About Organ Pipes

The swete Orgayne pipis comfortith a stedfast mynde
--Lekenfelde Proverb (ca. 1545)

Introduction

To students of musical acoustics in the 19th and most of the 20th centuries, the flue and reed organ pipes were iconic musical instruments; they represented the templates to which all flute and reed instruments were to be related. Restricting this discussion to flue pipes only, we may recall that a few of these pipes could be found in the back closet of every physics lecture hall. They demonstrate the effects of boundary conditions on hyperbolic partial differential equations. As such, there is some commonality between the equations that govern organ pipes and the Schroedinger equation that governs quantum mechanics. It follows that organ pipes sometimes mimic quantum mechanics. This all sounds very impressive. However, people still don't really know how they work! Pipe organs similar to modern ones have existed since Roman times. Organ pipes are pretty simple. How can this be that they are not well understood? The answer is twofold; it is a compound problem. First, there is the problem of the meaning of understanding. Second, given an answer to the first question, there is the problem of caring about understanding how they work enough to find out. Both of these problems have a cultural context.

To the minds of the 18th and 19th centuries, the first problem was completely answered with D'Alembert's wave equation -- which is derived below. It only explains how the empty tube part of the pipe works. It does not predict much more than the pitch of the sound and what harmonics could be present in it. With the advent of fast Fourier analysis and ubiquitous computers, a better answer is required to constitute understanding -- one that at least predicts timbre, attack and decay in detail. This means being able to predict the spectral and phase composition of the entire radiation field as a function of time of a pipe from first principles. Until the advent of computational fluid mechanics, such questions were nearly unanswerable. Here enters the second part of the "how it works" problem. Now that methods are available to solve the first part of the problem, who cares? Organ pipes are vanishing. Pipe organs are the most expensive and cumbrous of musical instruments. Their advantage of loudness has been obviated by electronic amplification. Their other special advantages are only really perceptible in live performances and only to those very near to the pipes. In fact, the subtleties of their spatial sound fields are best appreciated by sitting inside them! A rare experience. Live performances are in decline. Additionally, pipe organs have long been linked to European Christian religious rites. Religion is itself in decline and the European immigrant population is now quite small in the US. Not only is public interest low, few people are now being trained to play them. Fewer players mean fewer performances. Pipe organs may continue to survive in academic institutions. Evolution is possible. From the vantage point of my own laboratory, pipe organs can be constructed to possess the ictus, the expressive quality of the piano to touch. Popular interest might increase as a result; however the cost would also increase. Such a change would also engender a new literature. Many classical musicians believe that new instruments and new literature are undesirable. I discussed these ideas years ago with a talented young performer, Laurence Libin. He cursed such novelty and my suggestions. Eventually he became the curator of keyboard instruments for the New York Metropolitan Museum of Art, the antithesis of change. I also had this discussion with the famed Calvin Hampton. He too was very negative. He added that, regarding any advantage that might possibly accrue to a new instrument through the recorded performance, "anything you push through a hole is indistinguishable from anything else you push through a hole." (His became one of the first names on "The Quilt".)

The Silent Pipe

From harmony from heavenly harmony,
This universal frame began:
From harmony to harmony
Through all the compass of the notes it ran,
The diapason closing full in man.
-- John Dryden,"Song for St. Cecilia's Day" (1687)

The pipe functions as a sort of potential well to trap a wave. Very little of the wave can escape from the sides of the pipe. At each end, there is a sharp discontinuity, the opening there. These openings set boundary conditions on what waves can exist inside. Also, the pipe is quite long compared with its width. This combined with its cylindricality also sets constraints on possible waves. We might imagine cylinder waves and transverse waves in addition to longitudinal waves. These former would propagate over much shorter distances and thus correspond to higher pitches than that of those traveling lengthwise. At the top end of the pipe, these would correspond to dipoles and not radiate efficiently. They probably actually exist and do radiate from the mouth of the pipe. Experiment suggests that they are not significant in all but the oddest shapes of pipes. What remains are those waves that propagate lengthwise, the longitudinal waves.

Looking into the empty interior of the pipe, we find air. Air has mass and is compressible. These properties are analogous to those of a musical string. The string has mass and springiness. These are the qualities of which waves are made. The mass is in fact mass per volume or density. The compressibility (formally the reciprocal of springiness) of air is not linear. Worse, when air is compressed, it gets hotter and the heat travels away and the loss of heat lessens the pressure. This is a cumbersome complication. Fortunately, sound waves involve motions that are fast compared with the rate heat travels in air. If we pretend that the heat doesn't travel at all, we have a so-called "adiabatic" process. There is then a fairly simple relation between pressure and density that describes the springiness of air:
 
, where p is the ambient air pressure and rho is the ambient density, p' is the pressure at some instant and place in the wave and rho prime is the density at that same instant and place. Gamma is the ratio of specific heats, a thermodynamic constant for air. Density is an inconvenient measure for sound waves. Acousticians have developed one they like better, condensation, s. This is the ratio of the change in density at some instant and place to the ambient density. Absolute pressure is also inconvenient to acousticians. Hence they re-define pressure, p, as the difference between the pressure in the wave and ambient. The next step is to assume that p'/p as originally defined is nearly unity. Thus the above expression in terms of gamma can be linearized by expanding the ratio into a powers series and dropping terms. The result is:

 
where is now the ambient pressure. This relates pressure in a wave to condensation. Note that pressure increases with condensation. This defines the signs. These ought to be enough variables to solve the problem. But again they turn out to be inconvenient. We need some additional ones.

We can imagine the air inside the pipe as being divided up into cubes so small that we can identify the individual particles of air inside them. We will choose to draw a typical cube such that there are particles of air at its faces, since it's all imaginary anyway. The six sides of the box are moving with the wave. If one side moves faster than another the average of the others, the box gets bigger or smaller. As the box gets smaller the condensation increases. Whatever was in the box stays there after the change in size. To express this simple notion, we need to add spatial coordinates, x,y,z and corresponding particle velocities, u,v,w in these directions. Then we can write a "conservation of matter" expression in terms of differentials since the box is very very small:
 
 
 

We can test the veracity of this expression. Let the only movement be in the x direction. Let the partial differential of u be positive. In that case the right hand side of the box is moving faster than the left hand side. The box is getting bigger, so the condensation must be growing in the negative (shrinking). This is consistent with summing to zero.

The next step is to apply a result from vector theory. We do need any vectors however. Rather, we merely need to know that if the motion involves no rotation (and sound waves don't rotate) and no new material is generated in space that would violate the conservation of matter expression, we can define the wave we are trying to describe in terms of a scalar, that is a single number that varies over space and time. This lets us drop u,v and w from our calculations and come back to them later as follows:

 
 
 
 
  where is the scalar and is called the "velocity potential".

We are still not done. We have accounted for the compressibility of the air but not for its mass. We need to add Newton's Law about force equaling mass times acceleration. Consider the above box moving in just the x direction. Say the box has dimensions dx, dy, dz. The mass of the box is dx dy dz. The net force on the box is dp dx dy dz / dx, since we identify dp/dx as the rate of change of pressure. The acceleration of the box is du / dt. We get dp dy dz = - dx dy dz du / dt or simply dp/dx = (du/dt) which is written more properly:
 
 
 
 
 
when cast into three (actually four) dimensions. Similar expressions apply in the y and z directions. By substitution of the definitions of and integrating we get:

 
 
 
 
so long as doesn't vary much. In terms of s this is written:
 
 
 
 
 
 
Substituting this into the conservation of matter expression above gives D'Alembert's wave equation:
 

 
 
 
 
 
 
Fortunately for us in dealing with organ pipes we do not need the y and z terms. Furthermore,
 
 
 
 
 
 
namely, the square of the velocity of sound equals gamma times the ratio of the ambient average pressure to the ambient average density of air. Hence we need only solve:
 

 
 
 
 
 
As an aside, notice the similarity of this equation to the quantum mechanical Schroedinger equation in one dimension:
 

 
 
 
 
 
Since the V-term sneaks in boundary conditions that can otherwise be applied separately allowing V to be dropped, for some time-independent wave-in-a-box problems both equations can be solved with a similar series of sine and cosine terms.

D'Alembert's equation has been solved for tubes and horns of all sort of shapes. We now look at the solution for a pipe. We agreed to neglect radial and transverse waves since these contribute little if any to radiation from the pipe. That leaves waves that travel longitudinally in the pipe. We have made another assumption that introduces a boundary condition: the wave cannot escape from the pipe except at the ends. Spherical waves therefore cannot propagate in the pipe for its full length. This leaves only one spatial coordinate remaining, the pipe's axis. Hence D'Alembert's equation reduces to the "pipe equation"
 

 
 
 
 
as I said it would. This is the same as for plane wave propagation in free space. We have gotten off pretty easy compared with say sound waves in horns.

The first case we will study is that of the infinitely long pipe. Being infinitely long, this conceptual pipe has no ends to worry about. We know from the derivation above that pressure and particle velocity are related to derivatives of . We are looking for a class of function whose derivatives suggest sound waves and are also solutions of the pipe equation. Fortunately we are in luck. The culture of music has provided us with the answers, sine and cosine functions. If we insert a sine wave with a lot of undetermined constants into the pipe equation, equate the second partial derivatives as shown and then sort out what the constants become, we get
 
        
 
 
where , is the wavelength, c is the speed of sound and A is an arbitrary constant. If we do the same thing with a cosine wave, we get the same result. We can summarize these findings in terms of imaginary exponentials:
 
        
 
 
where j is the square root of negative 1. Imaginary exponentials are easier to work with albeit less graphic than sines and cosines.

Again we need to consider our variables. The velocity potential is a construct to facilitate writing D'Alembert's equation. We also have pressure and particle velocity. If we consult Lord Raleigh's The Theory of Sound (1894) we will find that he seemed largely content with finding the particle velocity and energy density. But times have changed; these were not the best variables for the 20th Century. Let me tell you an explanatory story.

When I was little, my family lived in several court buildings in Evanston, IL. We moved into ever-increasing quarters up until my sister and I graduated from high school. These buildings were arranged around a central court -- often with a fountain (that usually didn't work). These buildings were built between 1890 and 1928. They were heavily constructed and are all still standing in 2006. They were all equipped with internal communication devices. Some had mechanical-flag annunciators to signal the servants that were no longer there. The later buildings had telephones to communicate to the lobbies typically four lobbies per building. These telephones had mouthpieces fixed in the wall and earpieces that hung on hooks. The earlier buildings had been built with speaking tubes. A tube of about one inch in diameter connected between the foyer of each apartment and its lobby. There was typically a combined mouthpiece / earpiece and in some cases a whistle. Few of these were still in use. During renovations, the tubes were often used as a conduit in which to run the telephone wires. Although the practice of using speaking tubes in buildings was thousands of years old, the turn of the century saw the science of speaking tubes become the product of a telephone-oriented culture.

Communications engineers were already well established by the time the earliest of the court buildings were built. Communications engineering was a culture of electricity. One of the most historically significant pipe organ builders of the 19th Century was a communications engineer for a telephone company, Robert Hope-Jones. He introduced many electrical innovations into pipe organ building. He bucked the natural resistance of the music world to new instruments and literature. He did this at his own peril. Even in his progressive times, hate campaigns and physical sabotage were directed at his work. He was financially ruined and took many of his backers, including Mark Twain, with him. Eventually, after his death, most all of his innovations became accepted practice. Today he is memorialized as a saver of lives, the man that invented what was to become the fog horn.

By the 1930s the natural tendency for engineers of a type to have a set of units peculiar to them had evolved into a sort of reform movement. Since the units appropriate to electric circuits are voltage and current, an analogous pair of acoustic units was discovered, pressure and volume current. Their ratio was defined as acoustic impedance analogously to electrical impedance. Harry Olson of RCA Laboratories demonstrated the utility of electrical analogy units for both acoustic and mechanical vibrating systems. The idea became canon in WWII and remains fundamental today. Since we already have a way of finding pressure, all we need to find is volume current, U. It is simply the product of particle velocity with the cross-sectional area, S, of the pipe. Spelling these out:
 
From and our solution for we obtain,
 
.
 
 
From , we obtain,
 
.
 
 
These solutions to the problem of the infinite pipe illustrate what will be our basic acoustic parameters from now on, p and U.

   We may note in passing that the arbitrary constant, A, now has a prefactor of -j acquired in the process of finding p and U. This factor is not shown since these are still valid solutions. However they differ by this factor from that obtained for . Is this the only thing arbitrary about A in the picture? No. Seeing as how we have substituted a complex exponential for sines and cosines, A may be complex. When A is real we get a real cosine plus an imaginary sine. When A is imaginary we get a real sine plus an imaginary cosine. This is how we shuffle back and forth between sine and cosine terms. What about k? The constant k is also arbitrary at this point. It too could be negative. It has units of radians per distance. It is called the wave number in quantum mechanical contexts. The product, kc, is radians per time and is called angular velocity. It is two-pi times the frequency. This is the pitch of the sound. Notice that while A need not be real, we define p and U as the real part of whatever expression we derive for them. Before proceeding with the subject of complex numbers, I want to show you something else that is just as important.

    I mentioned earlier that the idea of looking at acoustic problems as analogues of ones in electrical engineering is facilitated by the concept of acoustic impedance. This is the ratio of p to U. It does not depend on p or U being real. For p and U as given above,
 

 
The parameter, z, is a constant in the infinite pipe since everything else divides out. In the case of free space, we may increase S to infinity while normalizing in terms of a unit of area. What we get is an impedance referenced usually to one square meter. This kind of impedance is thus "specific" impedance. Dividing this by S gives that of the impedance of a pipe. It turns out that when we examine the finite pipe, the impedances are not going to be constant.

    In the electrical case, one examines what are called 4-terminal networks. Two terminals go in and two come out.
 
 

 
 
 
 

The diagram above illustrates this convention. Voltage, analogous to pressure is measured across terminal pairs. Current is measured through terminal pairs. We can conceptualize a trans-impedance as the ratio of a voltage across say terminals 3 and 4 resulting from a current going through terminals 1 and 2. Electrical engineers will often conceptually hang another circuit off of one end and calculate what it does to the impedance of the other. This allows a complicated circuit to be reduced to blocks for simplicity. We draw the organ pipe analogously.
 
 

 
 
 
 
 
 
 

We define the impedance z₂ as that at the top end of the pipe. This may be open as in an open diapason, closed as in a stopped pipe (z₂ = ) or something else as in partially stopped pipes such as the lieblich gedeckt. The solution strategy for the open pipe will be to set z₂ equal to the radiation impedance of an opening the size of the end of the pipe. This will constrain z₁ to a single value looking into the pipe. In the next section, The Speaking Pipe, we will convert the acoustic impedance at the mouth to a mechanical impedance. Then, using the concept of the "wind reed" borrowed from the folklore of pipe voicers, we will calculate the mechanical impedance felt by this construct and the driving forces acting upon it.

The impedance z seems to lock the ratio of pressure to volume current to a fixed value. This cannot be the case since stopped pipes exist. If the z2 end of the pipe is stopped, the volume current is zero while the pressure definitely is not. This sets z2 at infinity. What decouples pressure from volume current is having waves traveling in both directions. The ct-x part of the argument of the exponentials would be ct+x for a returning wave. This is allowed because both k and c can be negative. Therefore, we are entitled to write a solution representing a backward-traveling wave using ct+x and keeping both k and c positive for convenience. An arbitrary backward-traveling wave can have any phase relationship to the forward-traveling one. As we will see later on, if we keep the coefficient A for the forward traveling wave and use B for that of the reverse traveling wave, allowing B to be complex in order to account for any possible phase difference, and consider the ratio of pressure to volume current at x = 0, we obtain this expression:
 

 
 
 
Clearly we can have any sort of relation between pressure and volume current. As I mentioned before, the development of the 4-terminal network transfer impedance model of the organ pipe was strongly influenced by electrical engineering where voltage and current are analogs of pressure and volume current. Electrical engineers have always assumed that voltage and current were independent. Their starting point is usually a pair of partial differential equations that have a lot more terms in them than the D'Alembert equation used by me here. No assumption is needed about anything like a velocity potential (in electric circuit theory). For instance the analog for telegraph lines is summarized in W.L. Everitt, Communication Engineering, McGraw-Hill, New York (1932). The starting point is different, but the results are identical to what we are about to derive together. In the case of microwaves, the starting point is Maxwell's equations. The result is again identical for both organ pipes and microwave plumbing. (This analogy proved important at the outbreak of WWII and plays a role later in this story.) The derivation which we now work out is, except for the constants, in fact identical to that given in K.C. Gupta, Microwaves, Wiley Eastern, New Delhi (1979) for microwave plumbing.
If we look at the solutions we obtained for p and U, we see that the dependence on time can be factored out. Thus we have:
 
 

 
 
 
 
This reduces the number of symbols we have to carry around with us when we do calculations. This approach will be robust only if we can relate P to U(x) without resorting to differentiating with respect to time. We can explore this relationship.
 
 

 
 
 
 
Because we can factor out the time-dependent exponential, it follows that:
 
 

 
 
 
 
Thus we have cleared out the dependence on time. We will need a name for this formula. Since I am too lazy to number all the equations, I will call this one "Gupta's"; it is analogous to equation (2.14) in Gupta [1979].
We can now consider the sum of a forward-traveling wave and a reverse-traveling wave. Since the same time exponential is common to both, we can write sum expressions for pressure and volume current.
 
 

 
 
 
 
Applying Gupta's formula, we obtain:
 
since the jk factor divides out.

 
 
 
 

Let us set up our coordinate system so that the top of the pipe is located at x = 0. This sets the exponentials in the expressions for P and U(x) to unity. We can specify boundary conditions, P = P_z2 and U(0) = U_z2 . We can now solve a pair of simultaneous equations to obtain P₊ and P_- . Gupta's equation tells us that P will be the sum of terms while U(0) will be the difference of terms. The result is thus:
 
 

 
 
 
 
This is easily verified by substitution. Meanwhile, some distance away, by inserting these values we obtain:
 
 

 
 
 
 
and

 

 
 
 
 

These expressions can be simplified by setting x = -s, where s is the length of the pipe.
 

 
 
 
 
and

 

 
 
 
 
These two equations can be readily formulated as a matrix product where the 2 by 2 matrix is called the scattering matrix, the transfer matrix or the ABCD matrix depending on the field of application. Since what we are interested in is the impedance at the mouth of the pipe as a function of the impedance at the top, we set U_z2 equal to unity and thus P_z2 becomes z2. Dividing P(-s) by U(-s) gives this result after converting from hyperbolic to circular functions:
 

 
 
 
 
This is the first major building block in describing the behavior of an organ pipe. Multiplying this equation "up and down" by S (G*d knows why anyone would want to make the expression more clumsy.) and re-arranging gives equation (5.78), the cylindrical horn equation, in Olson. [H.F. Olson, Acoustical Engineering, Van Nostrand, Princeton, NJ (1964)].

The above equation describes what goes on inside the pipe and for that reason it was worth deriving. The next consideration is for what goes on at the ends of the pipe, its interface with the outside world as it were. These are the radiation impedances and they depend on how the pipe is situated and the details of its construction. From a practical point of view these are minor considerations if what we want to predict is the timbre of the pipe. Yet, from a theoretical point of view the effects are considerable. For example, a pipe with large ears can be treated with an integral in one dimension whereas the same pipe with the ears removed requires a two-dimensional integral that is much more difficult. The presence or absence of ears affects the pitch and attack but not the timbre of the pipe (to a first approximation) other than some pipes will not speak stably or speak at all without them. Ears alter the flow of air and add a little to the inertial reactance at the mouth. Similarly, the presence or absence of a pipe stay is not of much concern to organ builders other than to keep the pipe from falling over. From a theoretical standpoint, the effect can be enormous. A pipe with a large pipe stay has a flange at the end (the stay). Its radiation impedance can be calculated in closed form. Removing the pipe stay results in conditions where computation is very difficult and impossible in closed form unless simplifying assumptions are made. In practice, adding or removing a pipe stay, even a large one, has so little effect that the pipe does not need to be re-tuned.

From what I have just said, it is obvious that computing the radiation impedance of either end of an organ pipe is a formidable task with very little intellectual reward unless one is a mathematician. For this reason, I will delay going into the matter until it is absolutely necessary. I will use the large-flange case for the top of the pipe and the large-ears condition for the mouth of the pipe since they yield formulas that are only somewhat much more complicated than the one just derived. These formulas are given in Olson cited above but are given incorrectly there, Eqns. (5.10) and (5.11). That for the open end of the pipe is (correctly):
 

 
where
 

 
and R is the radius of the pipe. Here, J is a Bessel function of the first kind and X is an approximation for a Struve function. (See M.A. Ezz-El-Arab, IEEE Trans. Sonics and Ultrasonics, SU24 (1977) 327-331 and R.M. Aarts and A. Janssen, J. Acoust. Soc. Am, 113 (2003) 2635-2637.) Inserting this into the previous equation gives the impedance seen at the mouth of the pipe from behind. This impedance appears in series with the radiation impedance of the mouth itself, viz. the two are summed. Unfortunately, no closed form expression has been found for the impedance of the rectangular opening at the mouth. Typically, the Helmholtz integral has to be solved numerically. There are some approximations, though, that we will explore. The simplest is to replace the rectangular mouth of the pipe with a circular hole. There are pipes constructed this way.

To test this approximation we need simply find the equivalent radius,
 
.
 
Here h is the height of the cutup and is the width of the mouth. In this case:
 
,
 
where z3 is the impedance of the mouth opening found as just described.

Below I have plotted the real portion of zm for a pipe 60 cm in length and 3 cm in radius with a one-quarter mouth and a one-quarter cutup. This roughly describes a fat open diapasion pipe of pitch a440.
 

An interesting experiment is to compare these results with a conjecture of mine dating back to the "Capitol Avenue Experimental Shop" days in Hartford, CT. To perform this experiment, I compare the resistance of the first four partials with the measured sound intensity (dB) of the first four partials of a presumably fat open diapason given in A. Douglas, The Electronic Musical Instrument Manual, Pitman Publishing, N.Y. (1949). The table below normalizes with respect to the fundamental of the pipe, the first partial.