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pipecalc Mathematics

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pipecalc (WebPipeCalc) is based on formulas by Dr. Ing. H. Ising, published in the German Publication "Walcker Hausmitteilung Nr. 42" from June 1971. In this post I want to summarize these formulas, because the original work is quite hard to read and for the most part doesn't even give the units for the formulas.

Isings formulas

Frequency

Open pipes

The first formula that Ising gives us is $$ f = \frac{c}{2 \left(\text{L} + \Delta\text{L}_\text{M} + \Delta\text{L}_\text{R}\right)} $$ which determines the frequency of the fundamental note of the pipe. \(f\) is in \(\text{Hz}\), \(c\) is the speed of sound in \(\frac{\text{m}}{\text{s}}\) and \(\text{L}\) is the resonator length in \(\text{m}\). \(\Delta\text{L}_\text{M}\) is a correction factor for the mouth of the pipe and \(\Delta\text{L}_\text{R}\) is a correction factor for the upper resonator opening.

The correction factors can be calculated as follows: $$ \Delta\text{L}_\text{M} = 0.73 \cdot \frac{\text{S}}{\sqrt{\text{S}_\text{M}}} $$ $$ \Delta\text{L}_\text{R} = 0.34 \cdot \sqrt{\text{S}} $$

Where \(\text{S}\) is the cross-sectional area of the resonator and \(\text{S}_\text{M}\) is the cross-sectional area of the mouth, both in \(\text{m}^2\).

In my experience this formula works pretty well, but in pipecalc the frequency is given, so we need to rearrange the formula to have \(\text{L}\) on one side.

$$ \text{L} = \frac{-0.73 \cdot \text{S}}{\sqrt{\text{S}_\text{M}}} - 0.34 \cdot \sqrt{\text{S}} + \frac{c}{2 \cdot f} $$

Stopped pipes

For stopped pipes the formula is $$ f = \frac{c}{4 \left(\text{L} + \Delta\text{L}_\text{M}\right)} $$

Again, rearranging it so \(\text{L}\) is on one side: $$ \text{L} = \frac{c}{4 \cdot f} - 0.73 \cdot \frac{\text{S}}{\sqrt{\text{S}_\text{M}}} $$

Speed of the air at the jet

The next formula Ising gives us is used to calculate the speed of the air as it leaves the jet.

$$ v_0 = 400 \cdot \sqrt{\text{P}_\text{F}} $$

With \(v_0\) in \(\frac{\text{cm}}{\text{s}}\) (???) and \(\text{P}_\text{F}\) in \(\text{mmH}_2\text{O}\). I don't use this formula in pipecalc, because it is just an approximation and it has really strange units. I use this (exact) formula:

$$ v_0 = \sqrt{2 \cdot \frac{\text{P}}{\rho}} $$

Where \(v_0\) is in \(\frac{\text{m}}{\text{s}}\), \(\text{P}\) is the pressure in \(\text{Pa}\) and \(\rho\) is the density of air in \(\frac{\text{kg}}{\text{m}^3}\).

Intonation number

Finally, we get the formula for the intonation number.

$$ \text{I} = \frac{v_0 \cdot \sqrt{\text{h}_0}}{f \cdot \sqrt[H]{H}} $$

\(\text{h}_0\) is the mouth width in \(\text{m}\) and \(\text{H} [\text{m}]\) is the cutup height. Note that in this formula all lengths appear to be in \(\text{cm}\), like \(v_0 \left[\frac{\text{cm}}{\text{s}}\right]\).

For us this works out as

$$ \text{I} = \frac{\sqrt{\frac{2 \cdot \text{D} \cdot \text{P}}{\rho \cdot \text{H}^3}}}{f} $$

With \(\text{D} [\text{m}]\) jet thickness.

The intonation number doesn't have a unit. It determines the efficiency of a pipe. At \(\text{I} = 2\) the pipe is working at optimum efficiency, which means that most of the energy is used to create the fundamental note of the pipe. This is usually wanted for pipes with few harmonics, like flutes. A principal stop would have an intonation number of \(2.5 - 3\). Nearing an intonation number of \(3\) the pipe tends to overblow. This is why thin, long pipes like the violins usually have a frein attached to them. It stabilizes the air flow.

Air consumption

Calculating the air consumption of the pipe is pretty straight-forward.

$$ \text{Y} = v_0 \cdot \text{A} = v_0 \cdot \text{D} \cdot \text{h}_0 $$

With \(\text{A} [\text{m}^2]\) jet area and \(\text{Y} \left[\frac{\text{m}^3}{\text{s}}\right]\) air consumption rate.

Air hole

Ising says that the air hole should be at least

$$ \text{A}_\text{air hole} \geq 10 \cdot \text{A} $$

Sound pressure and power

Ising also gives some formulas for calculating the sound pressure and power of organ pipes, but as they seem to be a bit off I won't explain them here. I included them as an experimental feature in pipecalc, look at the code if you want to know more.

Other important formulas

As there are still some unknown parameters. That's why I added some formulas based on other works, like calculating the pipes relative to the Normalmensur by Töpfer.

$$ d = 0.15555 \cdot 0.957458^{\text{h}} $$

With \(\text{h}\) in halftone-distance to C2 (international/scientific pitch notation) and \(d\) (pipe diameter) in \(\text{m}\).

The mouth width and cutup height are just set relative to the circumference of the pipe, which seems to be a regular way of doing it.