# The Physics of Open ended Tubes: Pennywhistles

### Essays on The Physics of Wind Columns by N. Drozdoff

Nicholas Drozdoff ndrozdoff@hotmail.com

### Mr. D's Music

The discussion of wind column instruments such as trumpet, trombone french horn or tuba requires that we have a basic concrete understanding of very simple wind columns first. We'll start off with penny whistle, a very simple cylindrical instrument with holes cut in it, open at both ends with a fipple at one of the open ends to create the notes we're to play.

First, let's consider a penny whistle with no holes in it or one in which all of the finger holes are covered up.

The mouthpiece of a penny whistle, called a fipple, has a flat, narrow tube that we blow through. This directs a high speed air stream at a slanted hole with a sharpened edge that splits the air stream causing a batch of eddy currents or vortices at the mouthpiece. These eddys have a lot of frequency content in them -- they're noisy! Some of the noise has notes in it that will just fit the tube, so to speak, and the tube begins to resonate. The resonance "talks back" to the fipple (again, so to speak) causing the resonant note to "capture" the system, and we hear a nice pure tone of a cylindrical pipe.

Now I have used several terms and voiced several ideas in the last paragraph that might seem a bit mysterious. Let's begin to dispel that mystery. It'll be easy!

First, let's make sure we see how sound or compression waves/pulses move in a pipe.

Let's consider our simple cylindrical tube. We'll hold our hand against one end of the tube effectively closing it off leaving the other end wide open. If we suddenly and vigorously pull our hand away from the end of the tube, we will leave a tiny region of low pressure right at the end of the tube. The low pressure we have left behind wants to suck air into its place in order to restore that tiny region to normal room pressure. Some air molecules from the room will move into that place, to be sure, but some will come from inside the tube. Since the tube is a closed environment, the molecules of air that moved into the low pressure point that we created leave a new low pressure slightly farther down the tube. This process continues right on down the tube until it hits the other open end. Now what happens here becomes a bit tricky, but I'll get to that a little later. For now let's make sure we understand how this low pressure pulse moved down the open ended tube.

If we think deeply for a moment we can see that the open ends of our tube want to stay at normal room pressure. The low pressure point that we created at the first end by yanking our hand away quickly moved down the tube effectively leaving that original end at room pressure with the pulse moving down the tube. The tube acts as a container the prevents air from the room from moving into the void. The air MUST come from further into the tube to fill the void, so we have a low pressure pulse moving down the tube until it hits the other end. Now, be very quite and think of this again -- re-read this paragraph. See? That shouldn't be too bad to get.

Now what happens when our low pressure pulse "hits" the other open end? Now hang in there as this will get a little wordy. The low pressure pulse has arrived at the open end. Now remember, the open ends want to stay at normal room pressure because there is no containment at the ends. The low pressure void sucks in some air molecules from the room right at the end. As soon as enough molecules move into the void, normal room pressure has been restored at the end of the pipe, as we would expect. But wait a minute, it doesn't end there! In order to see what happens I will have to get Newtonian on you! Now don't let fear of physics scare you at this point. It's not that bad!

As soon as pressure has been restored to normal room pressure at the end of the tube, we can see that there are no net forces on our air molecules at the end of the tube -- pressure is equalized on all sides of them -- no forces. OK. So far so good. But wait a second! Those molecules from the room had to MOVE into the void to cause this pressure equalization! Oh,oh... We've got to invoke Newton's First Law, or his law of inertia. If a moving object has no force to stop it or make move faster, they simply keep on moving! So those molecules that moved into the slight void at the end of the tube keep right on moving into the tube, even if only slightly. Now, there are other molecules in the way and as they try to move out of the way the find the walls of the tube holding them in place. Now work with me here!! What we now have is a HIGH pressure point at the far end of the tube!! The molecules the MOVED into the tube to restore normal room pressure at the open end caused a pile up just inside of the tube because of their inertia. This high pressure travels right back down the tube in the opposite direction in a similar fashion as did our original low pressure. The tendency is to preserve normal room pressure at the ends of the tube with high and low pressure pulses traveling back and forth in the tube.

OK. That's actually the toughest part. Now I will introduce some simple vocabularly. In physics we say that a reflection occurred at the open end of the pipe. In fact, most physicists start out by saying that. Of course, most students immediately and rightfully object to the counter-intuitive notion of a reflection coming off of thin air, but our discussion in previous paragraphs should give you the correct idea. What actually gives rise to this reflection is the fact that we went from a tube (contained environment) to the room (a completely open environment). In physics speak, we call this sudden change in acoustic environments a "discontinuity." In fact, you will probably want to memorize this notion: whenever any sound wave hits any type of discontinuity or sudden change in shape of a tube, a reflection of some sort will occur.

In our open ended tube the reflection is said to be inverted, that is the low pressure pulse that hit it originally was reflected back as a high pressure pulse. This high pressure pulse will now travel back down the tube, "hit" the original open end, and, again by virtue of Newton's First Law, send another LOW pressure pulse back down the original direction. This goes on and on, but not forever, in fact not for very long at all.

What happens is this: with each reflction at an open end, more and more air molecules in the ROOM are set into vibration (remember, not ALL of the molecules that are required to fill a void or move out of the way at an open end come from the tube -- some come from the room we're in.). The energy of reflected pulse is less and less until the whole process quickly peter's out.

Now let's talk about resonance. Resonance is a phenomenon that can only occur with events that occur periodically, that is, over and over again in a very regular fashion. In sound we are talking about the repeated compression and decompression of air (if the medium in which the sound wave is moving is air as opposed to say, water or metal).

Now we started this whole discussion with a decompression being sent down our tube. It reflects as a compression back down to us and then turns back into a weaker decompression form our end. But what if we help it along a little bit? Just as soon as the compression bounces off our end as another decompression, we quickly suck some air out again making the reflected decompression even more decompressed. Whats more is let's do this EVERY TIME THE PULSE REFLECTS! This is similar to giving a child in a swing a push every time he/she just gets back to you. As a result, we now have a wave that is stuck in the pipe building up in energy. HOUSTON, WE HAVE RESONANCE!!

Now we can see how an open ended tube can produce a note. This lowest note we have discussed here is called the fundamental pitch of the tube or sometimes the first harmonic. Now, we can produce many more than that one note, but let's get to that later.

At this point I want to discuss something that many musicians don't understand. In order to have resonance, we MUST HAVE A REFLECTION at the "business end " of the tube (the bell of our instrument). In fact, the quality of that resonance or how sharply "notched" in pitch that resonance is depends on having a very good reflection; that is, not much energy lost to the room. Think about it! Trumpeters love to refer to a good trumpet as "slotting up well." This is this quality of resonance they are referring to. However, the better the reflection, the better the resonance. A MUSICAL INSTRUMENT IS DESIGNED TO KEEP AS MUCH SOUND IN THE TUBE AS POSSIBLE!!! This is another counter-intuitive concept. Wind instruments are designed to NOT radiate sound too well. If they radiated sound TOO well, you would never achieve resonance and those nice, neat "notches" or "slots" (SPECIFIC notes) that we know, need and love.

Now let's talk about the other notes that we can get on an open ended pipe -- the overtone or harmonic series. In physics speak, we will now invoke BOUNDARY CONDITIONS. That is, we need to arrive at an understanding of what happens with resonanting waves as a result ot the end conditions of the tube. In our example, both ends are open.

In an open ended tube we have no effective containment at the ends of the tube. They must effectively remain at room pressure. In physics speak, we must have "pressure nodes" at the ends of the tube. For our fundamental, we have two pressure nodes at the ends, but a pressure "anti-node" right in the middle of the tube. What this means is this: at the center of the tube, the pressure is varying widely from a high pressure (over room pressure) to a low pressure (below room pressure) at the pitch of the note we hear. Remember, due to Newton's First Law, some of the sound leaks into the room -- this is what we hear.

The boundary condition that must be satisfied is the two presure nodes at the ends of the tube. We can still satisfy that with a note one octave (twice the frequency) higher. We now have two pressure antinodes in the tube and a node right in the middle, but the tube "doesn't care" about all of that. We must have a pressure node at both ends, and we do.

If we play around with geometry a bit we can see that this will occur with pitches that are 3, 4, and 5 times the fundamental. In fact, all integer multiples of our original note will fit these boundary conditions. This is what is called the overtone series of an open ended pipe. It turns out this way because the boundary condition at both ends is the same - open. We would get the same series if both ends were closed, but this would be a useless instrument as no sound could get out (in fact, getting our pressure waves in there in the first place is difficult - this is just theoretical stuff.). This is the same series that we get for a string that is fastened down at both ends as in a guitar or violin.

Now let's get to the real easy part and finish off our initial discussion of the physics of winds. In a pennywhistle both ends are open, so we get all integer multiples of the lowest note. The first interval we should hear as we blow harder on a penny whistle (casuing the higher overtones to "pop out") is an octave, which is indeed the case. In fact, that is our clue that we have open ended resonance - the presence of ocatves, or multples of powers of two. Not all insrtuments do that, but all open ended cylindrical pipes will. Now in addition to forcing the overtones to pop out by blowing harder on the fipple, how else can we change the notes on a pennywhistle? How about breaking out a hack saw and shortening it? That'll work. Think back to our original discussion as to when to hit the end of the tube again to help out the reflected pulse. How long we have to wait depends on how long the tube is. The shorter the tube, the sooner we'll need to give it another push. If we are doing it periodically, we'll need to do it more often - more frequently - higher frequency - higher notes! Get it? The shorter the tube the higher the notes.

Now there is a certain permance to using a hack saw to shorten the tube. We would need to get new instruments after every performance (every tune for that matter!). This is impractical. Let's just drill a bunch of holes (how about six) in just the right places along one side of our tube. Now remember, the reflection that gives rise to resonance occurs at any sudden change in the tube. A nice big hole qualifies niocely. Whenever, we lift our fingers off of the finger holes, the pressure wave is reflecting at that point, not the end of the tube any more. In fact, most of the sound that "leaks" into the room is coming from the side hole now. We have effectively shortened the tube by lifting up our fingers. Now by careful placement of the holes, judicious covering and uncovering and careful use of "overblowing" we can produce all of the notes of the major scale and, voila! You can now play all of the folk songs that you want!

Now this isn't exactly a "nut-shell" discussion, but it is by no means complete. I wanted you to get a bit of a picture of what is actually happening physically. However, I have left some things out. If you want a more thorough treatment, please see my "Further Reading" page. For example, the whole Law of Inertia thing gives rise to an effective lengthing of our tubes know as an "end correction." It wasn't necessary to discuss that here, so I didn't take the time.

One last thought in this first section on the physics of winds. Many of my readers will necessarily be brass players. Here is another misconception that I have heard imparted by many physics teachers as well as musicans. They think trumpets are open ended instruments because of the overtone series they produce. THIS IS UNTRUE!!! They are closed end instruments. Think: you clog up one end with your lips while the bell (in the absence of a mute) is open. This gets a little tricky. Check back for the next installments of my discussion of the physics of winds.

##### QED

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