Second post already! I know, prompt. I’m posting this now because my internet access will be fragmented next week until I get internet sorted in my new place.
The first of the series was an overview of melodeon acoustics – the second is about the reed itself. I will make a confession here: I have never been quite happy with the explanation of the melodeon reed. So the following is what I’ve been able to glean from what I can, but shouldn’t be taken as gospel. If at some point I discover more about it then I will let you know.
The free reed is a thin strip of metal, usually steel in a melodeon, but occasionally brass in older models. It is riveted to a plate and is placed in a slot such that it is unrestricted in movement perpendicular to the surface of the plate. When air is passed through the slot, the reed vibrates and produces sound. The most fundamental part of the melodeon is this reed. It is the reed which generates the sound and it is the reed which has the single biggest effect on the sound of the box.
The reed is a “self-excited oscillator”, meaning that a constant input (i.e. a flow of air) is turned into an oscillatory output (i.e. the vibration of a reed). All bowed instruments and wind instruments feature self-excited oscillators, as does the human voice.
The reed acts like a spring; it has an equilibrium position (from now on ‘Zero Point’) and if it is displaced away from it, it gains potential energy (the reed is in tension), acting back in the direction of equilibrium. If you give a reed a displacement (i.e. by plucking it), you give it potential energy. When you release it, the forces are unbalanced, so it moves in the direction of the zero point. At the zero point, there is no potential energy left in the system. However, conservation of energy tells us that this energy has to go somewhere. Some of it is dissipated in heat, but most of it is converted to Kinetic Energy, i.e. movement. So the reed keeps going. As it travels past the zero point, it starts to gain in potential energy, until the point is reached where it has no kinetic energy. The cycle is repeated until enough energy is dissipated in heat (a measure of how quickly this happens is referred to as the ‘damping’ present in the system), when it stops. This is a phenomenon which will be familiar to anyone who has watched a pendulum swing or a spring bob. The frequency at which the reed vibrates is the resonant frequency (specifically the fundamental) of the reed.
That is a GCSE level (or EBacc now apparently) explanation of why a reed vibrates when you pluck it. When you add the airflow, things become slightly more complex.
In order to explain the phenomenon, two concepts must be introduced. The first is conservation of mass. This effectively states that mass cannot be lost in the system. What this means in real life is that the mass flow rate (the mass flow per unit time) in to the system must be equal to the mass flow rate out of the system. This means that if there is a constriction, say a slot in a reed plate, then the flow must increase through that constriction. This is why a nozzle results in faster flow; why you can get champagne bottles to shoot jets of bubbly high into the sky if you are so inclined. Of course, this all goes a bit odd once you get above the speed of sound, however thankfully melodeons don’t break the sound barrier!
The second concept is Bernoulli’s Principle. This states that if a fluid speeds up, the pressure associated with it decreases. We can quickly see from the above that in a constriction, where the flow velocity increases, the pressure correspondingly decreases. The only other thing to note is that fluid flows from regions of high pressure to regions of low pressure.
Let us consider the reed when it is at its maximum above the slot, i.e. its maximum away from the direction of the flow. At this point the available slot area is at a maximum. As the reed begins to move towards the slot, the slot area decreases, so the velocity increases and the pressure in the slot decreases. This increasing pressure difference between the bellows and the slot pulls the reed further in, as the airflow pressure exerts a force on the reed. The reed’s inertia carries it past the zero point, after which the slot area, and hence the pressure, increases. This pressure increase means that the force on the reed decreases, until the tension in the reed overcomes it, meaning that it changes direction, moving back towards the plate. Now the pressure difference starts to increase again, but the inertia of the reed is sufficient to carry it past the zero point. As before, the further the reed gets from the slot, the less the pressure difference and so the less the pressure force on the reed, until the tension in the reed is enough to allow the reed to go back towards the zero point, where the cycle repeats.
However, it can be seen that this explanation will result in a swiftly decaying motion. This is because the pressure force of the flow acts as a damping force, in that for half of the cycle it is acting against the movement of the reed. Because the air itself has inertia, the velocity variation is not instantaneous. This means that the changing pressure lags behind the movement of the reed, meaning that for the majority of the cycle it is acting with the reed, rather than against it. This fulfils the criteria for a stable oscillation.
Hopefully the above demonstrates why the reed vibrates in a stable way. However, it doesn’t explain why it makes a sound. When a reed vibrates, it will not allow much air through it when it is at the zero point, as it should be a relatively good fit in the slot. So the motion of the reed acts to regularly interrupt the flow of air. The result of this is a series of what is usually referred to as “puffs of air”. I don’t think that “puffs of air” is a particularly helpful explanation, as puffs of air are not sound! Hopefully though the reader can see that the process described above also results in a roughly sinusoidal pressure wave in addition to the airflow, as the pressure in the slot is varying continuously. This pressure wave will propagate at the speed of sound. Sound propagates very effectively; the velocity of the flow will be slowed down and dispersed very quickly by friction, whereas the sound will continue.
This mechanism is also known as the Siren mechanism. Helmholtz, who generally had a good grasp on things, mentioned it in his “On the Sensations of Tone“. Then again, he did state in the same work that:
This whirring of dissonant partial tones makes the musical quality of free tongues very disagreeable.
This clearly shows that even genii can be wrong sometimes (especially if taken out of context for humorous effect by bloggers). His Siren was a rotating disk with holes on it, which passed over a pressurised tube. The frequency of the resulting note depended on the speed of the disk.
This isn’t the whole story, of course. Firstly the reader will note that plucking a reed forms a sound without airflow. This is because the vibration of the reed creates localised pressure differences on the surface, which form pressure waves. One might think that there might be a component of the sound coming from the reed itself, although it turns out that this is negligible (Helmholtz demonstrated this). More interestingly, air flowing past a perpendicular flat plate such as a reed may well be turbulent, which will create vortices and eddies that may generate sound. That is getting beyond my current level of knowledge, so I’ll stop there!
Next time I’ll be looking at ways that the sound of a reed can be altered. Until then, please enjoy this video of a Helmholtz double siren. The same principle as a melodeon reed, but a very different way of realising it.
Nothing in this post should be taken as scientifically proven. If you have good reason to believe that I am mistaken in anything that I’ve said here then please comment and I will gladly amend.
Amended 20/10/12 on the suggestion of Olav Bergflodt.