The physics regarding a siren's pitch

[magi]

Greetings siren enthusiasts!

I am currently working on an odd little project to create a siren with pitch I can change at will. I am already aware of the Yamaha music sirens, (viewtopic.php?t=26999) as they are in fact what sparked this project, but I have decided that I would like to go about this somewhat differently. From what I can tell, Yamaha simply used an octave's worth of sirens (give or take) that they would open/close with electronics. I would rather not take this approach, for various reasons, but rather create a siren whose pitch can be changed while at a constant speed. Currently, I have my ideas on how to accomplish this, but there are significant gaps in my knowledge of sirens and really sound in general, and so far my inquiries and research have yielded nothing. I believe that my problem is that I don't really know what I'm talking about. Hopefully, if I spout the nonsense I've had jumbling in my brain here, you kind folks would be able to help, or at least tell me if what I'm doing is actually possible.

I believe that the best way to accomplish this would be to have a mechanism capable of altering the shapes of the holes (through an aperture, sliding door, or other covering) to change the rate at which the air is vibrated by the rotor. In fact, I also believe that I have found a possible opening shape that doubles the wavelength from what the normal setup produces. I do not have physical evidence, but I have a lot of math. However, I do believe that I am missing an important factor, which I will point out once I reach that point of the description. I am still going to lay out my entire current solution, and I will be sure to clarify what is theory and what is fact.

fig. 1 & 2
1.1: A simplified depiction of a siren with four evenly spaced blades in the rotor, and four even openings in the stator.
1.2: A siren whose rotor is in a position where air cannot escape
1.3: A siren whose rotor is in a position where all air can escape
2.-: Each figure is the "cut and unfolded" form of its fig. 1 counterpart

me when im in a siren competition and my opponent is this guy.png (34.5 KiB) Viewed 620 times

fig. 3
The function representing the openings in the stator, with 1 unit being the width of the blades on the rotor.

RectangularOpenings.png (5.79 KiB) Viewed 620 times

In order to visualize this, I have devised a method of graphing the form of the holes in the stator. (Note that, at this point, I am not trying to create a siren design that can vary its pitch at will; I am only making a stator design that has a pitch different than that of the usual setup with an identical rotor.) If one were to take the circumference of a stator, cut it once and unfold it, they would get something not unlike what is depicted in fig. 2. Writing the x axis of a graph as this unfolded stator, and the y axis as the width of the openings, you get a representation of the stator as a function, which can be seen in fig. 3's representation of the siren in fig. 1. This function can be used to calculate the waveform of the sound the siren produces (in theory.)

fig. 4

Rectangles and also a wave.png (14.46 KiB) Viewed 620 times

Now is where my knowledge fails me. I'm not exactly sure how the rotor vibrates the air while spinning. If it is the buildup of pressure when the rotor is in a state similar to fig. 1.2, then I am limited in exactly what I manipulate; openings wider than 1 unit would lose that pressure, but also they might not. I have no clue what happens at this stage. Regardless, I will continue to explain my solution under the assumption that the sound wave the siren produces is identical to the function of air escaping the stator's openings over time, and the buildup of air pressure is negligible. A rather significant assumption, admittedly, but I do not know how I would have gone about this any other way, as I do not know the necessary equations. Fortunately, this assumption allows us to simplify the problem quite easily. The sound wave can be calculated from a function of the openings on the stator by integrating it from (x-1) to (x). From here, it is much easier to construct various sound waves from stator openings by writing down the ideal sound wave first, and reverse engineering it.

fig. 5
One may think of the "wave" function as how much air is able to escape from the space between the rotor blades through the stator openings, when the size of the blades on the rotor are 1 unit. In this diagram, the colored boxes can represent the rotor in different positions. For the red area, it contains 1 unit square, so the corresponding point on the graph is valued at 1. Yellow contains half a unit square, and green contains nothing, so their points are valued at 0.5 and 0 respectively. Of course, this visualization falls apart if you consider that air is in fact compressible, but problems cease to exist if you simply ignore them (citation needed.)

im trying my best ok.png (256 KiB) Viewed 620 times

This is how I created the configuration for a stator that creates a sound wave with twice the wavelength of the usual setup, as well as the "proof" for a configuration to produce a sinusoidal siren. I haven't done any other wavelengths, but I have reason to suspect that it is only possible to produce frequencies between these two extremes, so I may end up needing several sirens anyway, but this does mean I could replicate slides, like on a guitar. Of course, this is all speculative. I guess my plan by posting here is for someone smarter than I to come along and either say "yeah, I see no reason this shouldn't work," or "dude, I'm sorry, but that's not how this works at all."

fig. 6
Green: Staggered Rectangles, produces a triangle wave with a period of 4 units
Red: Circular Openings, (graphed as semi-circles stretched by a factor of 2,) produces a sin wave
Blue: Standard Rectangles, produces the normal triangle wave

i look insane dont i.png (41.58 KiB) Viewed 620 times

Well, that's everything I've discovered for this project. Thank you so much for reading all of my ramblings, and please do tell me what you think of this! I will be certain to accept any and all criticism, as I truly hope for this project to work out. I will be creating 3D printed models eventually, and I will be sure to share them with you all once (if) they are built.

[Eclipse DDS]

Wow, this is really interesting! I'm not clever enough to really know what sort of constructive criticism to offer, but all I can say is I really look forward to seeing how this turns out!
The only thing I would say is that the aperture mechanism would have to be designed to withstand the air pressure coming out of the chopper, you're probably already aware but some sirens have been likened to snow blowers due to the way they force out snow that has collected in their horns!

[magi]

Wow, this is really interesting! I'm not clever enough to really know what sort of constructive criticism to offer, but all I can say is I really look forward to seeing how this turns out!
The only thing I would say is that the aperture mechanism would have to be designed to withstand the air pressure coming out of the chopper, you're probably already aware but some sirens have been likened to snow blowers due to the way they force out snow that has collected in their horns!

Oh, that's neat! Don't sell yourself short; I hadn't heard the snow blower analogy before, and I totally see myself having made a weak, tiny plastic aperture that would have fractured and flew everywhere without knowing that
School is almost out, and Inventor has finished downloading, so I should be able to start producing the first prototype soon !!

[AcousticTheory]

It appears you are describing 'pitch' as a certain waveshape, but not as the frequency of the opening and closing of the ports over an interval of time. Frequency expressed in Hz is the number of times each second the ports cycle from open to closed again. Pitch is separate from "Timbre" which is the proportion of harmonic series content in the waveform (and its distribution among the various harmonics) relative to the fundamental tone, and that is what changes when you change the shape of the waveform.

Pitch is necessarily tied to the rotational speed of the siren's rotor relative to the siren's stator, which is normally stationary. It is a multiple of the rotational speed by the number of vanes, which normally corresponds 1:1 to the number of ports in the most efficient siren designs, which is most of them. If the ratio is not 1:1, then some ports will be opening while others are closing or are totally closed, and this will produce destructive interference between the output of adjacent ports, which integrate into an imperfect pressure sum at distances where the siren appears acoustically like a point source (that is, the flight time between adjacent sources is much less than 0.25 wavelength so that the phase shift between them is less than 90 degrees). As an aside, this means that alternative solutions that open some ports while closing other ports, such as the SoCal Edison JGP 120 or POW 120 siren, are not achieving maximum acoustic efficiency at significant distances from the source - which may be why such a large and powerful siren as the JGP 120 produced such a low SPL - the fundamental is largely cancelled between adjacent horns so that only the harmonics propagate.

Because pitch is necessarily tied to the rotational speed of the siren's rotor relative to the siren's stator, controlling the number of air bursts per second that produce sound at a certain frequency, the only way to vary the pitch while the rotor runs at a constant speed is to begin to rotate the stator in the same direction as the rotor but at a different speed. So what you have are two concentric rotors, instead of a rotor and a stator. By slowing the outer rotor from a speed equal to the rotor's speed down to zero while holding the inner rotor's speed constant, the siren pitch will increase, because the inner rotor and outer rotor ports go out of alignment (and thus close) at a greater rate as the outer rotor's speed is mismatched to the inner rotor's speed by a greater amount. There is also potentially an impact by having some ports moving toward the listener at a certain tangential speed while some ports are moving away, but I don't know if this would have a significant effect on SPL - if any, it would be detrimental because of ports experiencing greater delay while some ports experience lesser delay, putting some port outputs out of phase with other ports. This would need to be tested.

What I want to know most of all regarding the design challenge is, why? What do you hope to accomplish by holding the inner rotor's speed constant? It's much more practical to vary the rotor's speed relative to the stator than to create a separate spinning rotor outside the inner one so that the ports can open and close at varying frequency while the rotor spins at a constant rate. If you are concerned with the inner rotor producing a constant air pressure, consider other constant-pressure siren designs that use a separate blower such as the JGP 120, or the venerable Thunderbolt and Hurricane, or pneumatic sirens like the HLS from Germany. Using a separate blower or air supply enables these sirens to produce more consistent SPL as the pitch drops. They are mechanically more complex by necessity because of the need for a separate blower or compressor, however they would achieve what you want to achieve. A second motor or a variable speed drive system is going to be needed for the two coaxial rotor system I proposed above. Adjusting the open area of the stator ports, or the shape of the open area, is mainly going to impact SPL by impeding airflow when compared to conventional siren designs where the port open/closed time is 50% at the bladepass frequency of the rotor, but it could also impact timbre of the siren, although pitch will not be affected the way you think.

[Non-Zero]

The best way to achieve what you want is to have two very low slip motors running a set of siren choppers and two VFDs that are capable of alternating the frequencies quickly. It would be very challenging to alter pitch based on ports while the siren is in motion.

[magi]

It appears you are describing 'pitch' as a certain waveshape, but not as the frequency of the opening and closing of the ports over an interval of time. Frequency expressed in Hz is the number of times each second the ports cycle from open to closed again. Pitch is separate from "Timbre" which is the proportion of harmonic series content in the waveform (and its distribution among the various harmonics) relative to the fundamental tone, and that is what changes when you change the shape of the waveform.

Pitch is necessarily tied to the rotational speed of the siren's rotor relative to the siren's stator, which is normally stationary. It is a multiple of the rotational speed by the number of vanes, which normally corresponds 1:1 to the number of ports in the most efficient siren designs, which is most of them. If the ratio is not 1:1, then some ports will be opening while others are closing or are totally closed, and this will produce destructive interference between the output of adjacent ports, which integrate into an imperfect pressure sum at distances where the siren appears acoustically like a point source (that is, the flight time between adjacent sources is much less than 0.25 wavelength so that the phase shift between them is less than 90 degrees). As an aside, this means that alternative solutions that open some ports while closing other ports, such as the SoCal Edison JGP 120 or POW 120 siren, are not achieving maximum acoustic efficiency at significant distances from the source - which may be why such a large and powerful siren as the JGP 120 produced such a low SPL - the fundamental is largely cancelled between adjacent horns so that only the harmonics propagate.

Because pitch is necessarily tied to the rotational speed of the siren's rotor relative to the siren's stator, controlling the number of air bursts per second that produce sound at a certain frequency, the only way to vary the pitch while the rotor runs at a constant speed is to begin to rotate the stator in the same direction as the rotor but at a different speed. So what you have are two concentric rotors, instead of a rotor and a stator. By slowing the outer rotor from a speed equal to the rotor's speed down to zero while holding the inner rotor's speed constant, the siren pitch will increase, because the inner rotor and outer rotor ports go out of alignment (and thus close) at a greater rate as the outer rotor's speed is mismatched to the inner rotor's speed by a greater amount. There is also potentially an impact by having some ports moving toward the listener at a certain tangential speed while some ports are moving away, but I don't know if this would have a significant effect on SPL - if any, it would be detrimental because of ports experiencing greater delay while some ports experience lesser delay, putting some port outputs out of phase with other ports. This would need to be tested.

What I want to know most of all regarding the design challenge is, why? What do you hope to accomplish by holding the inner rotor's speed constant? It's much more practical to vary the rotor's speed relative to the stator than to create a separate spinning rotor outside the inner one so that the ports can open and close at varying frequency while the rotor spins at a constant rate. If you are concerned with the inner rotor producing a constant air pressure, consider other constant-pressure siren designs that use a separate blower such as the JGP 120, or the venerable Thunderbolt and Hurricane, or pneumatic sirens like the HLS from Germany. Using a separate blower or air supply enables these sirens to produce more consistent SPL as the pitch drops. They are mechanically more complex by necessity because of the need for a separate blower or compressor, however they would achieve what you want to achieve. A second motor or a variable speed drive system is going to be needed for the two coaxial rotor system I proposed above. Adjusting the open area of the stator ports, or the shape of the open area, is mainly going to impact SPL by impeding airflow when compared to conventional siren designs where the port open/closed time is 50% at the bladepass frequency of the rotor, but it could also impact timbre of the siren, although pitch will not be affected the way you think.

I am incredibly grateful for the long response, and I apologize for my late one; in my defense, you've given me a lot to think about.

As to why I have such an odd constraint, this project is to create something more along the lines of an instrument, rather than a warning siren. I have always loved the unique sounds created by sirens, but I've never been fond of the poor digital reconstructions or sampling, so the next logical step would be to make one myself, of course . (It is also worth noting that the openings/closing of ports stemmed from the hope to create something that "interrupts" notes when changing pitch, rather than smoothly transitioning from one to another.) This idea to make a small model with a completely different purpose than your average siren, thus removing the acoustic efficiency requirement mentioned in your second paragraph. Of course, it is still of note, but a design that compromises efficiency is not as detrimental here as I would imagine one would be in the warning siren industry.

With that in mind, I do think that your main criticism of my design (that being my faulty understanding of timbre and pitch) is certainly accurate. I am certain that achieving a pitch higher than that created with equal ports on both stator and rotor is impossible. Personally, I still have hope in a lower pitch, but I fear that any sound produced would either not change the pitch in the slightest, or create a sound too soft to hear properly. I will still be creating a physical model for many of my ideas, as I am still quite curious what they could sound like. I am in the process of creating a model with an easily interchangeable stator for testing, so if you have any port shapes you would like to see tested, please do tell me! Even if changing pitch this way proves impossible, I believe that some opportunities for unique sounds could arise with the manipulation of timbre. I am especially curious about the possible creation of a siren with a sinusoidal waveform.

Now, while the complexity would be too great for any normal siren, your proposition in your third paragraph of having two concentric rotors rather than the traditional stator-rotor design is fascinating, at least to someone like myself who has not delved quite as much into sirens as most others on this site have. While I would not be able to change notes as quick as I had intended, I do believe this concept may at least give a more fine control. As you said yourself, it requires further research.

After all this, it still seems that the best way to tackle this would be to take the route that Yamaha did with their music sirens, but I still think that a variable pitch from a single siren would be pretty damn cool . Oh yeah, and once I have some physical models, I'll post them, though if it will be on this topic or a new one I'm not sure.