Chapter 1

Oscillator Concepts

   An oscillator is a device of repetitive motion.  Musical instruments are oscillators.   They are as old as human history.  But for the last two centuries they have become sophisticated and stylized.  Their workings have been explained completely by Newtonian physics.  The history of the last century has been completely shaped by a new family of oscillators-electronic oscillators.  Not until the extensive development of electronic engineering was musical instrument engineering formalized.  These concepts were not introduced into books on flute engineering until (Fletcher and Rossing 1997) as based on the formalism first presented by (Kinsler and Frey 1942).  This is why the nomenclature, perspective, and concepts used now in flute engineering are those used in electronic oscillators.   This formalized treatment makes for easier understanding of musical instruments, especially flutes.

  Before getting into flute design, it is helpful to have a perspective of oscillator concepts and history.

 When stored energy switches from potential (stationary) energy,  force x distance, to kinetic (motion) energy, ½ mv²,  it is called an oscillator, or resonator. 

Some examples of oscillators are:

1.      A weight bobbing on a spring.

2.      A bouncing ball.

3.      Molecules.  For example, the Nitrogen of an (ammonia molecule) vibrating back and forth from one side of the middle of the triangle made by the three H atoms.  Ammonia is blown over the embouchure of a flute tube to make the most accurate tone there is.  It drifts by one cycle in 10¹³ years.  Charles Towens got the Nobel Prize for this flute, it is called the ammonia maser.  The flute has been taken on space journeys and checked when it returned to measure the Einstein prediction that it would “age”.  It did not.  It just kept on tooting at the same note. 

4.      A small tube with mirrors on the end and filled with fluorescent helium and neon.  This like any flute oscillates at multiples of  c/2L, in the 632,8 nmeter range (orange light).

5.      A clock.  Any clock, like the tuning fork watches made in the mid-20^th century by the Hamilton Watch Co (it hums).  A pendulum clock.  Today’s watches have a transistor in series with a coil and capacitor. 

6.      A musical instrument.

7.      Water swishing back and forth in a bathtub.

8.      A radio transmitter or the many electronic oscillators in computers. 

9.      A suspension bridge, sometimes when it is not designed right.

10.  A boat when it is rocking back and forth.

11.  A microwave oven.

12.  An ordinary playground swing.

A resonator that runs by itself is called a perpetual motion machine.  There is no such gadget. Oscillators all have universal aspects and in their design and study, engineers use well defined nomenclature.  The commonly used ones are:

• They all have a resonate frequency, sometimes more than one, the rate at which the kinetic energy and potential energy switches back and forth.
• They all have a quality factor, Q, which is defined as the energy stored divided by the energy lost per cycle.  For instance, the Q of a bouncing ball is the ratio of the height to the difference in height of two consecutive bounces. The quality factor of a perpetual motion machine is infinite.  Perpetual motion machine would be a perfect clock, but if you read it, you would have to extract energy, so it would not be a perpetual motion machine any more.  This is the origin of Heisenberg’s Uncertainty Principle, that every thing we measure, we change. 
•  Oscillators have to be “fed” .  The energy fed in has to be enough to make the Q infinite, plus enough to measure, hear in case of musical instruments.
• An oscillator is said to be lumped circuit if all elements for storing energy back and forth between potential and kinetic, are small compared to a wave length of the sound generated.  It is called a distributed circuit oscillator, like a flute or violin, if the elements are of the size of a wave length.  A good example of a lumped circuit oscillator, is a pendulum.
• Equal amounts of energy are stored as potential and kinetic energy.
• The two energy storage forms are always out of phase with one and other and switch back and forth at the resonate frequency.  Practically every equation and concept in flute design is a measure of the phase relation of the pressure (potential energy) and the current (oscillating wind or kinetic energy) and their phase relation.

Table 1 gives some examples of oscillators and their characteristics.

Oscillator	L/D*	Frequency Range	Potential Energy	Kinetic Energy	Quality Factor,Q	Feed Mechanism	Remarks
Weight on  a spring	L	1-20,000 Hertz	Spring con- stant x distance		50 to 1000	Push	v is speed
Basketball	L	5 Hz	Height x weight		10	Magic Johnson
NH3 maser	D	1.2 GHz	Spring con- stant x distance		10,000	Molecular beam over embouchure
Wind up Watch	L	1 Hz	Torque x angle		50	Spring-ratchet	is angular speed
Electronic oscillator	L	60 Hz to 1 THz			100-1000	Transistor	C-capacitance L-inductance
Microwave oven	D	3 to 10 GHz			1000	Electron beam embouchure	Z is impedance of space
Poorly designed bridge	D	1 Hz	Same as violin string	Same as violin string	30	wind
Rocking boat	L	1 Hz	Torque x angle		20	Waves and wind
Flute	D	3 octaves			80	embouchure	2 ends open
Woodwind	D	3 octaves			80	Vibrating end	One end closed
He-Ne laser	D	.6 wave- length			1000	Plasma induced amplifying	Red-orange light
Harmonica	L	4 octaves	Rod bend	Rod vibrate	100	embouchure
String Instrument	D	4 octaves	Stretched string	Vibrating string	10,000	Bow or pluck	Ring time indicates Q

                        Table 1 Examples of oscillators.  L and D stand for lumped and distributed elements.

Chapter 2

Elements of Flute Design

    Appendix A familiarizes you with the description of sound waves and how they are described mathematically.   The Appendix explains how waves are described in trigonometric functions sine, cosines, and tangents in a pipe.  It also explains how waves are described with imaginary numbers to keep track of phase relations.  If you are familiar with these concepts, we can start right in describing the impedance of a sound wave in a pipe.  Impedance is the ratio of pressure to current .  It was shown in Appendix A that a plane wave traveling in space has an impedance of .   We can think of u as the actual speed of a wind in meters/second.  Of course in a sound wave it is the speed of the wind going back and forth in phase with the pressure.  The impedance in a pipe either very long, or terminated with an absorbent material such as cotton wool,is

                                                                        .                                                                (2.1)  

where S is the area of the pipe.  When a sound wave is launched down a pipe  is the density of air in kilograms per cubic meterat room temperature or 1.293 at .  c is the speed of sound which is  meters/sec. u is the additional speed of the molecules in the air just as the pressure p is the additional pressure beyond atmospheric pressure of 14.7 and T is in Celsius degrees, usually 25.  At sea level and atmospheric pressure and 77F, it is about 344.86 M/s.  See Appendix B for details.  If S is the area of the pipe and u is speed in meters per second, then uS is the speed per unit volume.  So in a pipe, if then the pressure and current are always in phase.  With no reflection in the pipe, their description in time and space are exactly the same.   More specifically Equation 2.1 is,

                                                                                                      (2.1a)    

is the phase and determines what the pressure is at t=0 and x=0 and .  We can also describe the wave with out imaginary numbers, i.e. without j, as

                                                                      (2.1b)

or

                                     (2.1c)

Our next relation, Equation A28 from Appendix A is used extensively in flute design,

           .                                          (2.2)

It is the general transmission line equation.  It is applicable not only for sound waves in a pipe but for electro-magnetic waves in a cable, power transmission lines, and a wave traveling down a taught string.    Consider a pipe of constant area, S.  Choose a location down the pipe,.  Choose another location to the left of  , the “in” location or where.  The impedance “looking in” at x=0 is the of Equation 2.2.

We first illustrate the use of Equation 2.2.  With absorption material starting at .  There is no reflection and the load impedance at  will be  .  When substituting this for in Equation 2.2 ,  which is expected from Equation 2.1 if there is no reflection.

Figure 2.1  Pipe filled with absorbing material so there is no reflection of a sound wave launched for the left end. 

Open End Pipe

We now want to start talking about reflections in the pipe, i.e., situations where the pipe is terminated by something other than an absorbent material or is infinitely long.  Not only do we want to terminate it but we want to terminate is so that the pipe is a resonator, a flute.  In Chapter I  it was stated that all oscillators and resonate cavities are constructed so that the kinetic energy, current in our case, and the potential energy, pressure in our case, are out of phase and the energy switches back and fourth at the frequency of the flute.  A pipe open on both ends, as shown in Figure 2.2 satisfies this condition.  At a distance of 0.61a the pressure at the end of the pipe has become zero (we explain this later).  Therefore the load impedance, in Equation 2.2 so,

Figure 2.2  At a distance out from the end of an open pipe 0.61a, the pressure from inside the pipe has diminished to zero.  Actually, 0.6a is a fictitious number that assumes that it diminishes all at once. 

                                              (2.3)

From the definition of impedance as the ratio of pressure to current, Equation 2.3, is also

                                                    (2.4)

Figure A6 and an explanation in Appendix A shows that an imaginary quantity is out of phase with a real part.  Or another way to look at it is that  from the definition.   Now the  and , therefore, .  Note that p=0 when is and so on.  Or, in other words, when  is  because  This is the resonance condition, that the pipe has to be a multiple of a half wavelength. 

 We also want to show that the impedance of two pipes cascaded is the same as a single pipe of the same diameter.  If this were not true there would be something wrong with our equation for open pipe impedance.  This will be called series impedance, although it is not the sum of two impedances as one has with lump elements.  Sometimes we label this impedance as in distinction tofor  the “shunt” impedance which is the impedance of two branched pipes, i.e. parallel impedance. 

                        Figure 2.1. Impedance of a tube as a series of cascaded impedances.

If Equation 2.2 is valid, than the impedance at the location    is  .  Equastion 2.3 becomes ,

                                     (2.5)

Using as the new load impedance for the end of the pipe at , we have from Equation 2.2 for ,

.      (2.6)

Multiplying top and bottom by  gives,

    .                                   (2.7)

We note from the trigonometric identity in Appendix D that

.  Comparing this with Equation 2.6  results in

                                   (2.8)

which proves that the impedance of open pipes are additive if they have the same area S. 

We will have many occasions when the length is not more than the diameter.  I.e., when  and is a small angle.  Then  (if we use radians, not degrees)  For example a key hole of chimney height t is a tube of impedance .  In this situation the concept of a transmission line whose length is more than its diameter, where Equation 2.2 is valid, breaks down.  We will find out from experiments done by Keefe (1982) who measured the impedance of keys and key holes, where an experimentally determined   are valid concepts; where   for closed holes is for an open short tubes.  In other words can be used in the same manner as d is in Equations 2.3 through 3.8.  is a series impedance for a tone hole even though the main impedance element of a tone hole is a shunt (parallel) impedance which will be discussed under Branched Pipes below and depicted in Figure 2.6.

Closed End Pipe

                                             Figure 2.2  A closed end pipe of length l and area S =p a² .

The closed end of a pipe has the impedance of  because the pressure is high and the current is zero.  In this case Equation 2.2 becomes,

      (2.7)

For short pipes where  and ,

,                                                     (2.8)

where V is the volume of the closed pipe S and .     is called acoustic compliance or acoustic capacitance because a closed volume in acoustics behaves analogously to an electrical capacitance.  The impedance of an electrical capacitance is.  So the acoustical equivalent of an electrical capacitance, in phase and form is .  The analogy is not complete because you can not cascade two closed end pipes and you can cascade electrical capacitances.  It is a convenient concept in flutes because a closed key h