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.

Physics of Sound Series: The Waveform (or, What the Heck am I Looking At?)

We now know that a sound wave is made up of moments of compression and rarefaction, and we know a little bit about a waveform as well.  But there are other parts to a wave that we need to know about before we get into just what resonance really is.  Those parts are:  Period, frequency, amplitude, phase, and wavelength.

No doubt, you've heard of some of these before.  Frequency and amplitude, in particular, get a lot of attention in the music world.  A lot of times, we talk about frequency and amplitude as synonymous with pitch and loudness, and for most purposes they are.  However, when I talk about frequency and amplitude, I'm going to be referring to the actual physical properties of the sound wave (or waves), meaning the parts of the wave that can be measured with proper instrumentation and then studied.  Therefore, frequency and amplitude are objective measurements.  Pitch and loudness are usually associated with the perceptual properties of the wave, i.e. just how loud or how high/low a person perceives that sound differs from person to person (and from ear to ear, for that matter); so pitch and loudness cannot be measured per se, but rather discussed subjectively.  (And remember from my previous post:  Anytime I'm talking about perception, I'm talking about the interaction from a sensory signal with a person's higher cognitive functions and life experiences.  Therefore, perception is always subjective.)

Yup, those are the same sine and cosine functions from trigonometry that you see on your calculator.

When you look at the waveform of a basic sine wave (the red line) shown above, you'll see that it has a repeating pattern.  The number of repeats of this pattern in a given amount of time is called the frequency of the wave.  This is usually given in Hertz, but can also be stated as the number of cycles per second of the wave pattern.  (See, it was originally called cycles per second (cps), but then the International Electrotechnical Commission (IEC) decided to honor Heinrich Hertz's contribution to the field of electromagnetism, so they gave him a unit of measurement, cps, and called it Hertz (Hz).  Scientists are always re-naming units to honor the great contributors to the field.  Sorta like how medical terminology is also littered with the names of big anatomy contributors, etc.)  So, for the famous A440 that orchestra's (supposedly) tune to, the frequency is 440 cycles per second, or 440 Hz.  In math terms, frequency is shown by:  frequency = velocity over wavelength (f = v/λ).  Sounds pretty fancy, but the reason I'm putting this here is because the period of a wave is related to the frequency.  The period is how long it takes for one cycle of the wave pattern to complete itself.  So if the frequency could be shown as: 1/period, then the period is shown by:  1/frequency.  Seems like we're talking about the same thing, but in general, the frequency refers to how often the wave pattern is repeating itself where the period refers to how long it takes for one pattern of the wave to complete itself.  Why do we bother with this distinction?  Well, because it comes in really handy for mathematical analysis of wave patterns.  Why should singers bother to know about this?  Because...well...I'll get there for ya.  (Besides, if your ever playing around with PRAAT or some other spectrograph software, you'll probably see options for period or frequency change, and you might want to know what you're changing out as you play around.)

Wavelength corresponds to the distance one cycle of the wave travels.  Slower frequencies have longer wavelengths, so one cycle of A440 travels double the distance through the atmosphere than A880.  (In the above equations, wavelength is represented by that funny-looking symbol, which turns out to be the Greek letter lambda.  So now, when you poke around wikipedia and see frequency equations, you'll know some of what you're looking at.)

The amplitude of a wave corresponds to it's perceptual loudness, and is related to the amount of displacement the air particles go through in the sound wave.  Because it has to do with how far each particle is being "pushed," amplitude represents the atmospheric pressure of a sound wave, and is measured in decibels (dB).  In the waveform shown above, amplitude is represented on the vertical axis.  So if the sine wave had a higher amplitude, it would have taller peeks and lower valleys, going past the 1.00 marked above.  Even though amplitude of sound is represented by this vertical displacement on the waveform, in longitudinal waves, the displacement in the real world is happening horizontally.  This is different for other wave types, like light, but the math and the graphical representations are the same.  (I just want to point that out because it's easy to misinterpret the sound waves from your mouth as looking just like the waveform representation, but if we could see the sound wave, it would like more like the animation here.  That's kinda important to remember once we get to the anatomy of the ear and the role the ear drum plays in hearing.  And, oh yeah!  I'm going to get into the anatomy of the ear and how it plays a role in resonance as well, for both the audience and the singer!)

Phase is where we start to get into some important stuff when it comes to understanding resonance, and especially the phenomenon of standing waves.  The phase of the sine wave in the above picture basically is where in the cycle the wave starts when you're looking at the vertical axis.  Let's look at it more closely:

See how the sine wave is passing through the vertical axis where the horizontal line equals 0?  Now look at the cosine wave (blue, dotted line).  Cosine is passing through that vertical axis where a horizontal line equals 1.  So the phase of the sine wave is not the same as the phase of the cosine wave.  The fancy way of saying that is that cosine has a different phase shift than sine.  In fact, that's actually the main difference between sine and cosine:  The phase shift between the two.

Phase is really, really important because if you have two sound waves that are out-of-shift like this:

Look at the three middle waves to see the phase difference.

You'll see how when one wave has a peek in its amplitude, the other wave has a valley, or a negative amplitude.  This means that when one wave is in it's period of compression, the other is in rarefaction.  The result is that these two waves actually cancel each other out, because if the atmospheric pressure is equally positive in one wave while the pressure is equally negative from the other wave, the two pressure differences cancel each other out.  1-1=0, right?  Crazy, huh?

But what happens when two sound waves are perfectly in phase?  What if you've got 1 + 1 instead of 1 -1?  That's where the phenomenon of standing waves comes in, and that's what we'll start up with next time.  Stay tuned!

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.

Physics of Sound Series: The Acoustic Wave

I always have the hardest time starting up these series, because I spend a lot of time trying to figure out where to start.  I always know what the ending conclusion should be, but what's the beginning?  What's basic without being too basic?  So I'm going to start out where I think it should start out, but if I'm not being basic enough, please feel free to post any questions you may have.

Everybody always talks about resonance in the singing world.  Resonance, resonance, resonance.  Let's face it, as opera singers, we're pretty obsessed about it.  And why wouldn't we be?  It is, after all, the key to how opera singing works.  It is exactly how we are able to sing over an orchestra for hours at a time without hurting our voices.  The only issue I have with all this resonance talk is that it is painfully obvious that (some) singers have absolutely no clue what resonance really is.  It often gets talked about as a subjective thing that changes from person to person.  This is understandable given that so much of the sensation of singing is subjective, and therefore, how we teach singing is subjective.  It only makes sense that singers would start to think everything about singing is subjective somehow.  However, when we take something from the hard sciences, like resonance, and think of it as something that acts differently from person to person, as if it doesn't follow the laws of nature, we kinda sound like fools.  The other issue with all this resonance-as-subjective talk is that it makes what could be very clear pedagogy very fuzzy and confusing.  So, in order to fully understand what resonance is and how it can help us sing better, let's start with how a single sound wave works.

There are a lot of things in nature that function like waves:  Light, sound, the water in your bathtub...(okay, fine, ocean water too), but what exactly does that mean for sound to have a wave-like pattern of behavior?  Well, here's the definition of a wave from physics:  "a disturbance (an oscillation) that travels through space and time, accompanied by a transfer of energy...often with no permanent displacement of the particles of the medium (Wikipedia)."  Sounds pretty fancy, am I right?  But it does make a lot of sense.  If you drop a rock straight down into a body of still water, the rock disturbs the water's stillness causing a rippling of waves that travel out to the edges of that body of water.  Energy was transferred from the rock to the water which then traveled out to the edges of the body of water.  The water itself, though, will return to being still, i.e. it doesn't just keep traveling away from the rock until there's no water left, so there wasn't a permanent displacement of the particles of that water (you know, H₂O).

So what's the "medium" for sound waves?  Air particles!  All the lovely little air particles that make up our atmosphere is the medium for all the sound waves we hear, and the ones we don't hear too (i.e. ultrasound, infrasound, etc).  For our purposes, we'll think of a sound wave as beginning with air particles at rest.  An external force then comes along and sets those particles in motion (like when the electric slide is played at a wedding...sorry, couldn't resist.)  Anyways, let's imagine those particles are all lined up nicely next to one another.  The particles in row A, the ones closest to the external force, then get "pushed" up towards the particles in row B.  This is where we say the row A particles are "compressed" against row B, which then gets pushed up against row C, and so on.  (Anyone who's ever seen elementary-school kids line up for recess knows what I'm talking about here.)  So while each row is going into it's period of compression with the particles in front of it, the rows that have already been compressed then go into a period of rarefaction.  This would be when row A, after compressing with row B, swings back towards it's resting position.  But instead of landing at rest, row A actually over-shoots its resting position and ends up being spaced out farther from the row B particles.  If we want to get even more specific here, the property of inertia for those particles causes row A to compress with row B, then the property of elasticity over-takes row A's inertia, sending the particles back towards resting.  However, the property of inertia for that row of particles then over-takes elasticity and causes row A to over-shoot it's resting position.  But, don't fear, cause elasticity will over-take inertia and send row A back to towards resting.  This process will repeat itself until row A is again completely at rest.  Sounds complicated, but if you've ever set a pendulum into motion and watched until it came to rest again, you've seen this same action at work.  (*Edit to add:  This pattern of motion is called simple harmonic motion and is actually what pretty much everything in nature can be reduced to.)

This pattern of compression and rarefaction makes up what we call the sound wave.  This is why sound waves are sometimes called compression waves, but more commonly, they are called longitudinal waves.  (If you clicked on the link I had above on "compressed," you probably saw that coming.)  I encourage you to go ahead over to the link for longitudinal waves, because there are some very nice animations over there for you to see these waves in action.

One last thing before I sign off:  This pattern of compression and rarefaction is often graphically represented as a waveform.  Typically, waveforms are set on a typical Cartesian coordinate system (the graphs with x and y from math class), with the y, or vertical, axis representing the amount of displacement, which also happens to be the amplitude of the sound wave, and the horizontal axis representing the amount of time the wave has traveled.  We'll go over this all a bit more later, but I wanted to introduce it here for you just to get you more familiar with the terminology I'll be using.

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.