The old basics on acoustics restored

The works of Boëthius, Roman politicus, philosopher and writer of the V-VI centuries aC, were well known in Europe until the XVIIIth century. His book "De Institutione Musica", translated as The Fundamentals of Music, was until then "The Book" for all matters concerning music. Boëthius compares there sound propagation to the propagation of water waves. Physics indeed englobe both phenomena under mechanical wave propagation. 

Sound is mechanical waves

What is sound? Sound is the translation that our brain does of the information perceived by our eardrums, the sensitive membrane of the aural system. What our eardrums perceive is mechanical waves. 
Normally we perceive these waves when they do propagate through the air. But these waves can sometimes be perceived while they propagate through solids, for instance when you listen to the tone of a tuning fork, and as it vibrates you stick it to your ear; or when a doctor auscultate you with a stethoscope; or through liquids, as whales do in the ocean... But to make it now easier I will call sound mechanical waves propagating through the air, which is the case if you want to listen to music. 


Mechanical waves

Mechanical waves are the product of a perturbation of a medium caused very often by a vibration. What is characteristic of the different types of mechanical waves is that their propagation does not involve a real deplacement of the particles of the medium. The particles just oscillate a bit around their initial position, returning to the starting point as the energy goes away. That means that mechanical waves' propagation involves a deplacement of energy and not a deplacement of the substance of the medium where they propagate. From the point where the initial vibration occours mechanical waves will propagate in all directions. That means that as long as they don't reach any surface, mechanical waves will propagate in concentric spheres, loosing intensity with the distance. Whatever the density of the medium will be, mechanical waves will propagate always in the same way, changing only in the speed of their propagation: at 20ºC, sound will propagate in the air at 343 m/s, in water at 1484 m/s, and in brass at 4700 m/s.


What happens when mechanical waves find surfaces in their propagation? A surface is a change of the density of the medium where propagation occours: from a solid material into gas, or from gas into a solid material, or from gas into liquid, or from a more dense liquid into a less dense liquid, or from one solid into another solid, etc. When mechanical waves find a change of density in their propagation, a part of the waves will rebound on it –the “echo” effect– and another part will get refracted into it, crossing the surface and entering the new medium. The closer the two different densities are the bigger is the part that gets refracted in the surface and smaller is the part that gets reflected, and the opposite, the more distant the two densities are the bigger is the part that gets reflected on the surface and the smaller is the part that crosses it.

The angle of incidence of the wave on the surface is also important: with the same densities, the more perpendicular to the surface the trajectory of the waves is, the more waves will cross the surface, and the more oblique the trajectory is, the more waves will be reflected on it.

Depending on how a surface is, waves get reflected in different ways:
           straight surface                        convex surface                   concave surface

On a straight surface the angle of reflection is the same than the angle of incidence. A convex surface opens the angle of reflection, and a concave one closes it. Rough surfaces reflect waves in many directions.
We can see how mechanical waves propagate by throwing an object into calm water. From the point where the object touches the water will appear those characteristic concentric rings. These rings get wider as they spread away from the starting point, disappearing at the end as all the energy goes away.


The new basics on acoustics
Musical instruments produce sounds, and how does that happen? That happens by the refraction into the air of the mechanical waves that propagate within them, waves originated in the substance of the instrument by a vibrating corpus. This vibrating corpus can be strings, reeds, membranes, the lips of brass players and labiums, straight in organs and recorders, round in traversos.

In every instrument the sound production process is exactly the same, depending the timbre of each instrument on the type, shape and material of both vibrating corpus and the whole instrument: the vibrating corpus produces mechanical waves in the substance of the instrument; these waves will propagate through it, and in their propagation they will find lots of times the surfaces of the instrument, said a grosso modo the end of the instrument and the beginning of the air around and inside it, which is a change of density from a solid material into gas. One part of those waves will be reflected on those surfaces, remaining in the substance of the instrument, and another part will get refracted through those surfaces into the air, becoming sound.

Thanks to their cavities (ressonance boxes, and the inside of the pipes) musical instruments amplify the sound they produce, making it louder. This amplification occours as follows: the air in the cavities gets from every refracting point the same waves in the same frequency; so many refractions in the same and small volume of air result in the superposition of all those waves, getting amplified, which means that keeping their frequency, their peaks become higher and their troughs become deeper, being this translated into a louder sound. And these amplified waves go out of the cavity through the holes of the ressonance box of stringed instruments, and through the bottom and lateral holes of wind instruments.


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