Example 2 Silver Concert Flute

        First we establish the substitute   left of the embouchure as shown in Figure 2.12.  The impedance “looking into the left end” is the reciprocal of the reciprocal sum of the impedance of the two branches, the open embouchure pipe and the closed cork-end pipe as described by Equation 2.9 and Figure 2.3.  The impedance “looking into the left end” is also the impedance of the factious substitute end pipe.  This is the typical branch situation shown in Figure 2.3.

Figure 2.12.  The embouchure of a modern concert flute is rectangular.  Even more so when the player’s lip coverage is included.  Both for the embouchure height and for the open end substitute pipe are calculated values.  For completeness, is decreased by=-0.1 mm and so is the right hand pipe.

The impedance of the embouchure chimney, being an open pipe, is from Equation 2.3,

.                                                   (2.31)

Ditto for the substitute pipe,

                                                    (2.32)

The cork-end pipe is closed so the impedance is, according to Equation 2.7 and Figure 2.2,

                             (2.33)

So from Equation 2.9 we have for the substitute pipe,

                                                           (2.34)

which gives Equation 2.24 for the substitute pipe,

                                            (2.35)

is always in the 30 to 50 mm range so we cannot let unless we don’t mind being a few mm too large.  Otherwise D, k, and t have to be in meters.  Note that D are in the range.  This is why the position of the cork, D, does not come into play until the second or third octave and becomes large.  For  we get from Equation 2.15,

                (2.36)

and for the  note at 261.6 Hz that gives k=4.76, becomes,

                  (2.31)  

this gives radians and .  For completeness, we should take off .  But this gives an accuracy better than the precession, as Thom McGonnigal, an old NASA communications expert used to day.  Figure 2.13 shows the predictions for higher frequencies, when the cork distance D comes into play.  It is similar to a more detailed graph in (Fletcher & Rossing 1998 page 542-3 Figures 16.26 and 16.27) .  

Figure 2.13  Plot of the equation for the substitution embouchure length,  , as a function of cork distance to the center of the embouchure, D, area of the flute at the embouchure, , effective area of the embouchure, the wave number, k, and the substitute shunt chimney height of the embouchure,. 

We next determine the influence of closed holes on the length of the flute for the bottom note at 261.6 Hz. 

With /2 =345.1/2x261.63=659.9 mm; deducting the effective embouchure end effect of 40.2 mm and the other end effect of .3 (19)= 5.7 mm; the calculated embouchure-end distance is 614mm, as shown in Figure 2.8.  To check these results, we inserted the headjoint into a plastic pipe of 19mm internal diameter.  We then cut the pipe off when it sounded , as shown in Figure 2.13.  The measured length from the embouchure center to the end was 614 mm in agreement with the calculated value. The concert flute measures 608 mm; it is 6 mm shorter than calculated.  This is due to the series impedance of distance  in Equation 2.18 of a concert flute with 13 closed holes of chimney height t= 2 mm; 2b=13.5 mm average and  2a = 19 mm.  From Equation 2.18 we have,

=/key.          (2.37)

Figure 2.13  Bohem flute head with 17 mm cork distance connected to a 19 mm i.d. schedule 80 plastic pipe.  The distance from the center of the embouchure to the end is 614 mm for a note of 261.3 Hz. The same distance for a typical silver concert flute is 608mm.

Example 3: Rudell and Rose 1835 Wooden Flute

Prior to about 1850 flutes were commonly designed with a taper in the lower part as shown in Figures 2.13.  These flutes are now used for traditional Celtic music  and authentic orchestration of the old masters. 

A tapered pipe has an input  impedance, analogous to the input impedance of a cylinder as in Equation 2.2, of

                                          (2.38)

where  Figure 2.12 shows how these terms are defined for the impedance looking into either end.

Figure 2.12  The impedance looking into a conic section of length L uses the same equation, however the terms are different.  S₁ is always the end into which the impedance is defined.  L=x₂ – x₁ is negative when looking into the large end.   

We first wish to see how much shorter a  tapered flute is than a cylinder flute sounding the same note.  To do this we note that a conic section open on both ends will have an impedance on the end of zero, just as the cylindrical one does.  Therefore, setting in Equation 2.38 gives,

.                                         (2.39)

A more usable and enlightening form of this equation is,

                                              (2.40)

We get to Equation 2.40 from 2.39 by using the identity ; using the definition ; and noting that .  Because we used the boundary condition that =0 at the end, we have to take in the end effect that the pressure is only zero at a small distance  beyond the end, as we did with the cylinder in Equation 2.3.  So we have to rewrite Equation 2.40 such that L=,

                                                                                                              (2.41)

 Note the similarity between this equation and the one for a cylinder, which we repeat here.

                                                                  (2.3)

Therefore, a cylindrical substitute tube would be

                                                                  (2.38)

We measure, then make an actual substitute tube, Figure 2.14.  The difference in measured length of the two flutes was 24.2 mm as shown in Figure 2.13.  The calculated difference is,

Figure 2.13    Dementions of the end of a boxwood flute similar to one made in 1830 by Rudell and Rose of London, model # 742, owned and played by Chris Norman The measured difference in length of a straight pipe and the Rudall & Rose 1830 Model 742 was 24.2 mm.  Measurements were to ½ mm accuracy.            

.   

Figure 2.14  A substitute tube used to compare the calculated substitute length for the Rudell and Rose flute.  The headjoint used was not the one from the Rudell and Rose flute, but one of the similar flute made by Michael Chronnally,  of M&E Flutes.  Similar heads were tried with similar results. 

The dimensions for calculating are shown in Figure 2.13 and the frequency is 261.6 Hz.  This gives for c=343M/s, k=2p(261.6)=4.79.  So for this situation Equation 2.38 becomes;  k= .829.  You have to be careful because dividing k by k to get .829/4.79 = .172M(172mm) is the wrong answer.  The reason is that has an ambiguity as shown in Figure 2.15.  We have to chose the option other than  =172 mm because we know  has to be in the 470 mm range.  So we try the other option of k=(.829-p)=2.31; =.483 M.  The calculated difference between and L is 483-457=26mm.  This is about 2 mm off from the measured value of 24.2 as shown in Figure 2.13.  There is an end effect difference of 2 to 3 mm because of the difference in diameter of the end of the two flutes.  Remember the end effect as being =0.61a.  a is the radius.

Figure 2.15.  Even though the tangent of these two angles is the same, they have different arguments kL resulting in an ambiguity in the equation when solving for L

Appendix A

Mathematics of Oscillatory Motion

Below is the presentation my grandson made to run for Treasurer of his Fourth Grade Class.  What he said about him and money, I hope you will be able to say about you and sound waves after you finish this section:

          I think that I would be a good treasurer for a number of reasons. First I would like to say that if you do in fact find that this speech has either therefore convinced or changed your mind that will be a good treasurer I will like to say before hand thank you for this chance to tell you my reasons and for taking the time to listen.

            I think you will find these reasons rather good and will agree with me and maybe others that I can be a very good treasurer. Well first of all this may sound conceded  in fact I can grantee it will sound conceded but I promise you I am not conceded. So first up I say that I rock at keeping track of money. If I were to tell you about the times my mom ‘lost my money’ and it was right were I put it    would take up the school day. Maybe i exaggerated but man it happens a lot!

            Now to go to two. Adding money, subtracting money, doubling money, for Petes sake dividing money! I’m reminding you i told you i would say some things that will sound conceded but this is the truth and i don’t want to sound braggy. I am in the GT math and think that i can do the adding and subtracting needed. I don’t even know if we will even need to do any of this. But we might so thats why you need to know that i am in GT and probably can handle it.

            Now for the third. Last year the third grade teachers picked pupils to go for SCO meetings. I was one of those four kids. I mean yeah that was last year it may seem weird but why do i say this if i know that? Why? Because i am going to tell you what they said and what they do. Not the SCO together but the treasurer.

            “We have three hundred twelve dollars and thirty-four cents from our last lollipop sale.” is what they’d say. Maybe how much they owed someone or how much they lost in the lollipop sale for paying people to make them. See? They do a lot but i can handle it. i don’t see much more to say so i guess i have one more thing.

            Thank you so much for taking your time to listen to me. I hope that you listened to me and think i would be the right person for the job. I say thank you and remember my motto wish i convincingly came up with on the bus Monday afternoon: sometimes the best of all is the oddest of all. Thank you.

            Well, for Pete’s sake we have to learn to add, subtract, multiply and divide sound waves. 

Addition of Two Waves Traveling In the Same Direction

            A wave is the propagation or transmission of a pressure crest and rarefaction in a restorable or elastic medium.  An example is the transmission of a hump along a stretched wire.  Another is a water crest traveling on the surface of a pond.  Another is an electric field traveling as a radio wave.  These are all examples of waves that travel perpendicular to the direction of the pressure crest.  For example, in a water wave, the wave moves along the surface; but the motion of the water is up and down.   The same for a wave traveling along a taught wire.  The wave moves along the wire but the motion of the wire is up and down.  The force is perpendicular to the wire.   Unlike water waves or waves along a wire, sound waves are a pressure crest (and the associated rarefaction) along the direction of the traveling wave.  In order to understand how musical instruments work and to modify them, one has to be knowledgeable about waves and all aspects there of, pressure, current, phase, traveling waves, standing waves, boundary conditions, wave excitation, coupling, dampening, and resonators that generate waves.  Resonators are clocks, musical instruments, and molecular vibration.  In a larger sense, all measurements are made by resonators or repetitive motion.   Everything reduces to mass, length, and time, according to Einstein.  You really can not think of time without a clock, or distance without repetition of  yard sticks or mile stones or wavelengths.  You can not think of mass without thinking of bunches of things in grams, pounds, or pecks.  So before we go into describing musical instruments, we have to go into the details of describing waves and resonators.

             The mathematical construct used to describe waves is the trigonometric function, the sine and cosine wave. 

 Figure A1  Above:  Sine function in degrees.  Below: Cosine function in degrees.  Sin 30^o = Cos 60^o = 1/2.  Note also that the sine of an angle is the  sine of the remainder when  dividing  by 360^o.  For example the remainder of 810 when divided by 360 is 90 and so sin(810^o) = sin(90^o) = 1 and sin40^o = sin400^o = 0.642787609686539326322643409907263.  In engineering, one uses only the first three decimals, 0.642.  This is also the cosine of 50^o.  

Either the cosine or the sine wave can be used to define waves, they are the same thing with a different starting situation, called the phase.  Trigonometric functions have to be in some angular measure.  Before computer days, engineering books all had tables of trigonometric functions as in Figure 2, for convenience. These tables were worked out my Christian monks in the Middle Ages.

Figure A2 Table of trigonometric functions from the back of a surveyor's field note book.  I still use surveyor’s field books as rugged journals with 50% rag paper. 

Trigonometric functions used for repetitive motion and wave study, use radians as the unit of angular measure of choice, not degrees.  Because the circumference of a circle with a radius r is r, and there are  radians of angular measure in a circle, the angle in radians is the ratio of the cord to the radius of a circle.   Figure 3 shows a sine wave in both degrees and radians. 

Figure 3.  This is a plot of signal strength ,s, as a function of angle, a.  It is written as .  The lower graph shows the angel in radians and the upper one is in degrees.  There are 3.141x2 radians in 360 degrees.  If radians are used as the angular measurement the  sine of the angle approximately equals the angle for small angles.  

            The waves in Figures 1 and 3 do not describe any thing real, they are just mathematical functions. 

Let's first describe a real wave, say a water wave (Figure 4).  This wave is frozen in time and so is the skier.

Figure A4.  A water wave of wavelength and crest height of H_MAX with a water skier located at 120⁰ .

If we want to describe where the skier is in terms of distance, instead of angle in degrees, we can say that,

.  If we want to describe where the skier is in terms of distance, instead of angle in radians, we can say that,

where is the angle in radians, in this case,

  radians.

The wave of Figure 4 along a direction X with time frozen is,

            or or                  

,

where  is the wavelength, x is the location on the wave where the height H is being described with maximum for a wave of maximum height of  _.  The phase angel  or  gives the value of the wave when X=0.  Note that the .  If you wanted to express a wave, stationary in time, and in degrees instead of radians, it would be,

                                                ,

which is never used for many reasons; one is that, for small angles, the angle in radians is approximately equal to the sine of the angle, as shown in Figure 5.

                        Figure A5.  As the angle  becomes small and become equal.

This approximation is very handy with flutes design.  For example, a keyed hole is a closed or open pipe of short length X, therefore is a small angle and the sine of the angle is replaced by the angle.    is commonly denoted as k, the wave number and, finally, a pressure wave in the direction x in a pipe, frozen in time, is usually written as,

,

where p is the pressure in the pipe, a distance x from some reference point and is uniform across the pipe (except close to the walls).  In the three dimensions, this equation takes on the vector form,

,

With flutes we use mostly the one directional form.

            Keep in mind that this describes a “still” picture of a wave that starts at P_o  amplitude when x is zero.  How do we describe a moving wave, a wave moving in the negative X direction (to the left) in time?  First we note that in Figure 1, the top wave is the same as the bottom one except the bottom one has moved (to the left) backwards by 90^o, or in radian measure it would be.  So the top one would be written as 

The bottom would be written as

.

Therefore, you can see that if you want to describe the wave moving backwards in time, you can just change the phase element biger and biger in time.   In other words, would be twice as far to the left as  is.  So if we change the phase angle to

,

where ? is, so far, just a proportionality constant indicating how fast the wave moves backward in time.    must have units of radians (of angle) per second because t must be angle.  There is another unit commonly used in place of radians per second.  There are 2 p radians of angle for each circle of 360^o.  So for our angle measurement we used, not degrees and not radians, but the number of whole cycles per second the wave has gone through, the above equation becomes,

,                                                       (A 0 )

where f is our old friend, frequency.  In other words, if a wave goes backwards by 20 cycles per second, then it has gone through an angle of (20)• (2p) radians of angle.  is commonly called the angular frequency of a wave and f is the linear frequency, or just frequency.  When the  wave has traveled backwards a distance of one wavelength, , in a time of 1/f seconds per cycle, then the speed of travel backwards is distance divided by  time, or , the speed of travel of the wave is.  Just one more thing to complete the description of the backward moving wave, we need a term to describe what the value of the wave pressure is when the time and x are both zero; in other words, the all important phase of the wave.  So we just add a term  for the initial condition and now our final description of a backward moving is,

                                                                        ,                                             (A.1)

and of course a forward moving wave would be,

                                                                                                                      (A.2)

where  is the initial condition for the backward moving wave and  is the phase of the forward moving wave, i.e.,  and  are the values when t and x = 0. Another way of thinking about the wave being forward or backward traveling is to remember that must be constant for p to stay constant. In Figue A4 if the skier is to stay at the same place on the wave, i.e. if   is to stay constant; then kx - t can stay constant .  This means that kx and  getting positive together.  Conversely, for  it is just the opposite.  When time increases, x must be getting smaller for to stay constant.   A wave cannot be moving forward and backward at the same time.  So either equation (A.1) describes the wave or equation (A.2) describes the wave.  Always look for the + or – sign between the position description, x, and the time description, t.

            These are perfectly good description of forward and backward moving waves but they are not used because they are awkward and time consuming when adding, multiplying and dividing waves and otherwise manipulating wave functions.  It is also very difficult to keep track of the phase of two waves in their sine and cosine form.  And you will soon find out that in flute analysis, or analysis of any resonator, we will be obsessed with phase.  A more convenient and rather ingenious way to represent the wave is to use the definition,

                                                .                                             (A.3)

j acts like and has the properties of .  Of course there is no such thing as , but ;; and; just as  does; because •= -1;; and ••= - . 

Figure A6. The graphical depiction of a wave in the imaginary representation.  The vertical axis is the imaginary axis and the horizontal axis is the real axis.

In this representation the backward traveling wave becomes,

                                  (A.4)

and the forward traveling wave becomes,

                (A.5)

We can see from Figure A6 that what pressure we measure is .  It is important to get used to this representation because it is used extensively in disciplines where waves are described.   A second advantage of this representation is that you can solve problems graphically, especially when you are adding, dividing, and multiplying waves and want approximate answers.  Also, we will see when we describe waves in flutes, the real part of the ratio of pressure to current (impedance) in a flute must vanish to define the resonance condition.  As we will soon see, when describing oscillators, the current and pressure have to be 90° apart, i.e., out of phase. 

To help understand this representation, we will add the two waves of Figure A1, i.e. the sine and cosine waves of the same frequency.  This is done graphically in Figure 7.  Then we do it mathematically with the complex representation.

Figue A7  Addition of a sine and cosine wave of equal frequency

In Figure A7 the distances from the base line for sine and cosine waves (shown in black) are just added to give the wave for the sum of the two.  Note that at 45^o, 225^o, and 405^o the arrows show the sum of two waves at their equal magnitude to give the peek of the summed wave. 

So now let’s calculate what we did graphically in Figure 7.  The two waves in Figure 1 are

                                                                                                      (A.5)                                                                ,                                                       (A.6)

Where the sine has been changed into a cosine by changing the phase by - 90 ^o. We can now put them into a complex form by adding a j sin ? term.  Equation A5 becomes,

                                            ^(A.7)

and Equation A6 becomes,

                                                                                    (A.8)

From algebra we have, we can now separate out the phases in Equations A7 and A8 as

                                                                                                (A.9)

Now the two waves can be  added in the complex forms as Equations A8 and A9 as,

                                                     or,                     (A.10)

The first term can be substituted to be and so equation (A10) becomes,

We now draw a picture of the two phase components in Figure 8

Figure A8  A graphic depiction of Equation A10 at x=0.  As the angle kx increases, the vector rotates in the counter-clockwise direction.  If you stop the rotation at some value of X and let t increase, the victor  rotates in the clockwise direction.  Just the opposite is true for a backward traveling wave.

From the diagram of Figue A8 and Equation A10,  we can see that the description of the sum of a sine wave and a cosine wave of equal amplitude P_o can be re written as,

,

or,

.                                 (A10) 

   In review, two waves with the same frequency (and of course wavelength), of amplitude  P₁  and  P₂ and phase  traveling backwards,

                                                                       (A.11)

or,

            ,               (A.12)

where is the vector sum or ₁ and   ₂ as shown in Figure A8.         

Another way to look at it is to add projections on the two axes in the usual way of vectors as,

                                                                                                                (A.13)

and

                                                                                                  (A.14)

and,

As shown in Figure A8.

 Figure A8   Phaser P, ?, for the addition of two waves with phasers P_1, ?₁ and P_2, ?₂  

Note that if P₁=P ₂  and ? ₁ = ?  ₂ ± 180^o

                                                                                         ^(A.15)  

In other words, two waves of equal strength traveling in the same direction out of phase will cancel each other.  This never happens in the real world.  Do you care to venture why?  How about conservation of energy?  You would have two sources in the same place.  Two things that occupy the same space are the same thing.  Cancellation of beams is common, but you have to have them reinforcing some place else.  It is worth doing a little experiment just to show this important concept.  Take two common lasers.  Set them so their beams cross.  At the intersection put a piece of partially reflecting glass at 45^o to both as is shown in Figure 9. This results in two beams of the same wave length traveling in the same direction.  If you make a careful adjustment of the beams, one set will vanish and the other will be twice as strong. 

Figure A9  Two lasers shining on a partially reflecting plate.  Low cost helium-neon lasers at 0632.8 nM work fine.  If one beam is weakened because the two halves are out of phase, the other one is reinforced, otherwise, conservation of energy would be violated.  You can also do the experiment with one laser and two mirrors.

            It is interesting to note that a helium-neon laser is a gas filled tube with a mirror on each end.  One mirror is partially reflecting where the beam comes out.  It is a miniature flute only the gas builds up the wave inside.  But like a flute there are modes generated harmonically at frequencies spaced by c/2L i.e., (/2 = L).  For example a 3 cm long laser at a wavelength of 632.8 nM (6.328 x 10^-4 cm) and a speed of light at 3 x 10¹⁰ cm/sec; gives a spacing in frequency of the laser light of  c/2L = (3 x 10¹⁰ cm/sec )/2 x 3 cm = 5 G Hz separation between the laser modes.  There are about three or four spikes in frequency of laser light centered around 632.8 nM (orange-red) light.  In other words, the flute is ½ wavelength long or 6 cm wavelength.   Therefore one gets a beat of 5 G Hz between radiation in the same laser beam with a detector.  So in summation the adding of two waves of the same frequency going in the same direction will be a wave with an amplitude and phase determined by the vector sum of the two phasers.

            Speaking of conservation of energy, a wave traveling in a pipe will not go on indefinitely without attenuation.  The pipe warms up and so does the air.  The pipe warms up due to the friction of the air rubbing the pipe.  The air warms up because of its viscosity (thickness).  We will have lots to say about this later in Appendix C, but for completeness a plane wave moving forward in a pipe is described by:

,                                                            (A15b)

describes the diminishing of the wave power as it goes down the pipe a distance x .  Often we express the diminishing in time and not position by using where ct=x and is used just as often, for example is the 3db bandwidth of an oscillator, we will learn when we get into flute losses and sound levels.   We mention this because authors use for both and as described here. 

Standing Waves in a Flutes

            With the phase equal zero at t=0 and x=0, we can describe two waves of equal ampliltued, A,moving “through each other” as,

                                                            .                                    (A16)

or,

                                                                                                              (A.17)

in the complex representation.  To put this in a usable form in the triganometric representation, use the identities from Appendix D,

and Equation A.16 becomes,

                                                                        .                                               (A.18)

To put Equation A.17 in a usable form in the complex representation, note that,

Equation A.17 can be written as,

.

Use the identity  from Appendix D to get .

The real part of this is exactly Equation (A.18), the triganometric represetation. 

Figure A17.  Depiction of a standing wave resulting from two traveling waves of equal amplitude passing through each other.  The dotted lines show the standing wave at different times, ,, and  .  Note that this pressure standing wave between two null points is what we have in a flute.

            The amplitude of the forward traveling wave and the backward traveling wave are seldom of equal magnatude due to attinuation and leakage from the ends.  So for the more realistic case, we rewrite Equation A.17 replacing the amplitude of the backward traveling wave from A to B, to wit,

                                                            .                                                        (A.19)

We put the real and imagionary parts together to form,

                                                .                                          (A.20) 

Compare the part of Equation (A.20) with the equation depicted in complex space in Figure A7.  We show it in Figure 18. 

                        Figure A18.  Shown is a depiction of Equation (A.20) in complex space.  Note

                        that a coeffecient of  e^j?t rotates counterclockwise at an angular rate ?t .

When the reflected wave amplitude is smaller than the incident wave, then the standing wave pattern shown in Figure A17 changes to the one in Figure A19 as a depiction of Equation (A.20), or

                                                .                                   (A21) 

                        Figure A19. Depiction of a standing wave resulting from two traveling waves

                        of unequal amplitude passing through each other.  The dotted lines show the sand-

                        ing wave at different times, t₁,t₂, and t₃ .  This is similar the the situation where the

                        two waves are of equal ampliltued, as shown in Figure A17. 

Curent Impedence and Admittance

            As with the design of any oscillator, it is the ratio of pressure to current that we have to deal with because their ratio must be imaginary, i.e. out of phase, and the energy switches from one to the other at the resonant frequency, as explained above.  We get the relation between pressure and current from Newtan’s law, the force equals the mass times the acceleration.  Figure 20   shows our arrangement for considering this.

               Figure A20.    S is the area of the flute and ? is 1.21 Kgm/M³ and force is in Newtons.  It is pressure times area.

From  we get ,gives the all important,

                                                                        .                                                               (A22)

c is called the impedance of the medium in which the sound wave is traveling.  p is in Newtons per square meter and U is the “speed up” caused by the pressure p in meters per second.  .  For air at room conditions, ?= 1.2 Kgm/M³ (about 2 lbs. per cubic yard, heavy stuff) and c= 343 M/sec,giving an impedance of 415 Kgm/M sec.  For flutes of cross-sectional area S, we want to use the characteristic impedance of the flute which would be, for a one-way traveling wave; i.e. the pressure and current are in phase.

                                                                                                                 (A.23)

The units of U are ?x/?t, the speed of the sliver of Figure A20.  The units of u = ?x S/?t is the speed of the volume unit ?V=S ?x.  Some chose to refer to the characteristic impedance of a pipe as Z_{o =}?c/S but I chose to use Z_o = ?c= 415 Kgm/M sec and keep ?c/S as a symbol to emphasize the importance of pipe area, especially for those designing conical bore flutes.  If we go through the same process for a backward traveling wave, the current is out of phase with the pressure, to give,

                                                                                                              (A.24)

            Finally, we can get to the impedance of a standing wave in a pipe.  Equation (A.19) is what we used for the standing wave pressure in a pipe with a forward wave of amplitude A and a weakened reflected wave amplitude B, we repeat,

                                                            .                                                        (A.19)

The impedance of the standing wave will be,

                                                                               (A.25)

or from Equation A.19 and A.24, the impedance becomes                                                                                                                                            .                                                      (A.26)

If we write this equation for x=0, it is commonly labeled because it is the impedance that you “see” when “looking into a pipe from x=0 and it is,

                                                                        .                                                     (A.27)

For clarity and convention, we will relabel Equation (A.26) replacing x = L, indicating that is the impedance of a place down the tube a distance L.  So Equation (A.26) is rewritten as,

                                                            .                                                    (A.26)

If we change Equations A 26 and A.27 into the form,

                                                                        .

This allows for the elimination of B/A and gives an impedance for a pipe that is independent of a signal strength.  In other words, it gives an expression for the impedance “looking” into the pipe that is dependent on the pipe geometry alone.  This is the, all important equation,

                                                .                                                  (A.28)

This is the equation used for transmission of power on high voltage transmission lines at a 60 Hz frequency (5000 Km wavelength).  It is the equation used for sending and receiving signals from an antenna along a coaxial cable; and it is the equation for sending sound waves along a pipe.   The early work on sound in a pipe done by Helmhertz and followed up by others was not expressed in the form of Equation A.28.  It was not until formalization of electronics and the work of Lawrence Kinsler and Austin Frey(1942); Neville Fletcher; Thomas Rossing(1998); Arther Benade; and John Coltman and others, using that formalism; that the engineering was expressed in an intuitively obvious way.  The resonance frequency  occurs at  the value of kL, for which the real part of Equation A.28 vanishes.  For the flute we do not need to bother going through separating out the real and imaginary parts because one can see two cases where this occurs. The first is the resonance condition for clarinets, saxophones, bagpipes, and most woodwinds with reeds or lips closing on end.  When one end of the pipe is closed thenbecause the current has to be zero.  Rewriting Equation A.28 to see what vanishes, and temperedly neglecting the loss part,

                                                                                                                 (A.29)

you can see that when ,

                                                                                                          (A.30)

       Our second condition, the one for flutes, that provides for to be imaginary, i.e., for the real part of Equation (A.28) to disappear is for the pipe to be open on the end, giving .  In this case  of Equation (A.28) becomes

                                                                                                           (A.33) which becomes the classical impedance for a flute open on both ends of effective length .  when or any multiples of half wavelengths. All other woodwinds are constrained by the boundary condition or Equation A30.  In this case when and or .