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Background

I'm currently writing a book on music theory and I'd like to include some background information on the physics of sound waves. My example of choice is to explain compression and rarefaction using a tuning fork example (seeing as this is a simple device that musicians will be familiar with). I understand the vibrational motion of the tines once the tuning fork has been struck however I'm confused with how it starts.

Question

When the first tine is struck against an object (table) it would move towards the second tine. My assumption is that this would increase the pressure in-between the tines and move the second tine outwards but this would mean the tines are moving in phase. So how do the tines end up moving together and apart (out of phase)?

Diagram

My second thought is that perhaps the vibrations are induced by the wave moving up and down the stem and not the air particles... however my ability in physics runs out after this thought!

I would really appreciate any diagrams to help my understanding too if possible

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    $\begingroup$ Answers below say, "it's complicated," But picture this: Imagine you are weightless in space, and floating in front of you, two equal masses connected to each other by a spring. Give the one on the left a bump toward the right and two things will happen. (1) The whole assembly will start moving toward the right. We'll ignore that motion and focus on (2), The system starts oscillating, with the two masses alternately compressing and extending the spring. If you can see why (2) must happen, then you'll be one step closer to understanding the more complicated vibration of the tuning fork. $\endgroup$ Apr 16, 2020 at 13:03
  • $\begingroup$ Essentially all the "communication" between the tines is via the metal not the air. $\endgroup$ Feb 6 at 17:35
  • $\begingroup$ ... and if anyone would like a quick proof, just consider the concept of impedance matching (only you have to know what that is of course) $\endgroup$ Feb 6 at 17:37

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The exact mechanics of how the tuning fork vibrates is complicated*-however once set vibrating , the equilibrium motion is easy to understand-the to and fro movement of tuning fork prongs.

Your emphasis is on explaining compression and rarefaction-for this how the tuning fork reaches its equilibrium vibrational motion isn't important. Infact you don't even need the second prong(buzz of fly wings for example). You are right in thinking that the second prong, the one not struck, is not set in motion by the interveining air. In fact the prong would vibrate in pretty much the same way in vacuum too.

Once the prong starts vibrating at a fixed frequency, it moves rapidly towards and away from its nearby air molecules. Air is a fluid and compressible. So the rapid movement compresses and "stretches" the nearby air volume. These generates local pressure variations which are what we call rarefactions and compressions. Its the pressure variation which travels away from the fork towards the listener.

Here are some generic images illustrate the point:

fig 1.Two vibration modes two modes of vibrations

fig 2. Compressions and rarefactions comps and rarefactions


*A fork is a elastic body. An absolutely rigid body comprises of constituent atoms/molecules that absolutely don't budge from their initial positions. A very nearly rigid body or elastic is the one which can be slightly deformed under applied stress(~force). This is characterized by the moduli of elasticity of these bodies. Things which are nearly but not perfectly rigid are capable of vibrating. When struck, the atoms/molecules are set abuzz gyrating slightly about their equilibrium positions. Macroscopically, we see this as propagation of sound through the material. In fact the whole body is set vibrating. The exact vibration state is determined by how much of each normal mode of the body the initial impulse "awakens". For each body, there exist some natural ways in which it can vibrate. These are called normal modes. They depend on object geometry. Each normal mode has a characteristic frequency and motion. When struck, objects vibrate in a superposition of these normal modes so the motion can be quite complicated and the sound produced multi-tonal.

Fortunately for high Q systems like tuning forks, when struck, the fork settles into one particular frequency-the normal mode its designed for.

Further Edits

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On how the second prong is set into motion after the first prong is struck

The short answer is that its connected to the first.

This question is in general about how motion spreads to other parts of a body when impulse is applied to only a particular part.

Elastic solids like a tuning fork, comprise of atoms (or molecules) in a certain structure. Most solid materials we encounter in daily life have some sort of equilibrium arrangement of atoms. This arrangement is responsible for stability of matter. For a metallic tuning fork, its a metal lattice.

The constituents are held in equilibrium positions by inter atomic forces. At low temps**, the atoms can be approximated as staying in place.

When an impulse is delivered to any point on such an object, the atoms at the place of impact get displaced from their equilibrium positions. The displaced atoms being linked to other atoms via inter atomic forces, push on them in turn. This sets of a chain reaction throughout anything in contact with the point of impact: the solid itself, the air around it too.So the vibrations from one tong of the struck tuning fork travel to the other part via its bulk and set it in motion.

This displacement wave which is set off in the material is the sound that travels through it. This sound reflects off of the object's boundary and traverses across it again and again and so on. In this transient phase, the exact motion is, at least in theory, just the evolution of certain superpositioned normal modes. By design, in case of a tuning fork, eventually the body is left vibrating in just one dominant mode while the energy in most other modes has decayed away faster.

This treatment of analysing transient motion in terms of normal modes(eigenmodes) of interfering sound waves has the benefit that one doesn't have to track down the individual motion of typically $10^{23}$ atoms to make predictions about sound frequency. Only geometry is needed.


**compared to melting point

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  • $\begingroup$ I appreciate the effort of your answer however as I mentioned I'm happy with my understanding of the physics of the tuning fork once in stable vibrations. My question is asking for an explanation of this: "The exact mechanics of how the tuning fork vibrates is complicated* ". In a nutshell, once struck, how do the vibrations transfer from one prong to the other? $\endgroup$
    – andowt
    Apr 16, 2020 at 12:16
  • $\begingroup$ Your expansion has perfectly answered my question, thanks! In essence vibrations from the struck tine spread to the 'fork joint' in turn vibrating both the stem and second tine. The vibrations will largely be a superposition of the normal modes and as the energy transferred decays into different forms i.e sound and heat, the higher energy eigenmodes decay leaving the fundamental vibration at the intended tuning frequency. $\endgroup$
    – andowt
    Apr 17, 2020 at 9:44
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The tuning fork has a number of eigenmodes, elementary ways it can stably oscillate. The lowest frequency and energy eigenmode (the fundamental mode) is the one where both tines are moving in opposite directions, with the biggest displacement at their ends and nearly no displacement at the forking point. When struck, the vibration will first be just across one tine. This is not itself an eigenmode and it will tend to transfer energy to other eigenmodes - in particular, the low frequency eigenmode with both tines moving. The transfer presumably happens because the vibrating tine will cause pushing and pulling on the forking point that stimulates the oscillation in the other tine.

See also the images at this blog post (dealing with the other odd issue of why the frequency doubles when you press down the fork on a hard surface).

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  • $\begingroup$ Those links are definitely a good read, thanks for that. Would you be able to expand on this part of your answer: "The transfer presumably happens because the vibrating tine will cause pushing and pulling on the forking point that stimulates the oscillation in the other tine." as I'm trying to understand how this transfer happens. $\endgroup$
    – andowt
    Apr 16, 2020 at 12:20
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I think you may envision this, in a simplified way, as a system of two masses $m$ (the tins) connected to a bigger mass $M$ (the fork handle) with springs. Imparting an initial force to either of the $m$ will utimately set the whole system in motion according to one of its eigenmodes. During the transient, movement is gradually transferred to the second mass $m$ via the movement of the bigger mass $M$ and the generated forces in the springs. Of course in the real case, mass are ditributed and springs are "flexural " springs, and what is transferred are bending moments.

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Coming in a bit late. Perhaps, for non-scientists, this less-precise perspective might be useful.

  1. Activation: When one tine/arm of a tuning fork is struck, or is itself struck against a heavy object, that arm is pushed toward the mid-line. (Since most tuning-fork arms are fairly stiff, displacement of the tine tip is perhaps a few tenths of a millimeter.) At the same time, inertia of the other tine effectively accelerates it (in the CoM frame) toward the mid-line by an amount determined by the distributed mass, sectional stiffness and length of the tine. Someone must have measured and published this, but I'm guessing that under usual condition, the "passive" tine will end the strike with between about 0.2x and 0.5x the displacement of the struck tine. So the tuning fork has a significant fraction of the struck displacement in each arm from initial contact.

  2. Quenching of non-symmetric movements: When a fork is struck on one side (per usual), the establishment of stable symmetric modes is preceded (and sometimes followed) by excitation and propagation of traveling modes. (In fact, a useful way to envision the establishment of stable resonances - hum, prime, tierce, quint, nominal, etc. - in a conical-symmetry bell is as paired waves traveling in opposite directions around the circumference, joining into even-ordered standing waves as they meet.) In free space OR if one end is firmly mounted (as in "chime fences" like this famous one in the USA: www.paulmatisse.com/the-musical-fence), these traveing waves may bounce back and forth along the instrument, making dynamics more complex. For a hand-held tuning fork, however, the base is in close contact with lossy materials and structures - fingers and hand - that damp non-symmetric modes and traveling modes in just a few cycles (Q ~ 1). So a "bright" struck tone almost instantly relaxes to the instrument's fundamental(s), with higher symmetric modes not as efficiently supported by the fork's shape.

  3. Role of nonlinearities: Very few real physical systems show precisely linear behaviour over a wide range of excitations. Nonlinearities allow coupling between even orthogonal modes of a system. (An analogy in optical physics is emission of phosphorescence, often via quantum-mechanically forbidden transitions, when local symmetry is disrupted by lattice defects or thermal excitation.) So in a structure as complicated as a tuning fork or bell, vibrational energy will move quickly between not only eigenstates but lossier and often dissonant "forced" modes and traveling waves: equipartition of energy, or ergodicity. So with asymmetric delta-function excitation, even a large bell, chime or tuning fork will reach dynamic equilibrium with the most energy settling into the most stable modes.

Hope that's useful to someone, or at least entertaining. Corrections welcome.

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