Power Law Absorption Research Articles

Numerical simulation of ultrasound-thermotherapy combining nonlinear wave propagation with broadband soft-tissue absorption

Ultrasound (US) thermotherapy is used to treat tumours, located deep in human tissue, by heat. It features by the application of high intensity focused ultrasound (HIFU), high local temperatures of about 90 ∘C and short treating time of a few seconds. Dosage of the therapy remains a problem. To get it under control, one has to know the heat source, i.e. the amount of absorbed US power, which shows nonlinear influences. Therefore, accurate simulations are essential. In this paper, an improved simulation model is introduced which enables accurate investigations of US thermotherapy. It combines nonlinear US propagation effects, which lead to generation of higher harmonics, with a broadband frequency–power law absorption typical for soft tissue. Only the combination of both provides a reliable calculation of the generated heat. Simulations show the influence of nonlinearities and broadband damping for different source signals on the absorbed US power density distribution.

Power law absorption in polymers and other systems

The theoretical treatment of acoustic absorption which goes as a power of the frequency is quite subtle. In this paper, it is shown that some of the difficulties can be resolved by using a wide distribution of relaxation times which is cut off at a short time limit τ1>0, and a finite long time limit τ2<∞, without altering the general form of the acoustic response. This formalism is used to show that quadratic, linear, and fractional losses can be consistent with the Kramers–Kronig (KK) causal relations. The linear loss in polymers is associated with a weak velocity dispersion, so the group velocity and the phase velocity are quite close. The implications for space charge measurements in insulating polymers are discussed. Using the Helmholtz–Kirchhoff tube absorption as an example, it is also shown that the local approximation for the KK relations is unsatisfactory.

Effects of frequency-dependent absorption on the propagation and attenuation of high-intensity acoustic waves containing shocks

Shock waves used in extracorporeal lithotripsy and ultrasonic surgery are greatly influenced by thermoviscous losses in the tissue. The influence of the frequency dependence of the absorption coefficient on nonlinear wave distortion and energy attenuation has not been investigated in detail up to now. For absorption frequency dependence other than quadratic, the corresponding nonlinear evolution equation is often complicated and thus frequency domain methods are more effective. The presence of the shock, however, makes this analysis time consuming due to the large number of spectral components needed in the numerical calculation. To avoid this complexity, a semianalytical method that enables calculations with relatively few harmonic components is used. Propagation in media with various power law absorption is considered. It is shown that energy attenuation of an initially sinusoidal wave in a medium with linear power law absorption is quite similar with that in a medium with quadratic power law absorption, even though the waveforms are quite different. The effect of selective absorption of a certain harmonic component is also investigated. It is shown that second harmonic selective absorption does not prevent shock formation, but results in much less energy attenuation over large distances. [Work supported by NIH, FIRCA, and RFFI.]

Acoustic pulse effects for media obeying a frequency power law absorption

The effects of phase velocity dispersion accompanying attenuation can alter both the shape and time delay of finite length acoustic pulses. For attenuation described by a frequency power law with an exponent y, a time causal theory [T. L. Szabo, J. Acoust. Soc. Am. 93, 2277 (A) (1993)] has shown that dispersion is greatest [in the range (0≤y≤2) when y=1 and is zero for y=0 and y=2. These results are in contrast to the approximate nearly local Kramers–Kronig theory [M. O’Donnell etal., J. Acoust. Soc. Am. 69, 696–701 (1981)] which predicts increasing dispersion as y increases from 0 to 2. The consequences of dispersion from the two theories are compared for simulated finite-length pulses for different values of y. For values of y=1, where the theories agree, the effects of bandwidth on pulse shape and delay are illustrated. An approximate general scaling law for pulses propagating in lossy media is presented.

Extreme ultraviolet spectroscopy of PKS 2155-304.

We present the extreme ultraviolet (75-110 angstroms) spectrum of the BL Lac object PKS 2155-304, the first spectrum of an extragalactic source obtained with the Extreme Ultraviolet Explorer. The spectrum shows a generally smooth continuum, which can be modeled by a single power law plus interstellar absorption, and possibly an absorption feature at approximately 80 angstroms. The best fit to the data suggests that the EUV spectrum can be interpreted as a simple extrapolation of the X-ray continuum, with an energy index alpha approximately 1.6; however, shallower or steeper power laws with indices between -0.4 and 2.7 cannot be ruled out by the existing EUV data alone. The data provide strong constraints on the interstellar neutral H and He along the line of sight. Using a column density of 1.36 x 10(20) cm-2 for the Galactic neutral hydrogen along the PKS 2155-304 line of sight, the neutral helium column density is constrained to be 9%-10% of the hydrogen amount.

Causal time domain parabolic wave equations for power law absorption

The classic parabolic time domain wave equation describes acoustic propagation in a medium in which absorption is a quadratic function of frequency. For the general case of power law absorption, α(ω)=α0‖ω‖y, where α0 and y are arbitrary real positive constants and ω is angular frequency, generalized time domain parabolic wave equations are presented. The differential loss operator in the original classic parabolic equation is replaced by a single propagation convolution operator that accounts for both absorption and dispersion. These operators, based on new time causal relations, have different forms for y as an even or odd integer or noninteger. The new equations are compared to those in the literature corresponding to the cases y=0.5 (acoustic duct), y=1.0 (medical and underwater applications), and y=0 or 2 (classic forms).

Burgers’ equation generalized for absorption obeying a power law

Burgers’ equation applies to finite-amplitude waves propagating in a medium with absorption that has a quadratic frequency dependence. Numerical solutions of a modified Burgers’ equation have been obtained in the frequency domain for other types of losses; however, a complete set of time domain nonlinear equations corresponding to power law attenuation has not been available. Power law attenuation is defined by the equation, α(ω)=α0‖ω‖y, where α0 and y are arbitrary real constants, and ω is angular frequency. Blackstock [J. Acoust. Soc. Am. 77, 2050–2053 (1985)] has suggested that Burgers’ equation could be generalized if an appropriate operator L could be found such that the equation could become pz−L*p=Bppτ, where p is pressure, B is a constant, τ is delayed time, and the subscripts denote derivative operations. The L operators have been derived based on a new causality principle and parabolic wave equations for power law loss. The resulting time domain equations extend Burgers’ approach to finite amplitude propagation in media with arbitrary power law absorption.

The X-ray spectrum of the superluminal quasar 1928 + 73

EXOSAT observations (0.1–10 keV) of the radio-loud quasar 1928 + 73, which displays superluminal motion, were made at four epochs between 1983/283 and 1984/009, and spectra of this source are obtained from the EXOSAT database. The X-ray spectrum (0.1–10 keV) can be explained by a power law and uniform absorption model. No significant low-energy absorption or soft excess were detected. Both soft (0.1–2 keV) and hard (2–10 keV) X-ray-flux variations of this quasar were found on a time-scale of one month. Spectral index values obtained from the power-law model suggest that they exceed the ‘canonical’ value. A thermal bremsstrahlung model also fits well. Uncorrelated variations of the soft and hard fluxes are found in this source which indicate that they have different origins. A soft excess is apparent when the spectra are fitted with a canonical photon index ~ 1.7. The spectra presented here are the first in the X-ray region for 1928 + 73.

X-ray spectrum (0.1-10 keV) of the broad line radio galaxy - 3C 390.3

Broad line radio galaxy, 3C 390.3, was observed with EXOSAT on several occassions between 1984 and 1986 using the low (LE) and medium (ME) energy detectors. Detailed results of the spectral analysis of the LE (0.1-2 keV) and ME (2-10 keV) spectra which were obtained from the EXOSAT archives, of this source are presented. Simple power law and uniform absorption model provided good fit to the combined LE and ME spectra but the hydrogen column density (N H ) values are much smaller than the galactic N H value. However, the above model with the fixed N H value also provides good fit to the datasets.

Einstein spectra of quasars

The results of the initial stage of the CfA survey of Quasar Energy Distributions (QEDs) are reviewed. Einstein Imaging Proportional Counter (IPC) spectra of 33 quasars have been studied by fitting a single power law slope and absorption by an equivalent column density of neutral hydrodgen, N H . A wide range of energy index, α E , is seen and α E increases as the radio loudness of a quasar decreases. Comparison with the higher energy HEAO-A2 data leads us to a two-component model for th x-ray spectrum. The x-ray column density is systematically lower than the 21 cm measured Galactic column density along the same line of sight. This discrepancy is not present in a comparison sample of Bl Lac objects and so the cause must be intrinsic to the quasar. A third component, an ultra-soft excess (<0.28keV) above a single power law in the quasar spectra, provides a plausible explanation. This excess could be the high energy tail of the blue bump in the optical and ultra-violet (UV).

Coulomb and nuclear inelastic excitations and the optical potential for heavy ions

We present a simple, exact numerical method of obtaining the complex, $l$-dependent potentials $\ensuremath{\Delta}{U}_{l}(r)$ induced in the elastic channel by coupling to inelastic channels. For $^{16}\mathrm{O}$ + $^{152}\mathrm{Sm}$ at 72 MeV, we consider pure Coulomb coupling and find that a simple power law absorption or the imaginary $l$-dependent polarization potentials of Baltz et al. gives good agreement with coupled channel cross sections out to 100\ifmmode^\circ\else\textdegree\fi{}. For larger angles a more accurate representation of the polarization potentials for the low partial waves is required, both real and imaginary parts being significant. We also examine the $^{13}\mathrm{C}$ + $^{40}\mathrm{Ca}$ system at 68 MeV including a number of excited levels in both projectile and target. Here, nuclear coupling is dominant. The strength of the imaginary part of $\ensuremath{\Delta}{U}_{l}$ is found to vary quadratically with deformation length and inversely with excitation energy for excitations above about 5 MeV. The states above 8 MeV lead to a real potential of strength comparable to the imaginary in the surface region. The effect on the cross sections is discussed.NUCLEAR REACTIONS $^{152}\mathrm{Sm}$ ($^{16}\mathrm{O}$, $^{16}\mathrm{O}\ensuremath{'}$), $E=72$ MeV; $^{40}\mathrm{Ca}$($^{13}\mathrm{C}$, $^{13}\mathrm{C}\ensuremath{'}$), $E=68$ MeV; coupled channel analysis; deduced effective elastic optical potential.

Some optical properties of amorphous and crystalline antimony trisulphide thin films

The change in optical properties accompanying the amorphous crystalline transition has been studied for antimony trisulphide thin films. The real and imaginary parts of the dielectric constant are found to be much lower for amorphous films at lower photon energies, possibly because of a large number of defect states which would mostly disturb the top of the valence band and the conduction band comprised, respectively, of chalcogen lone pair and antibonding states. The crystalline material shows some structure in the imaginary part of the dielectric constant that corresponds to interband transitions, and apparently the direct band edge is at 1.88 eV. In the edge region the power law absorption has been observed in the amorphous material from which the extrapolated optical gap has been found to be 1.7 eV.