The Physics of Sound: Resonance and Standing Waves

So what does happen when two sound waves are in phase with one another?  The two waves constructively interfere with one another to result in one wave that this double the amplitude of the two waves.  Basically, they both add up like some awesome crime-fighting team...and they...help people hear and stuff.  (Yeah...I don't know where I was going with that metaphor.)  Anyway, to better understand this, let's talk a little more about the phenomenon of interference.

Interference is when two waves sorta "line up" together.  Depending on how they "line up," the two waves combine to form one wave that is either of lesser or greater amplitude than the two waves were just on their own.  Think of it like this:  If one wave is going along with an amplitude of, let's say, 2 dB, and it meets up with another wave that's out of phase with this first wave, and the second wave's amplitude is 1.5 dB, then the resultant effect will be the 1.5 dB wave "canceling out" some of the amplitude of the first wave.  So you'd get a net result of a 0.5 dB sound wave.  If, however, the 2 dB sound wave meets up with a wave that's totally in phase with it, and this wave is going along at 2 dB, the resultant wave will be 4 dB.  So, yes, interference is very much like when you hang out with that soul-sucking person you really shouldn't be around (destructive) or that person who just makes you feel great (constructive).  (That's a super-basic way to represent interference mathematically, and the real math is much, much more detailed and complex, but it's just there to give you an idea.  So please don't go around thinking it's just addition and subtraction when scientists are figuring out interference.  It'd be a bit like those people who think a graduate degree in vocal performance just means you sing karaoke all day and get a degree for it.)

Sound waves travel along just fine until they hit a boundary.  When that happens, the waves bounce off the boundary and become reflected waves.  The initial wave, called the incident wave, can meet up with the reflected wave where the two waves interfere with one another to form a new wave that is the sum of the other two waves.  This is called the principle of superposition.  (I know I'm getting a bit redundant, but hang with me here.)  If, during superposition, two waves meet up that are completely in phase, the result is a standing wave.

As you can see above, one type of standing wave doesn't travel anywhere.  It stays in the same place constantly.  This results in areas where the displacement is zero, called nodes (shown by the red dots above), and areas of maximum displacement called antinodes (the tall peaks and valleys above).  So the constructive interference of an initial wave meeting up with a reflective wave to form a standing wave looks something like this:

The red and blue waves meet up to form the standing wave in black.  Other cool animations can be found here and here. 

But how do standing waves that don't go anywhere contribute to a singer's resonance?  Well, that question is kinda jumping a bit farther ahead than where we are now.  For now, just think about standing waves on a medium that is fixed on both ends, like a string.  Ever played with a string or  necklace where you bounce the string up and down?  If you have, you actually formed a standing wave at the string's first resonant frequency, called the fundamental frequency.  But the string does have other frequencies it could resonant at, called overtones.

First fundamental and first six overtones of a string

But why am I talking about strings?  What do strings have to do with vocal resonance?  Think about it a second:  What acts like vibrating strings with fixed ends when we speak or sing?  Yup, the vocal folds.  But, the vocal folds vibrate in patterns that are much more complex than just a single string.  Remember how the air opens them from the bottom to the top, due to subglottal air pressure, and then the folds get sucked in laterally because of the Bernoulli effect?  The resulting wave pattern is very intricate,which results in a complex waveform (multiple simple sine waves going out at once) being produced at the level of the vocal folds.  The fundamental frequency and all of the overtones of the human voice originate at the level of the vocal folds.

*That last bit is very, very important, and it seems to be where a lot of singers get very confused...usually not due to any fault of their own.  The vocal tract absolutely cannot create sound waves or overtones to those sound waves:  Not the singer's formant, not the harmonics, not any of it.  All of the frequencies picked up by a spectrograph originate from vocal fold vibration.  The vocal tract only acts as a filter for the frequencies sent out by the vibrational pattern of the vocal folds.  And that's where we'll pick up next time!

Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.