Interference

Interference due to Phase Relationships

Here are two sine waves, both at 440 Hz, the second one shifted over by an amount that will cause the two waves to almost cancel each other out when they are played.

Plot[ {Sin[2 Pi 440 t] ,Sin[ 2 Pi 440 t + 3 ]},
	{t, 0, 1/440}]

Here are the two waves played together, one through the right channel, the other through the left.

Note: The following examples will only sound correctly on machines capable of stereo output.

Play[{Sin[2 Pi 440 t], Sin[2 Pi 440 t + 3]},{t, 0, 1}]

Here are the same two frequencies,  played again, but without any difference of phase between them.  Note how much louder this is, due to the absence of phase interference.

Play[{Sin[2 Pi 440 t], Sin[2 Pi 440 t]},{t, 0, 1}]

Note: The difference in volume is not due to the amplitude settings of the two sine waves, but due to the interference they cause each other when coming out of the speaker.

Here is an example in which one of the sine waves is phase-shifted by the value Sin[t] * 3.  Effectually, the amount of shift in the phase is moving from -3 to +3, which generates the changing magnitude of the change in amplitude.

Play[{Sin[2 Pi 440 t],Sin[2 Pi 440 t + Sin[t] * 3]},
	{t,0,Pi},SampleRate->44100]

Interference due to Beating

Beats occur when two sounds are close to each other in frequency.

The resulting sound periodically waxes and wanes because the two waves are periodically in and out of phase, hence, they periodically reinforce and interfere with each other.

The frequency of the beats can be calculated from the following formula:

	beat frequency = freq2 - freq1
	
	if freq2 - freq1 <= 15, beating will be heard
	if freq2 - freq1 > 10 to 15, we hear the two as separate frequencies
		
In music, the tones we hear are complex, so many possibilities for beating arise, for example among the upper partials of notes played by instruments.

Beating between pitches is frequently used for tuning instruments (the guitar), and for setting certain organ stops (the voix celeste).  Beating also helps explain the timbre of massed strings, choral groups, and others, that not all tuned  to exactly the same pitch.

Here is a plot of the relationship between the frequencies 440 and 455 Hz,  showing them moving apart.  Later, they come back together.

Plot[{Sin[2 Pi 440 t],Sin[2 Pi 450 t]}, {t, 0, 15/450}]

Note: This example will only sound correctly on machines capable of stereo output.

Here is the same relationship  between the two frequencies, presented in sound:

Play[{Sin[ 2 Pi 440 t], Sin[2 Pi 450 t]},{t,0,4},
	SampleRate->22050, SampleDepth->16]