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Even though this is filed under the "reality-check" tag, I'm looking more for semi-plausible explanations as this is almost assuredly impossible using real world physics.

In my world people use special non-electronic instruments (as this is set in a pre-modern time period) that emit sounds that get progressively louder the farther away you get from the source, within a certain radius.

In practice this means that up close and personal with the musician the tunes will sound of normal amplitude but as you get away from the musician you shift and enter a middle distance where the sound progressively gets louder the farther away you go.

That is only until you reach the maximum distance the sound waves emitted by the instrument can travel while remaining somewhat audible. Where the sound gets progressively quieter the farther away you go until it becomes completely inaudible.

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    $\begingroup$ Making this sound plausible for an instrument alone is going to be really hard. Doing it for an instrument played inside a specially constructed building is more feasible; will that do? $\endgroup$ – John Dallman Nov 20 at 13:46
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    $\begingroup$ Sort of possible, with constructive/destructive interference. But will not give soft/close vs loud/far pattern, instead a radial checkerboard of loud and soft zones. $\endgroup$ – user79911 Nov 20 at 14:44
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    $\begingroup$ Does this effect have to be omnidirectional? $\endgroup$ – Alexander Nov 20 at 17:32
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    $\begingroup$ @Theexurbmm10a There are already answers suggesting that the sound may somehow be amplified at a greater distance - in some areas. But if you want the sound to be amplified in all directions, that's may be unrealistic. $\endgroup$ – Alexander Nov 20 at 19:25
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    $\begingroup$ This is already true for must church pipe organs, because sound-producing pipes are high in the air and there is almost no direct sound to people who are "close" and almost underneath the instrument. In fact the organist can't hear what the music sounds like in the body of the building, because he/she is almost sitting "inside" the base of the instrument. $\endgroup$ – alephzero Nov 21 at 5:50
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It's probably at least theoretically possible to some extent, but you need a rather large instrument. Or rather, a rather large reflector.

When sound disperses freely in 3 dimensional space it decays roughly quadratically with distance. The easiest way to see why is probably by imagining a large sphere around the sound source, and realising the sound should sound equally loud around all the sphere. As the area of the sphere increases like the square of the radius r, the loudness must thus decay like 1/r^2. This is free space though, and any kind of reflecting materials could complicate this.

Probably, the simplest example is to build a large wall; if we imagine an ideal wall (as I will henceforth do for all surfaces) it's reflection will double the loudness of the sound in the non wall direction, but it would still decay with distance.

If we are inside a very wide building, with a floor and a low ceiling, we could use the same argument as above to see that the perceived loudness should be equal along a circle, and thus decays like 1/r. Though still decaying, this is a huge improvement if the distance is large.

Now, to get our instrument to sound louder when we move away, we need to do better. Enter the Ellipse!

An ellipse, in the mathematical sense, is a precisely defined closed curve with many interesting properties. The one of interest to us is that it has two so-called focal points: Focal points of an ellipse When something, like light or sound, is emitted at one of the focal points it will bounce off the walls and come together to focus at the other. This is true no matter the length of the ellipse.

This gives us a First answer to your question: In an elliptical room with the instrument at one focal point, a listener could move away and first hear the loudness decay, only to increase again when she arrives at the other focal point.

Now, I presume you want this to work outside. The key observation is that you could take away some segment of the ellipse and still have the sound bouncing off what's left be focused on the other focal point! For listener midway between the points, most of the sound energy will, so to speak, go around her.

If you only care about the loudness in one direction, this is fairly straight forward.

Second answer: Make a large (10m?) bowl-like reflector and place your sound source at the closest focal point.

For large distances, this will be very similar to a Parabolic reflector, but with the important difference that the reflected sound would not be "parallel" but focused. For a very close listener the sound is loud. Moving away it first gets quieter but then increases as the focal point is approached.

Now, I assume you would prefer this to work in any direction. I'm not sure how feasible this would actually be, but here goes:

EDIT: I've updated the third answer, and I'm much more confident it could work now. The old "mushroom" from my previous answer is replaced by another shape, perhaps more resembling the underside of a "flower".

Third answer: Build a large round ceiling, shaped like this: enter image description here The cross-section is shaped like segments of two intersecting ellipses with a shared focal point at the same height the instrument would sound (preferably near ground level to get as much gain from reflections from below as possible) at the centre. The other focal point would form a circle around the player. Like I've drawn it, the "focal ring" would be at the ground, but you'd probably want to tilt the ellipses to get the ring at ear level.

You MIGHT even get away with using a reflector small enough to be almost portable! The more of the elliptical arc you cover the more amplification near the ring. Most of the reflection happens near the centre, so once you reach a certain size you don't gain enormously from making it just a bit bigger (until the edge gets close to the other focal point) The proportions shown in my image should give a noticeable effect, as long as it's large enough that much of the sound goes above the heads of near by spectators.


If you only care about certain directions, and especially if you want a portable solution, my second answer is probably the most relevant. It could be adapted further depending on precisely what context you have in mind.

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    $\begingroup$ Good answer. Another term for an elliptical space with these unusual acoustic properties is a whispering gallery, as it allows even whispers to be heard over large distances in an otherwise noisy space. $\endgroup$ – Nuclear Hoagie Nov 20 at 18:24
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    $\begingroup$ What if you had a straight wall of speakers, playing slightly out of sync? You could align them so that they constructively interfere at an arbitrary point in front of them, which... vaguely matches the goal? $\endgroup$ – Mooing Duck Nov 20 at 23:39
  • $\begingroup$ I have been inside one such reflector built for a mathematical exhibit. It works exactly like you described, and there are rooms built like this. Moreover, there are setups where the room is built as two or more intersecting ellipses, so that the sound is spread over two or three foci. php.math.unifi.it/archimede/archimede/curve/visita/… $\endgroup$ – LSerni Nov 21 at 23:25
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    $\begingroup$ @MooingDuck You could do it with speakers, but not probably not with instruments. The problem is that not only the location and frequency, but also the phase of the sound matters. You would have to control the vibrations with millisecond precision, which is possible electronically but not with a trumpet. With "real" instruments, you'd likely get the sound amplified in SOME direction, but you don't know which, and it changes all the time. Ash's answer shows a clever way to to sort-of this in a low-tech setting. $\endgroup$ – EdvinW Nov 30 at 20:44
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Yes - but only if the stars align acoustically.

So this is very concocted setup, but hopefully it tells the idea of what's happening.

enter image description here

An instrument played at the yellow point, observed by a listener at the green point, will be heard to come from the two red points that are the openings in the wall.

Because the geometry isn't perfect - the sound travels a slightly shorter distance to one side by a precise amount, when the waveforms re-join at the green point, they're out of phase, and cancel each other other out. This results in a significant reduction of volume of a particular frequency, which due to luck in the measurements is the precise note that is emphasised most in the peice the instrument is currently playing.

Eg for middle C, the distance from yellow to both red points must differ by about 66cm.

As the green observer walks away down the line, the phase difference becomes less significant, and the apparent volume becomes louder and louder, towards the end of the line, it will become softer again unfortunately. But for a bit of his journey, it will increase.

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    $\begingroup$ There's a subtle mistake in my diagram. This is good enough - but unfortunately I slightly tilted the camera when I rendered this diagram, and mistakenly closed and didn't save so the path doesn't quite appear to intersect the peaks of the sound waves. Rotate the path about 2 degrees anti-clockwise (so all 3 lines intersect at once) for a perfect diagram. $\endgroup$ – Ash Nov 20 at 16:55
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    $\begingroup$ This was my first thought, too. There's a recently invented sound system that can beam audio to a particular point in space, allowing one person to hear the audio without headphones, while no one else around them can. This only occurs at one particular point in space and requires proper head tracking and aiming of the audio beam, but the principle of a sound being louder further away from the source is totally plausible. consequenceofsound.net/2020/11/beam-music-speaker-headphones $\endgroup$ – Nuclear Hoagie Nov 20 at 18:00
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    $\begingroup$ Out of curiosity, what did you use to render this? $\endgroup$ – Daniel B Nov 21 at 0:56
  • $\begingroup$ @DanielB openscad $\endgroup$ – Ash Nov 21 at 5:13
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What you describe looks like the acoustic equivalent of an optical amplifier.

How does an optical amplifier work?

You need a medium with inverted population, meaning that all of its molecules are in an excited state, and you need a beam of light travelling through the medium. The photons of the light beam will induce stimulated emission in the active medium and therefore more photons will add along the way. Since you have no cavity to select the frequency, you will be basically amplifying all the wavelengths in the beam: the more the beam has travelled in the inverted medium, the more photons will have been added to it, and the growth of the intensity will be exponential.

I am not aware of any general mean to produce inverted population in the acoustic domain, therefore strictly speaking what you describe cannot happen unless you use a combination of microphones and speakers.

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You need an elliptical dish that has a second focal point that is far away from the noise source, and a block that sits between the source and observer.

If the observer is standing right on the other side of the block they will hear nothing if the block and dish are perfect. If only a partial block then they will hear some sound as they sit close. As they move away they will exit the dead zone created by the block and begin to hear more sound. As they move towards the focal point the sound will get louder and louder. Once they pass the focal point it will again begin to get quieter.

So in you don't need a special instrument, but rather a special setting. Whatever instrument they hold, the musician must sit in front of a curved surface of some kind that acts to focus sound at the audience. This surface could be a portable set of wood panels that assemble into a reflector of say 10 foot diameter or so, or it could be a permanent feature of a stage.

enter image description here

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Using conventional physics, no. Simply because once the sound wave is produced by the string in your example, it dissipates into the surrounding medium.

However...

If you're willing to go with an instrument that maintains loudness the farther you get from it, then kindly allow me to introduce to you to two concepts: the stroh fiddle and the speaking tube.

The stroh fiddle is an instrument that amplifies and projects the sound of a stringed instrument, channeling said sound through a tube & horn. Vibrations from the strings activate a grammophone diaphragm which sound is then amplified by the horn.

enter image description here

The speaking tube is an instrument that transports sound over a distance, channeling it through a tube.

enter image description here

Your instrument will simply connect the two pieces together: replace the flare of the horn with a long tube, and then, add a grammophone horn at the end of the long tube. Sound waves, once emitted from the diaphragm, will propagate through the length of the tube without dissipating and without much loss of energy.

The sound will appear to become louder with distance simply because it will be louder than the unaided sound travelling through the surrounding air.

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An individual wave is always going to be loudest at the source, but there are options:

Route the sound above the audience: An instrument that does this when played in a grand hall, or from a specialized gazebo could be designed to bounce the sound off a high ceiling while muting direct transmission. The sound at ground level is loudest a specific radius from the instrument and gets quieter as you approach until you are right on top of it.

Cheat and use ultrasound: While the design itself would be anachronistic a non-electronic directional speaker should be possible. Something like an adjustable dog whistle could be constructed long before its time. That gives you a means to produce ultrasound at a controlled frequency. A mechanical attachment converts the source note into two inaudible ultrasound waves which are directed at a target. When the waves collide with the target a audible wave forms at the differential frequency. Adjust the wave frequencies and aim the waves at the right target and sound should appear to emit from that location.

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  • $\begingroup$ Ultrasonics were my thought as well. $\endgroup$ – supercat Nov 21 at 21:14
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Your instrument is the room

We know, that a greek amphitheater is a marvel of acoustics: With the right setup, there are spots on the stage where a whisper can be heard as if the speaker stood next to the listener yet he is some 50 meters away! The straight back wall and the dish-like shape of the ranks result in constructive resonance from the speaker's position to the auditorium.

Now, you also can design a room in such a way that soundwaves that would have escaped or have been directed to the front ranks before now get channeled to the back ranks more. With the right setup, first, the back ranks get an equal sound as the front ones, and with strategically placed reflectors upwards, the back ranks can get more volume than the front rows.

enter image description here

The front half of this reflector setup's ower half reflects the sound up into the dish, which then focusses the sound into the indicated upper area on the right - the very first sound wave gets reflected 3 times inside the dish before ending on the higher edge of the benefiting sector, meeting with those waves that come from the middle of the lower reflector. If way the sound has to travel is a multiple of the wavelength of the sound, then we get positive wave interference and as a result, the amplitude (which we feel as loudness) increases. If the sound waves however come in such a way that it is an odd multiple of half the wavelength, then we get a spot where these sound waves cancel out - and that spot is suddenly silent.

Sound acoustics are really hard to make right, yet the ancient Greeks managed to build their amphitheaters without a computer and knowledge of the underlying physics! It took us more than 1000 years to recreate this feat of sound engineering.

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Create a density gradient in the atmosphere above you using carefully controlled temperature and humidity that refracts and bends sound waves downwards. The beam your sound up into the air. It's quiet near the instrument (at ground level) and gets louder further away. You could do the same underground and refract seismic waves upwards.

Or build a circular instrument completely surrounding the audience. The sounds become more concentrated towards the centre, further from the instrument.

Or build the instrument on a very tall vertical tower, phased such that the waves cancel out near the foot of the tower but reinforce further out.

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It is possible for particular sinusoidal frequency to use standing wave principle. The sound will be periodically louder in arrow points and silent in nodes of standing wave. The distance between reflecting surfaces, walls in building, need to be adjusted for frequencies, and location of listener need to be focused to arrow points. This will be effective for low frequencies and long waves. High frequencies have short waves that better for which ray acoustic is fare more better when solving such problem, as @EdvinW shows.

Another principle which is really possible is density gradient, rise earlier by @Nullius in Verba" The source amplitude need to be large (as in open air festival). If specific environment conditions, namely cold ground and warm upper air exist, a sound will travel slower near to ground and faster in air which gives bending of sound waves to ground which focus in some distance from source. This can be observed at late summer dusk in distance about 2-10km from concert place, and its mostly audiable in low frequency beats.

In both solution the sound will be loudest at the source, if no electrical amplifying device is used.

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