Understanding Interfaces and Waves: A Physical Explanation

[Tsunami]

Right, so this is odd:

In my course there are derivations of transmission for 2 cases, normal incidence of an (acoustic) wave on an unbounded interface, and oblique incidence of an (acoustic) wave on a flexible unbounded interface.

So my idea was, the second will be an expanding case of the first.

However, they seem to contradict. (Skip below for the mathematical derivation. Stay here if you feel more capable of giving a purely physical, intuitive explanation.)

Whereas the first case seems to find the mass factor of the interface to be more important for higher frequencies, and the stiffness of the medium (=the interface suspension) for lower frequencies, it's the other way around for the second case!
This seems to be so because in the second case, instead of a constant stiffness s, we have a bending stiffness which is linked to 'free flexural waves' (I don't know what this is?). This bending stiffness seems to be frequency depending somehow, and this is probably what causes the different effect.
But how can this be? In my eyes, nothing seems to have changed from case 1 to 2, except that the direction of incidence has changed. Surely, this cannot give a wholly different behaviour?

At the beginning of case 2, my course says:

Having established the principle of applying the blocked surface pressure as the forcing field on a fluid-loaded structure, we may now apply it to the case of an unbounded, thin, uniform, elastic plate upon which acoustic plane waves of frequency omega are incident at an arbitrary angle phi.

This seems to suggest it behaves by the same principle.






In the first case, the derivation is done by modelling the interface as a mass m, linked to a suspension medium, modeled as a damper with factor r and a spring with factor s.

This way, I get a transmission of the form :

tau =   1/[     [(omega*m-s/omega)/(2 rho c)]² + [1+r/(2rhoc)]²  ]

So, when my frequency omega is smaller than the resonance frequency of the medium, then the stiffness term with s is the most powerful.
For higher frequencies, the mass term becomes more important.

In the second case, something similar is done, only here the stiffness is being replaced by something called the bending stiffness D, and there's seems to be an equivalence in the formula's:

D kz^4 ~ s , where kz is the normal factor of the wave number.

However, since kz is frequency dependent, this gives us a totally different picture:

When omega is smaller than the 'critical frequency' (=the equivalent of the resonance frequency), the mass term is the most important. When omega gets higher, the stiffness term takes over.

 

[LightHero]

This is exactly the opposite of what I got in the first case, and it doesn't make sense to me. Why should a different angle of incidence change the behaviour so much?

 

[piercas]

I would first try to understand the physical principles behind the derivations and equations presented in this content. I would also look for any underlying assumptions or simplifications that may have been made in the derivations.

One possible explanation for the contradictory behavior observed in the two cases could be the difference in the boundary conditions of the interfaces. In the first case, the interface is assumed to be unbounded, meaning that the acoustic wave can travel through it without any restrictions. In this case, the stiffness of the medium would have a greater impact on the transmission of the wave.

However, in the second case, the interface is assumed to be a flexible, thin plate. This introduces a boundary condition where the plate can vibrate and transmit its own waves. This is where the bending stiffness comes into play. As the frequency increases, the plate becomes more rigid and can transmit the acoustic wave more efficiently, leading to a higher transmission coefficient.

Another factor to consider is the direction of incidence of the wave. In the first case, the wave is incident normal to the interface, meaning that it is only affected by the properties of the medium it is traveling through. But in the second case, the wave is incident at an angle, and this introduces a component of the wave's energy being transferred into the bending vibrations of the plate. This could explain why the bending stiffness becomes more important at higher frequencies.

In summary, while the two cases may seem similar at first glance, the differences in boundary conditions and direction of incidence can have a significant impact on the behavior of the interfaces and the transmission of waves. It is important to consider all these factors when analyzing and understanding physical phenomena.