I was pondering about the following 'experiment', and was curious about the formulae behind it.
Imagine having a glass filled with a liquid. On one side there is a sensitive receiver that records all the vibrations and is attached to the outside. If I would then tap on the opposite side I would in send an impulse to the glass side, which would transmit the waves to the liquid, which would travel to the opposite side and will be transmitted to the glass and finally the receiver. Of course the glass is higher density so some waves will reach the receiver earlier.
I have some questions here:
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I am used to dynamical mechanical systems involving eigenmotions and the likes. This is no different I presume? I am sending an impulse and recording the output, so in fact I have a transfer function. What I am wondering is is what the eigenfrequency of a solid/fluid is when being excited. Will this be the speed of sound through the solid/fluid?
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What happens at the interface between the glass and the fluid? I can imagine the wave moving with $V_{glass}$ through the glass, however how is this transmitted to the fluid? What will $V_{fluid}$ be? Will it be equal to $V_{glass}$? I can imagine that the fluid will attenuate the higher velocity movement, because high frequency sound is attenuated by a fluid, but that is sound… and is this sound?
Can someone point me into the right direction?
Best Answer
I think the question is too complex to give a satisfactory answer without deep study of the problem. The biggest problem is that there are several ways how vibrations (sound waves) travel from the source to the receiver. The path of the excitation: the source - glass - fluid - glass - the receiver you've described is only one one possible path. Some of the excitation shall travell through glass only, e.g. bottom or side of the glass.
So if you want to have exact solution you should find standing sound wave (eigenfunction) solution of the whole system glass + fluid, which is far from being trivial. Of course, in this case you must also consider dissipation, because you have constant excitation and without the dissipation the vibrations would continue to grow to infinity.
The second possible, more engineering way to get the answer is to disregard sidepaths of excitations and calculate in terms of traveling sound wave. At one end you have the source, excitations first travel through glass, then difract on the border between glass and liquid, travel throught liquid, again difract on the border between glass and liquid and finally travel through the glass to the receiver, where all of the wave energy is absorbed.
In both cases you need speed of sound in medium, which is not hard to obtain:
$$v^2 = \frac{E}{\rho} = \frac{K}{\rho},$$
where $\rho$ is density of the medium, while $E$ is Young's modulus (solid) and $K$ is bulk modulus (liquid). In the second case you also need transmitivity, which is of course dependent on the incident angle and you can find the right equations by peaking at Fresnel equations.
I hope I understood you right and that my answer helps.