8. What is phase?

Phase denotes the particular point in the cycle of a waveform, measured as an angle in degrees. It is normally not an audible characteristic of a single wave (but can be when we using very low-frequency waves as controls in synthesis). It is a very important factor in the interaction of one wave with another, either acoustically or electronically.

The Flash example below traces a sine wave by plotting the height of the tip of a spoke (or radius) of a rotating wheel (y-axis) against time (x-axis). The height of the spoke-tip corresponds to the waveforms relative amplitude, here plotted between an imaginary +1 to -1. To measure the phase angle, we start with the reference of the spoke pointing completly to the right and refer to that as 0º, with a relative height or amplitude of 0 as well. A the wheel rotates couterclockwise, the sine wave reaches its peak positive amplitude when the spoke has traveled 90º from its starting point (click on the 90º button), with a relative amplitude of +1. At 180º from the starting point, the amplitude of the sine wave has returned to 0 (click on the 180º button). At 270º, the sine wave reaches its peak negative amplitude of -1 (click on the 270º button) and then returns to 0 as it returns to its starting point of 360º or 0º.1

Another common sinusoidal waveform used in synthesis or measurement is the cosine wave, which is exactly the same shape as a sine wave and would sound exactly the same all by itself, but is distinguished by the fact that it’s cycle begins 90° out of phase to a sine wave, or at +1.

When waveforms of either the same or differing phases are combined, they interfere with each other and their instantaneous amplitudes are summed to create a new composite wave. This is a fairly simple-to-predict process with electronic circuits, but is far more complex in the realm of acoustic sounds as we will see below. For waves of differing phases, the fraction of a period between peaks is the phase difference and is also expressed in degrees, as above with the sine and cosine waves. Two waves with the same frequency and phase will combine to create a single sound of greater amplitude—this is called constructive interference.

Two identical waves 180 degrees out of phase will completely cancel each other out in a process called phase cancellation or destructive interference.

Often there is a combination of both destructive and constructive interference as pictured below in a mix of a sine wave and cosine wave of equal amplitude and frequency. Notice how the result is a sinus-shaped wave of slightly greater amplitude than either component, but slightly out of phase with both.

In real-world acoustic environments, constructive and destructive interference occurs constantly due to room acoustics and other factors. In fact, interference between the sound source and reflected waves are key to producing standing waves. Sometimes taking a small step to the side may completely change the timbral characteristic of a sound because it alters the phase relationship of the source and its reflections. Stereo microphone pairs that are improperly placed can inadvertently lead to unwanted phase cancellations at certain frequencies.

Two sounds with a small difference in frequency, say two piano strings of the same pitch, may be perceived as a single sound, but as the waves evolve, they move slightly in and out of phase with each other. The resulting constructive and destructive interference produce a pulsation of amplitude. This pulsation is known as beating. The rate or pulsation or beat frequency is the difference in frequencies. A string tuned to 440 Hz and one tuned to 441Hz will produce a pulsation once per second.

Use the interactive example below to create beats for yourself. Turn on the 440 Hz sine wave, then turn on the 441 Hz tone (see if you can time your click to the 1-second timer). This will produce one beat per second (441 - 440 = 1). Turn off the 441 Hz tone and click on the 442 hz tone. This will produce two beats per second (442 - 400 = 2). What happens when you activate all three tones?

 Use the interactive example below to create beats for yourself. Turn on the 440 Hz sine wave, then turn on the 441 Hz tone (see if you can time your click to the 1-second timer). This will produce one beat per second (441 - 440 = 1). Turn off the 441 Hz tone and click on the 442 hz tone. This will produce two beats per second (442 - 400 = 2). What happens when you activate all three tones?

For further study, see Hyperphysics->Interference, Hyperphysics->Beats

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1 Dodge p. 29-32

An Acoustics Primer, Chapter 8
URL: www.indiana.edu/~emusic/acoustics/phase.htm