Effect of impedance changes less than a wavelength

[BOYLANATOR]

I am interested to know what is the impact of various length scales of impedance changes on wave propagation.

From undergraduate physics (a few years ago for me) I roughly remember how to derive reflection and transmission coefficients for a wave traveling from one medium to another with a different impedance; e.g. a wave traveling along a thin string connected to a thicker string.

I can also vaguely remember solving the wave equation in 2D (maybe 3D) for waves in a box.

In both the examples above the boundary is clearly defined and the media on either side of the boundary can be thought of as extending away to infinity. I am interested in cases where there are perturbations of the impedance over different length scales. Let's say the thick and the thin string interchange every λ/8. How does the wave "see" the changes"?

Practically, I am thinking about seismic exploration for hydrocarbons where the dominant frequency of the sound pulse may be around 20 Hz, a typical rock velocity may be 2000 m/s therefore the "wavelet" is about 100 m long. However the rocks can be layered on a scale much less than 100 m (e.g. look at Grand Canyon pictures) and therefore the impedance medium varies on all length scales, both shorter and longer than the wavelet length. In the hydrocarbon example a hole may be drilled and the rock properties can be observed on a vertical scale of less than a metre. However typically we apply a moving average of the rock properties over a scale roughly equal to the wavelet length/duration, and roughly guess an attenuation factor before modelling this as an "effective medium". Is this theoretically sound?

Can anyone provide a discussion of what the key concepts to read up on are, or point me in the direction of a paper, online description etc?

Thanks,
BOYLANATOR

 

[BOYLANATOR]

An update... this looks to have the answer but it's not as simple as I had hoped: https://www.crewes.org/ForOurSponsors/ResearchReports/2011/CRR201159.pdf

The answer seems to be the Born Approximation. Using this approach the classical equation for the boundary between two half-spaces,
T = 1 + R,
can be derived in just 7 pages of maths...great

The linked paper also has an example for a short perturbation between two infinite half spaces which is similar to my conceptual examples.

In summary, it seems the answer isn't simply described by basic undergrad physics.

Feel free to disagree...

 

[jasonRF]

Your question has two parts.

First, you want to model your media as being a sequence of layers, each of which is a homogeneos medium with an effective impedance/velocity/density. Second, you want to calculate the propagation of waves through this layered medium.

For the modeling problem, as long as the inhomogeneities within each layer are much much smaller than a wavelength and consistent throughout the layer then you are probably okay with your approach. Note that inhomogeneities that are even 1/4 of a wavelength within the layer are much too large to model this way.

Assuming that your layered model makes physical sense, then you need to predict the propagation through it. I learned about this in the context of electromagnetic waves, where layered media are used to build microwave filters, anti-reflective coatings, etc. There are two standard ways to do this. Here I am assuming that the layers are parallel; if they are not then this is much more complicated.

The first approach sets up and solves the entire problem directly. Let ther be N boundaries at $z=z_1$, $z=z_2$, ..., $z=z_N$, with $z_1<z_2<\cdots<z_N$. So there are N+1 media, with medium 0 in $z<z_1$, and in general medium $\ell$ in $z_\ell < z < z_{\ell+1}$, and the top medium N in $z>z_N$. We assume that the incident wave is from medium 0, and let all quantities vary as $e^{i\omega t}$. I will do the case of linear acoustic waves because the wave quantity can be expressed as a scalar, but if you have some kind of elastic wave your wave may be vector valued. We will assume that the wavevectors in each layer are in the x-z plane: $\mathbf{k}_\ell = \mathbf{\hat{x}}k_{\ell,x} + \mathbf{\hat{z}}k_{\ell,z}$, and write the velocity potential in layer $\ell$ as

$$\psi_\ell(x,z) = A_\ell e^{-ik_{x} x - i k_{\ell,z} z} + B_\ell e^{-ik_{x} x + i k_{\ell,z} z} .$$

In writing this I have already used part of what you learned in your undergrad physics, in that the phase matching condition at the boundaries forces $k_x$, the component of the wave-vector parallel to the interfaces, to be the same in all layers. This means the wave-vector component perpendicular to the inverfaces is $k_{\ell,z} = \sqrt{k_\ell^2 - k_x^2}$, where $k^2_\ell = k^2_{\ell,x}+k^2_{\ell,z}= \omega^2 c^2_\ell$ and $c_\ell$ is the phase velocity in layer $\ell$. If you have normal incidence then $k_x=0$. Anyway, now apply the two boundary conditions at each interface. Continuity of the z-component of velocity yields:
$$ \partial_z \psi_\ell (x,z_{\ell+1})= \partial_z \psi_{\ell+1} (x,z_{\ell+1}), $$
or
$$ -i k_{\ell,z} A_\ell e^{-ik_{\ell,z} z_{\ell+1}} + i k_{\ell,z} B_\ell e^{-ik_{\ell,z} z_{\ell+1}} = -i k_{\ell+1,z} A_{\ell+1} e^{-ik_{\ell+1,z} z_{\ell+1}} + i k_{\ell+1,z} B_{\ell+1} e^{-ik_{\ell+1,z} z_{\ell+1}}, $$

and continuity of pressure gives,
$$\rho_\ell \omega \psi_\ell (x,z_{\ell+1})= \rho_{\ell+1} \omega \psi_{\ell+1} (x,z_{\ell+1}) $$
or
$$ \rho_\ell A_\ell e^{-ik_{\ell,z} z_{\ell+1}} + \rho_\ell B_\ell e^{-ik_{\ell,z} z_{\ell+1}} = \rho_{\ell+1} A_{\ell+1} e^{-ik_{\ell+1,z} z_{\ell+1}} \rho_{\ell+1} B_{\ell+1} e^{-ik_{\ell+1,z} z_{\ell+1}} , $$

where $\rho_\ell$ is the density of the material in layer $\ell$. Now we have a set of 2N linear equations for the $A_\ell$ and $B_\ell$, $\ell=0 \ldots N$. We can set $A_0=1$ and know that $B_N=0$, so we have the right number of equations for the number of unknowns. Now just set up the system of equations (use matrix-vector formulation) and solve (numerically, in general!).

The second standard way to calculate the propagation through a layered medium uses a sequence of impdedance transformations, working top to bottom. That is the way I first learned to do these problems in undergrad engineering electromagnetics class. Many engineering electromagnetics books should have this, for example "fields and waves in communication electronics" by Ramo, Whinnery and Van Duzer.

Chapters 5-8 in the following notes discuss reflection at boundaries and propagation in layered media in gory detail for the electromagnetic wave case:

http://eceweb1.rutgers.edu/~orfanidi/ewa/

If you are actually working with elastic (seismic) waves that are vector valued so have a polarization, then those notes will provide a template of how to work with that since electromagnetic fields also have polarization (although perhaps different boundary conditions).

jason

 

[jasonRF]

By the way, 1/8 wavelength layers can indeed matter. 1/4 wavelength layers with the right properties can be used for very simple anti-reflective coatings (or in the transmission line case, can be used to impedance match over a narrow bandwidth).

 

[BOYLANATOR]

Hi Jason,

Thanks for your response.

I have a few conceptual questions.

For the modeling problem, as long as the inhomogeneities within each layer are much much smaller than a wavelength and consistent throughout the layer then you are probably okay with your approach. Note that inhomogeneities that are even 1/4 of a wavelength within the layer are much too large to model this way.

Reference https://www.physicsforums.com/threads/effect-of-impedance-changes-less-than-a-wavelength.947909/

This assumption isn't strictly true in the case of modelling sound through rocks but it is generally made. See the "blocky model" used to separate out different impedance layers with small scale perturbation within each layer based on this measurement from drilling into the rock: https://www.researchgate.net/figure/1D-seismic-elastic-model-based-on-sonic-and-density-logs-of-the-Ktzi202-observation-well_fig7_277343916
You can see that the separation of a large scale layering from short scale perturbation is not a clear one. What do you think of assuming a layered model here?

A side question: what happens if a wave is propagated along a string with constant impedance with a very short perturbation in impedance of ,say, λ/50 before moving back to the constant impedance. Do we observe a reflected wave or a difference in the transmitted wave?

UPDATE: The answer to my last question must be that, yes, reflections are generated at the interfaces as per usual. Bu the reflections off each interface will be reverse polarity and similar amplitudes and separated by only λ/50 so will virtually cancel out. But we should still see a slight loss in amplitude in the transmitted wave?