The Dyson equation is exact. ( Though we will need approximations for M )

Transcription

1 Lecture 25 Monday Nov 30 A few final words on Multiple scattering Our Dyson equation is or, written out, < G(r r ') >= G o (r r ') + dx dy G o (r x)m (x y) < G(y r ') > The Dyson equation is exact. ( Though we will need approximations for M ) It is important to recognize that M and Go and <G> are all translationally invariant, functions only of x-y, not x and y separately. G itself did not have this property, because the actual medium before averaging was not homogeneous. Because of this translational invariance, the Dyson Equation involves only simple spatial convolutions; it therefore succumbs to a FT in space: < G(q) >= G o (q) + G o (q)m (q) < G(q) > ( because the FT of a convolution is the product of FTs) ie. < G(q) >= [G o (q) 1 M (q)] 1 = [µq 2 ρω 2 M (q)] 1 Except for the M, this looks just like the familiar form for G in the q,ω domain. It is clear that M somehow modifies the dispersion relation; now we have ρω 2 = µq 2 M (q) ; a wave propagation with a modified speed and attenuation. The result is sensible in a way that doing Born approximations was not. This also implies a modified modal density related to Im < G (0) > The remaining issue is the evaluation of M. Our infinite series for M can be truncated; it is clearly a series in powers of ε. The simplest approximation is to write M(x-y) = = E(x-y)G 0 (x-y) = ω 4 <δρ(x)δρ(y)>g 0 (x-y), probably pretty good if ε is small. Some call this the "first order smoothing approximation". Karal and Keller ~ 1980(?) By way of specific illustration, let us take the density fluctuations to be delta- correlated <δρ(x)δρ(y)>= R δ(x-y) (i.e large but very short range ) Then M is R δ(x-y) G o (0) and M's spatial FT is M(q) = Rω 4 G o (0). The dispersion relation is then 209

2 ρω 2 = µq 2 irω 4 ImG o (0) Rω 4 ReG o (0) It is unfortunate that ( except in 1-d) that ReG o (0) is divergent and our idealization of delta-correlation breaks down. One concludes that the random medium has a very different wave speed in this limit, and has acquired some attenuation. If we want to know the real part, we'd best relax the delta-correlation and give the fluctuations some finite correlation range. If we only wish to estimate attenuation however, this analysis perhaps suffices. What does this attenuation represent? The medium has nothing that absorbs energy, so there cannot really be attenuation. Recall that we have calculated the ensemble average G. Each realization of the medium has its density fluctuations in slightly different places, so each realization of the response to a fixed source will have different phase. When we average across the ensemble, these different phases lead to cancellations and therefore some apparent attenuation. Is this relevant to any experiment? If an experimenter does indeed do ensemble averaging ( as it might if the scatterers move around as perhaps they would in a fluid) and the experiment is repeated over and over, then this theory applies ( depending on the approximations and their legitimacy of course ) In solids, however, there is no averaging of that sort, and few experimenters will repeat their experiments with an ensemble of samples. We are forced to hope that the transducers average across their face(s), or averages as we move a transducer to different places, is somehow equivalent. It is a much harder problem to calculate mean square G, i.e energy, in a multiply scattering medium. Diffuse Wave Theory. On time scales long after a source has acted, the resulting wave field if not dissipatedmay have reflected many times off the boundaries, or scattered many times off internal heterogeneities. If wavelengths are short compared to the size of the sample, then it is virtually impossible to make detailed predictions about responses, or about the signal received at a detector. Nevertheless there is lots we can say about these things statistically. We can make estimates of the mean square signal (related to energy density) and its fluctuations ( related to how well we can infer energy density from a sample signal to within some error bars) and how that mean square signal may vary slowly in time and/or space and/or frequency. The key to it all is the principle of equipartition, that we may take all normal modes to have equal mean square excitation ( manifestly absurd and untrue though it be; caveats apply! ) in order to make predictions for the 210

3 mean square signal at a detector or the work done by a source. The topic has not yet been put on a firm theoretical basis, though there are a few theorems. It is related to Statistical Mechanics (Thermal Physics) in which each mode gets an energy proportional to temperature k B T, and to "Statistical Energy Analysis" used to describe the random vibrations of large structures. Please do not confuse the word "diffuse" with a diffusion equation. The energy density of a diffuse ultrasonic field may, or may not, be well described by a diffusion equation t ~ D 2. See below. Here we mean by the adjective 'diffuse' that the waves are randomly going in all directions with no obvious phase correlations. A good example from optics is how a frosted glass makes the rays go in random directions. The frosted glass is a 'diffuser.' By way of motivation, consider the pictured ultrasonic experiment in which a broad band pulse is inserted at one point on the surface and the resulting wave is detected at another. The detected signal might look like this: LL LLLL P R LS? time (µsec) Conventional ultrasonics would seek to interpret the detected waveform. It would identify a surface skimming P arrival, a Rayleigh arrival ( identified in part with help from knowing the wavespeeds ) then the feature labeled LL is identified as a bottom reflected P wave. The feature labeled LS is a mode converted wave reflected off the bottom. There is another feature labeled LLLL that is a twice reflected P wave off the bottom. After that, the signal gets too hard to interpret; multiple ray arrivals (some of which may be due to reflections off the lateral walls) overlap too much to be distinguished. If we study the signal at times beyond this 100 µ sec, we see the following: 211

4 The 50 or so interpretable µsec are followed by over µ seconds of what looks like slowly decaying noise. (It is so long lived because this is done in aluminum in air for which losses are weak.) It is, however, not noise in any conventional sense; if you repeat the measurement, the wiggles at late time are reproduced exactly (if the system has not changed, e.g by temperature fluctuations that change wavespeeds.) The signal level at negative times ( equal to zero to the precision of the above plot) gives us an idea of the level of the true noise. Clearly the signal will remain above true noise for at least another couple of 100,000 µsec. What might measurements of such "noise" be good for? Can the level of that noise be predicted by theory? How about its fluctuations? Its time-dependence/decay? How it might it be different at different receiver locations? Or if the source were moved? How robust are the details of such signals against changes in the solid body? Before beginning that theory, let us recap the domain of expected applicability: -> when rays are hard to use because of many reflections/scatterings -> if absorption time is long ( much longer than transit times L/c ) -> when size L >> λ Room acoustics is a ( rather simple compared to elastic waves ) example of diffuse field acoustics. In a "reverberant" room ( reverberant is defined as the case in which the decay time is long compared to transit time, so that every ray reflects many times off the walls, or scatterers, before being dissipated ) we expect the energy density E to be uniform across the volume and for energy to be headed in all directions uniformly. A standard calculation then seeks to know how fast energy is delivered to the walls ( where it may be absorbed or reflected ) 212

5 E = energy density = total E / Volume Consider a small volume element a distance r from the wall on which there is a patch da. We ask how fast acoustic energy from the interior of the room is delivered to that patch. The volume element is (in 3-d) dv = r 2 sinθ dθdφ dr, with dr = c dt. The energy in that dv is the energy density E times dv. A fraction da cos θ /r 2 of that energy is aimed towards the patch. Put this all together and we find that the direction θ,φ delivers energy to the patch at a rate (power per area da per angle dθdφ ) equal to sinθ cosθ dπ = E c da dθdφ The integral over directions ( φ from 0 to 2π, θ from 0 to π/2) gives dπ = E c da / 4 A patch of size da on the wall receives a power equal to the energy density in the interior times c/4 times da. In room acoustics one then posits that this patch absorbs a fraction α of the energy it receives, and reflects the rest. The absorptivity α depends on the material there; painted concrete absorbs very little, plush carpeting absorbs lots, an open window absorbs most. Typically in room acoustics one assumes α does not depend on angle θ, but in practice it does. The energy E = ΕV in the room then diminishes at a rate de / dt = E c αda 4V (the integral is over the area of the walls) that scales inversely with the volume and scales proportionally to the surface area of the room and the absorption strength α. The energy in the room diminishes exponentially. E = E o exp( t / τ ) with 1 / τ = c αda / 4V In practice -except for the very largest theaters - losses are dominated by losses at the walls, 213

6 not in the interior, so intrinsic losses are usually neglected. If they were to be included one would write E = E o exp( t / τ ) with 1 / τ = c αda / 4V + 1 / τ intrinsic If we apply this kind of reasoning to a solid, there are differences: 1) depending on the solid, losses may be more important intrinsically in the interior than at the surfaces. (especially in concrete or plastics ) 2) losses at surfaces tend to be much weaker than they are in air-borne acoustics. ( It will be an exercise to calculate the rate at which a diffuse field in a solid loses energy to the air. It is in general very weak, because the impedance mismatch is so high. ) and 3) the wall reflections will come with modeconversions so we have to keep track of energy in both P and S waves ( and maybe Rayleigh waves too ). It is not apriori obvious how a diffuse field will partition its energy amongst various wave types. One of the first interesting results in Diffuse field ultrasonics is that this partition is universal. Regardless of how the source distributes energy initially, after a few reflections a diffuse field will have a set partitioning, independent of the source or boundary conditions. One way to study this is to posit a volume with diffuse energy densities E P and E SV. (and no significant differences from place to place or in S-wave polarization or in direction of propagation) Each of these delivers energy to a wall patch da from direction θ,φ at a rate (using v for the group velocity that may differ from the c ) dπ P or SV = E P or SV v P or S da sinθ PorSV cosθ PorSV dθ PorSV dφ θ P and θ S are related by Snell's law and the ratio of the phase velocities: sin θ P = (c P /c S ) sin θ S. The energy mode conversion coefficients α P->S ) from P to SV and α SV->P ) from SV to P could be calculated from the methods of page 100. In terms of them one can state that the time-rate of change of the energy E P =E P V in P waves is given by 214

7 de P dt = E P V v da sinθ cosθ P P P α P >SV ) dθ P dφ + E SV V v da sinθ cosθ S S S α SV >P ) dθ S dφ and the time rate of change of energy in SV waves due to this patch da of the surface is the opposite of that ( because energy is conserved) de SV dt = E P V v da sinθ cosθ P P P α P >SV ) dθ P dφ E SV V v da sinθ cosθ S S S α SV >P ) dθ S dφ Those waves that are SH at the patch of course do not mode convert. If we integrate these over all φ, we get de P dt de SV dt = E P 2V v P dasinθ P cosθ Pα P >SV ) dθ P = E P 2V v P dasinθ P cosθ Pα P >SV ) dθ P + E SV 2V v S dasinθ S cosθ Sα SV >P ) dθ S E SV 2V v S dasinθ S cosθ Sα SV >P ) dθ S Given an original deposition of energy by some source, we see that the energy density, and its partition between P and S may evolve under the influence of mode conversions. A dynamic steady state may be achieved in which the net rates of conversion de/dt in the two directions (from S to P and P to S) are the same implies some steady state ratio of the energies E. The above suggests that this occurs when the energies E are in inverse ratio to their coefficients above. Now the trick: rather than calculating the mode conversion energy coefficients α using the methods of p100ff, notice that the reflection matrix between the amplitudes of incoming P and S plane waves, and outgoing P and S plane waves is an S-matrix (recall the discussion of S matrices in scattering a couple of weeks ago) If the plane waves are normalized to unit incoming power ( not intensity ) then, as argued there, S is unitary SS *T = I. We also argued there, using time-reversal invariance, that S must obey S S* = I. Therefore S = S T. I conclude that, if the plane waves are normalized to unit power, then S PS =S SP. The power conversion coefficients S PS 2 = S SP 2 are therefore equal. The two factors α in the above expression are equal α P >SV ) = α SV >P ) A steady state is achieved when each expression is zero, i.e when (dropping the common factor of α) E P 2V v P sinθ P cosθ P dθ P = E SV 2V v S sinθ S cosθ S dθ S But sinθ cosθ dθ = d (sinθ) 2 and the sin θ are related by Snells's law. So steady state implies E P 2V v P c 2 P = E SV 2V v c 2 S S 215

8 If we also assert E SV = E SH =E S /2, we then conclude that the steady state demands E S 2 = 2v c S S 2 E P v P c P regardless of the boundary conditions ( as long as the α were not zero at all angles ) Presumably one can make a similar argument if the mode conversions occur, not at the walls, but at scatterers. I have not yet worked that out. The first paper on the topic (Davis Egle 1982) presented a numerical evaluation of these integrals and obtained a numerical value of the steady state ratio of energies in accord with this more general treatment. Rather strikingly perhaps, this ratio is large. For a Poisson ratio of ( as in aluminum ) it is 16. Thus most of the energy in a diffuse field is in the form of shear waves. The result can also be obtained more abstractly by arguing as one does in thermal physics and statistical energy analysis that all normal modes get equal energy. (This is the principle of equipartition, of which more below) We may also notice that this ratio is precisely the ratio of energy depositions for a point source in an unbounded medium. ( see page 90 ) 216