelength is 166 inches. One-eighth (0.125) of that wave length is 20 inches. If we look at the tables above, a 0.125 wavelength misalignment results in the sum being 5.65dB greater in amplitude, rather than 6dB. So if we made a measurement error of 20 inches - which would be a very poor measurement - we would fail to attain that potential 0.35dB!

We do not have to be overly precise when we apply delay to the treble frequencies, either - since these wavelengths are impossibly short, making misalignments appear and disappear with every minute movement of our heads. The wave length of 5000 Hz is just under 2.7 inches. The half-wavelength - where phase would be 180 degrees different - is 1.35 inches. We move our heads that much (and more) all the time, without noticing acoustic aberrations. Our hearing systems learned long ago to ignore narrow deviations once we get above the frequency where a sound would be 180 degrees out of phase at one ear relative to the other ear (and that is around 1500 Hz).  The result is that treble misalignments are not audible as individual cancellations (they are audible as lost amplitude, however, and that amplitude can be made up for in other ways).

When it comes to achieving fidelity in a car, using delay to overcome path-length differences is an important tool, but resolution in these measurements past a certain point is certainly not a big deal. It’s far more important that we verify that our signals are in phase when we start, especially OEM signals. OEM signals are rarely aligned in phase and time any longer, so testing and correcting these signals is a vitally-important topic for another day.

Does that mean that delay “puts everything back in phase”?

While that would be great, delay doesn’t magically fix every phase misalignment.

Some in our industry have even complained that distance-based delay doesn’t work, because after it’s applied, there are still phase misalignments remaining.

Properly-applied distance-based delay eliminates the comb filter cancellations caused by path-length differences, in one listening position. That’s all it does.

Does that mean that tape measures are not useful in predicting phase cancellations? Of course not!

What else causes phase misalignments?

  • If the signal you’re starting with has phase and time nonlinearities
  • If you set your crossover parameters problematically
  • If your OEM signal has uncorrected phase and time processing
  • If you wired an signal input out of polarity in error
  • If you wired a speaker output polarity in error
  • If there are reflections (which there always are)
  • Any change in amplitude (such as a crossover filter)

Distance-based delay won’t address these issues. If one or more of these problems exist in your system, and you apply distance-based delay, those problems will still exist in your system. That doesn’t mean delay’s not a powerful and valuable tool in achieving great sound, or course - it simply is a great tool to address the problems caused by path-length differences.

What about reflections?

“Direct sound” travels directly from a speaker to our ears, taking the shortest possible path. “Reflected sound” takes a longer path - it travels first from the speaker to a reflective surface, reflects off of that surface, and then travels to our ears. For this reason, reflected sounds arrive later than direct sounds. Once the same sound is arriving at two different times, we have multiple arrivals!

Fortunately, the farther sound travels, the more it is attenuated (that is, some of the initial amplitude is lost). So, reflected sounds are usually less powerful than the direct sound, and that means that phase cancellations are not as severe as they can be when both sounds are the exact same amplitude. For the worst cancellations to occur, the two sounds must be similar in level. This is one reason that reflections are the least crucial cause of multiple arrivals.

Reflections are a part of every listening room. We expect reflections - if we magically eliminated them, the sound would be unpleasant to our ears. For the purposes of this exercise, we will accept the effects of reflections as a cost of doing business. It turns out that we can achieve wonderful sonic results without worrying about the phase effects of reflections very much.

Best Practices

So, best practices include:

  • Verify the phase and time integrity of the signal you’re starting with
  • Correct the signal’s phase and time nonlinearities before tuning
  • Set up the system in such a way that QC catches any wiring errors
  • Configure crossovers so that phase errors are not baked into the result
  • Do not go out of your way to create reflections by getting overly creative on your speaker locations. (Some complex speaker installations end up making reflections worse than the OEM locations!)

Once you follow these best practices, setting delay based on speaker distances can be a very powerful - and very simple - tool to deliver great sound in a car.

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