Some questions regarding Acoustic wave reflection

[itamar123]

TL;DR Summary

What are the general governing equations for Wave Reflection at boundary in solid-solid case?

I'm new to this subject so bear with me and I have a salad in my head. I have some fundamental question regarding wave reflection/transmission. Lets say I have a flexible steel cable connected to an end support (pillar) and waves are induced in that cable. My goal is to get the maximum amount of reflection from this setup. Picture for this case:
https://www.imagebam.com/view/MEWPXEX

<a href="https://www.imagebam.com/view/MEWPXEX" target="_blank"><img src="https://thumbs4.imagebam.com/f0/5b/80/MEWPXEX_t.png" alt="guitar.png"/></a>

I tried reading bits and pieces on this case and this is my current understanding:

The cable's waves are transverse, and so they induce shear waves (and not longitudinal) waves in the Pillar (end support).
the reflected wave amplitude is given by the reflection coefficient, and the higher the acoustic impedance of the pillar (Z2=sqrt(Rho2*G2)) compared to the cable (Z1=Rho1*C1), the higher the coefficient of reflection.

I have a couple of questions,
How does the geometry of the end support play a role? From what I read, it is recommended that the length of the pillar be at least a quarter of the biggest reflected wavelength, so assuming amplitude of 8mm of the cable, the height if the pillar h should be 2mm up and down from the connection point.

Also, How does external constraints on the pillar affect the reflection? meaning, lets say I induce some force on the sides or top of the pillar to constraint its vibration, intution tells me it stiffens it and so increases the reflection, how can I calculate this?

One last question, how does the normal modes of the pillar affect this interaction? Intution tells me that the pillar acts like a HPF or LPF, and the frequencies of the cable that match the normal modes of the pillar are not reflected back as much as the ones that dont.
In that case, will a different, maybe weird geomery reflect waves better?
Whats the connection between the normal modes of the pillar and the coefficient of reflection?

Dont refrain from Math and equations,

If someone can point me to a book/article/paper that covers this topic thoroughly, ideally, a way to make an FEA analysis on this (Using Solidworks Simulation, preferably, ansys is good as well) I'd very much appreciate it.

 

[Baluncore]

My goal is to get the maximum amount of reflection from this setup.

You need a rigid and heavy pillar, that is set in a solid foundation. Consider tungsten carbide as a pillar material.

The pillar should be shaped like a pyramid or a cone. The pillar shape may be optimised for some modes of vibration.

 

[itamar123]

You need a rigid and heavy pillar, that is set in a solid foundation. Consider tungsten carbide as a pillar material.

The pillar should be shaped like a pyramid or a cone. The pillar shape may be optimised for some modes of vibration.

Thanks for the reply,
Im looking for a more in depth answer in case one of the parameter changes.

 

[Baluncore]

Lets say I have a flexible steel cable connected to an end support (pillar) and waves are induced in that cable. My goal is to get the maximum amount of reflection from this setup.

What type of instrument are you contemplating?

How do you intend to attach the cable to the pillar, so transverse waves are reflected cleanly, without fatiguing and breaking the wire?

Maximum energy reflection suggests you would also benefit from minimum loss in the cable. So why do you propose using a cable when you could use a solid wire with less internal friction.

 

[itamar123]

Actually I'm contemplating regular Electric guitars, there's a lot of misunderstading surronding that topic and lots of myths so I wanted to investiage.
Your suggestion of tungsten/tungsten carbide seems to be accurate, I just want deeper understanding.

 

[tech99]

The string carries a transverse wave which involves an interchange of energy between displacement and KE. Notice however that tension increases at every peak of the displacement wave, so has a component at twice the frequency, so the pillar is excited at twice the frequency of the generator. In principle this produces a transverse wave on the pillar, although we would usually make the pillar short and fat to make this negligible. It is possible that the pillar would have a resonance at this double frequency, but I think we would usually expect the pillar to simply absorb some energy. Unlike an electrical wave, I don't think we can just assume the approximation that the wave on the string turns through 90 degrees and travels down the pillar.

 

[Baluncore]

The long neck of an electric guitar generates more flexibility than the pillars. The neck is designed to counter and balance the string tension. There is little advantage in making the pillars more rigid, without first fitting a deeper and more rigid neck.

 

[itamar123]

I know, but now I just want to understand the saddles part of it.

 

[itamar123]

The string carries a transverse wave which involves an interchange of energy between displacement and KE. Notice however that tension increases at every peak of the displacement wave, so has a component at twice the frequency, so the pillar is excited at twice the frequency of the generator. In principle this produces a transverse wave on the pillar, although we would usually make the pillar short and fat to make this negligible. It is possible that the pillar would have a resonance at this double frequency, but I think we would usually expect the pillar to simply absorb some energy. Unlike an electrical wave, I don't think we can just assume the approximation that the wave on the string turns through 90 degrees and travels down the pillar.

Awesome insights! didnt think of it. how short and fat is enough though? finding the resonance frequencies of the pillar is pretty easy and I understand the physics of it, but not the other parameters.

Lets say theortically the first fundamental of the pillar is far higher than the 20th fundamental of the cable and im neglecting higher frequencies, what happens to the energy those fundamentals carry? they dont all fully reflected back, dont they?
How does the geometry (and therefore the fundamental frequencies) of the pillar play a role?
Would a Tungsten pillar that has some matching frequencies be better than a stainless steel or cermaic pillar that has much less matching frequencies?

 

[tech99]

If the pillar has a self resonance far above the twice frequency of the string, then each cycle it will remove a small amount energy from the string and store it in its elasticity. We can say the pillar will add shunt elasticity and friction to the string, lowering its frequency and reducing its efficiency as an oscillator. Of course, the pillar can also transmit energy to the mounting surface, which might have its own resonances and possess radiation resistance, so the loudness will increase, and the quality of the sound will alter.

 

[Steve4Physics]

Hi @itamar123. Each end of the vibrating part of a guitar string rests on a support. That means the vibrating part of the string is not attached to the side of a pillar.

An analysis of a string attached to the side of a pillar doesn't seem relevant. Or maybe I've missed something?

 

[itamar123]

Hey, I didnt want to get into the specifics of it as I wanted to keep it simple to understand the basic physics first, so I described an idealized system.
If you want to get more specific, it depends on the type of saddle or nut the guitar (or any other stringed instrument for that matter).
A locking nut is more similar to the problem I described.