next up previous
Next: Clarinet reed

Negative Resistance or Negative Damping

Most things when set vibrating gradually loose their energy and thus loose their amplitude of vibration and gradually die away. This is damping, and is caused by the resistance of the environment on the thing that is doing the oscillating. To get a feeling for what is happening, let us examine the example I used in class, namely the child on a swing. If we give the child a big push, the child will swing, but the amplitude of the swinging will gradually die away until finally the child stops swinging. To understand what is happening, imagine that you are standing near the center point of the swing, and loosely stick out your hand. Each time the child swings by, it will hit your hand. Each time this happens, you hand will exert a force on the child briefly and lightly (assuming your hand is held out loosely) you will give the child a push back opposite to the direction in which the child is swinging. This light push will reduce the speed of the swinging of the child slightly, and after a while, your holding out your hand will bring the swinging to a stop.

This is the way in which damping usually behaves. The rest of the world exerts, to a greater or lesser extent, forces on a vibrating body which oppose the direction in which the body is moving at that instant. For example, air around the vibrating body is pushed aside by the vibrating body, but it pushes back on the vibration, slowing down the vibration.

Clearly in the continuous sounding instruments, something different must be going on. In pulling the bow across the string in a violin, or blowing through the reed of a clarinet, or across the mouthpiece of a flute, the vibration (of the string in the first case, and of the air inside the instrument of the second and third) does not die out but keeps on vibrating as long as the bowing or the blowing lasts.

The physical interaction between the agency causing this continuous sound and the vibration I call negative resistance or negative damping - ordinary damping which causes the decay of the vibration being taken to be positive.

How can we see this in the case of the child on the swing. In order to make the child swing higher ( larger amplitude) or to keep it swinging, one must behave in a very different way from the above case where one just holds out one's arm. To keep the child swinging, one must instead push the child, exert a force on the child, every time it passes by, not in the opposite direction to the velocity, but in the same direction. One's force must try to increase the velocity at each passing, rather than decreasing it. It, one must act in the opposite way to the way in which ones arm would naturally act on the child.

In the case of the child, the swinging is slow enough that one can consciously decided to push with one's arm each time in the same direction as the swing, instead of just letting the child run into the arm each time. However for a string vibrating at 440 Hz say, in the case of the violin, one can clearly not decide which direction to push the string each time it vibrates. Instead one must set up a situation such that the motion of the string itself causes the bow to push on the string in the correct direction to increase the vibration, rather than decrease it. Or one must cause the reed to operate in such a way, that air will flow into the instrument when the pressure inside is high, and out of the instruments when the pressure inside is low. If the air flows in when the pressure is high, this will tend to increase that pressure. If the air flows out when the pressure is low, this will tend to decrease the pressure even more, just like pushing the child to the right when it is moving to the right, and to the left when it is moving to the left.

What is astonishing about musical instruments, is that people, though trial and error I guess, have come up with an astonishing number of different ways of accomplishing this counter intuitive behaviour- of developing this negative resistance or negative damping.

next up previous
Next: Clarinet reed
Bill Unruh 2002-04-12

Copyright W G Unruh