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Classical Physics
Effect of impedance changes less than a wavelength
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[QUOTE="jasonRF, post: 6001788, member: 192203"] 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 [I]much[/I] 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: [URL]http://eceweb1.rutgers.edu/~orfanidi/ewa/[/URL] 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 [/QUOTE]
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