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Basic hooter whistle

By Johan Liljencrants

This note provides a few design parameter details for a basic single note whistle. The scope islimited since a couple of those parameters are very specific, based on custom and experience.

Namely that the whistle diameter is one third of its length, and that the mouth extends all around

and has the same area as the bell.

The design is illustrated with an air blown prototype, constructed from PVC drainage pipes. A

simple trick is shown, by which you can simulate the pitch bend characteristic of steam blowing.

1. The whistle parts

This general drawing

includes two ways to hold parts together. These are

examples of how you could

do, but they can be varied at

will depending on fabrication

facilities and taste.

The most basic part is the

bell , a tube that is closed at its

top end, open at bottom. Its

bottom rim forms the upper

lip of the mouth of the

whistle. The upper lip is

usually somewhat sharpened

with a bevel typically at an

angle of 30 degrees. Neither

this angle or the sharpness of

the edge is critical.

Blowing air or steam enters

through the bottom bowl , or

foot . The upper rim of thefoot has the same diameters

as the bell and forms the

lower lip of the whistle. Most

of the foot top is covered by a

round disk, the languid ,

normally with a slightly

beveled edge. The diameter

of the languid is somewhat

smaller than the inside of the

foot, leaving a narrow slit , the

flue. Through this the blowing air/steam forms a jet across the mouth, heading toward the outside of the upper lip. This

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jet is the driving agent for the whistle as its direction alternates between inside and outside of the

upper lip.

In most classical whistles the bell is held in place by a rigid central post , the right alternative here.

The lower anchor point of the post is complicated in that it must stay clear of the flue and leave a

free path for the blowing medium. The upper lip must be accurately aligned above the slit. For thatreason it may be good to put a ruggedizing spider inside the lower part of the bell.

The left design has no post, popular when whistles are made from tubes. Instead the bell is

anchored to the foot by three or more wings, running outside as bridges across the mouth. Another

variation may be to reduce the mouth width to occupy e.g. half or a quarter of the circumference,

like in organ pipes, such that the bowl and bell are parts of the same tube. This however reduces

the sound volume.

2. Critical dimensions

2.1 Length

The most basic measure is the length L, internally from the closed top of the bell down to the

languid plate. This sets the pitch of the whistle, it closely equals a quarter wavelength of the note

produced. The pitch, alternately note, or frequency F, is measured in Hertz, the same as cycles per

second. Knowing F you find wavelength as c / F, where c is the speed of sound. The nominal

length L of the bell actually has to be made shorter than a quarter wavelength, because air outside

the mouth will take part in the resonance motion of the bell air column. For this particular design

with 3:1 scale this 'end correction' is empirically about 6%, so the whistle will speak one semitone

lower than you might believe from L alone.

For air at 20 centigrade (69 F) sound speed c is 343 meter/second, same as 13500 inches/second. It

increases about 0.17 % per degree centigrade (0.10 % per degree F). With steam c is notably

higher. Blowing a whistle with steam will heat it gradually, and this together with the expulsion of

its air produces a characteristic pitch bend at the tone start. For steam you can not tell an accurate

value, but we can assume an approximate average of c to be 400 meter/second, or 15750

inches/second. This represents a upward shift of about 2.7 semitones relative to air.

The following set of scales relate note to length L, valid for air, and including the mentioned end

correction. Put a vertical line at some quantity you know and read corresponding values of the

others. For example, for a keyhole C frequency is 260 Hz, and L is 12 inches or 305 mm. The little

top right indicator shows how much to shift the upper L scales to the right in order to use them forsteam. The same length will then rather produce a 305 Hz #D. Or that same C will then require L

to be 14 inches, or 356 mm.

2.2 Diameter

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The ratio of length L to diameter D is called the scale of the pipe. This may be varied at will by

the designer. A narrower scale makes the tone contain stronger harmonics to the fundamental, a

feature much used in organ pipe design. In the same time this increases its tendency to overblow,

that is, the fundamental is lost and the pipe will speak at a harmonic instead. But with whistles a

main concern is loudness and no overblowing. Experience then tells that a good rule is to select a

3:1 scaling, diameter D should be one third of the length L, D = L/3. We stay with this rule here.

2.3 Mouth

The next general recommendation for a loud whistle is that the mouth area should be about the

same as the bell cross section. For the basic design with 3:1 scale and a mouth extending all

around the bell this automatically means the cut up H should be one twelwth of the length, H =

L/12 = D/4. With a lower cut up you must hold back blowing power to avoid overblowing, and

with a higher one the jet loses relatively more of its energy into turbulence under way to the upper

lip.

2.4 Flue

In the flue the static energy of the blowing medium, represented by its pressure P, is converted

into kinetic energy in form of speed V. This conversion obeys the Bernoulli equation P = 0.5 ** V2, where r is the density of the medium. In his research on organ pipes Hartmut Ising once

defined an 'intonation number', a formula to determine a best slit width T, depending on cut up H,

jet speed V, and pitch F (for a general dimensioning chart see [1]).

Using the Bernoulli equation and our present specific assumptions of a 3:1 scale, a 360 degree

mouth of same area as the bell, and with intonation number 2.5 this boils down to the simple

formula PT = 25L, using SI units. This tells two essential things. One is that the slit width T

should be set in proportion to L, the other that it should be inversely proportional to pressure P.

The formula is graphically shown by the black sloping lines in the following diagram.

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For a particular whistle, draw a horizontal line at its actual L, displayed at left. On this line, the

black lines form scale marks for slit width T against the bottom pressure P scale.

From the pressure, slit width and diameter one can also compute the power delivered by the

blowing air and its flow rate. Supply power is shown by the red lines. In an efficiently working

whistle the typical acoustic output power is about 1 % of that input. Flow rate is shown by the blue

lines.

This diagram, derived from the Ising equation, may rouse some discussion. For instance, its

description of T vs. P differs very substantially from what is given in [2]. It is unclear if there is a

theoretical background to that practice, but obviously the pressures in classical steam whistle

applications are vastly higher than in organs. No doubt sound level can increase with higher

blowing pressure, but most likely efficiency (the fraction of supplied power that is actually

converted into sound) goes down. With pressures approaching and beyond atmospheric, then

common linear acoustics relations are no more accurately valid. It is also very common that high

pressure pipes are overblown. To cure overblowing one remedy is to decrease slit width, but this

may soon reach a practical limit for reasons of mechanical tolerance. A more drastic means is to

increase cut up.

2.5 Inlet and valve

When the blowing medium passes a constriction its pressure is reduced in proportion to its linear

speed squared, as described by the Bernoulli equation above. One issue may be that the mediumshould find no essential constriction on its way to the flue, because that will reduce the power

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available to the whistle in relation to what is supplied by the blower. If we assume that the passage

has a single constriction area A while the flue has area B (slit width times circumference), then the

fraction of pressure lost can be formulated as B2/(A2+B2). If A=B, then the loss is 50%, but

increasing the constriction area A e.g. three times will reduce the loss into only 10%. One can

adopt the latter as a rule of thumb: any constriction in the feeding path, like inlet or valve

diameter, should have a cross sectional area that is at least three times larger than the flue area.

Presumably, with steam driven train whistles and the like, the foot pressure is substantially lower

than the boiler supply pressure, due to the throttling action of a narrow valve passage. As a

counter example using the formula above, if valve area A is one third of flue area B, then foot

pressure is only 10% of the supply pressure.

3. Note on pitch and end correction

It was mentioned that blowing with steam will cause a gradual pitch rise as the pipe heats and the

medium is changed. A similar rise comes with air, but to a lower degree, see e.g. [3]. The cause of

this can be attributed to two factors.

One is related to the energy transfer from the flue jet to the resonator. The resonating air column

in the bell makes for a flow oscillating in and out through the mouth. This flow controls the

direction of the flue jet to tend alternately toward the inside and outside of the upper lip. This

mechanism incorporates a delay that depends on the speed of the jet, hence also to pressure, it is

related to the transit time for the jet across the mouth. The delay corresponds to a certain phase

angle at the oscillating frequency. At stable oscillation there is a match between this 'jet phase'

angle and the phase of the resonating column i the bell, such that energy is transferred from jet to

column. Precisely at the bell resonance its phase angle is zero, by definition. The 'jet phase' makes

the oscillation frequency deviate somewhat from that bell resonance, to an extent varying with

pressure. - This complicated theory, here briefly outlined, is one base for the Ising formula.

The other factor may be coined as 'the blown away end correction'. In classical acoustic theory the

length of the resonator tube is apparently incremented by 0.6 times its radius. This is because the

medium outside the open tube end takes part in the resonance motion. Now, with the jet traversing

the resonator end this external air is partially swept away, the oscillation energy in the outward

bound puffs is lost to the resonator. This means the end correction becomes smaller because of the

crossing jet flow, and also the resonance Q lowers. Since the end correction is proportional to tube

radius this effect is larger in wide scale whistles.

This mentioned classical end correction for a tube comes when mouth area equals the tube area. Insome whistles and most organ pipes the mouth area is however considerably smaller. Then one

might believe the end correction would also get smaller. But actually the opposite will happen - a

narrower mouth will act as an additional acoustic mass to make the tube virtually longer and lower

the pitch produced. Ingerslev and Frobenius (1947) gave an empirical formula for the effective

end correction dLm when the mouth area Sm is much smaller (<0.25) than the tube area S, namely

dLm = 0.73*S/sqrt(Sm). For larger mouth areas you gradually reach the classical straight tube end

correction dL = 0.6*r = 0.34*sqrt(S).

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6. References

[1] 'Pipe cut-up and flue pipe dimensioning chart', in Mechanical Music Digest, Technical

gallery: http://www.mmdigest.com/Tech/isint.html

[2] 'Slit width vs pressure .xls', inYahoo steam-whistles group:

http://groups.yahoo.com/group/steam-whistles/files/ [3] 'Scale v frequency stability.jpg', inYahoo steam-whistles group:

http://groups.yahoo.com/group/steam-whistles/files/

[4] Ingersev, Frobenius (1947): Trans. of the Danish Academy of Technical Sciences, No. 1.

Johan Liljencrants 2006-02-24

http://www.mmdigest.com/Tech/isint.html






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