A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

/pdf/modeling-nonlinear-ultrasound-propagation-in-heterogeneous-4mrm4bt7mx.pdf

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

Mathematical Methods of Physics.

A new method is presented for the analysis of complex dynamic systems which is based on partitioning the system degrees of freedom into a “global” set and a “local” set. The global equations of motion are formulated and solved in a standard deterministic manner, although due account is taken of the presence of the local degrees of freedom via an approach which is analogous to fuzzy structure theory. The local equations of motion are formulated and solved by using statistical energy analysis (SEA) with due account being taken of power input from the global degrees of freedom. The method can be considered to encompass a number of existing analysis techniques and to form a flexible framework within which a number of detailed modeling strategies can be devised. A simple rod system provides an initial application, and it is shown that the method yields very good results from low to high frequency excitation.

A hybrid method for the vibration analysis of complex structural-acoustic systems

Acoustically sensitive emulsion droplets composed of a liquid perfluorocarbon have the potential to be a highly efficient system for local drug delivery, embolotherapy, or for tumor imaging. The physical mechanisms underlying the acoustic activation of these phase-change emulsions into a bubbly dispersion, termed acoustic droplet vaporization, have not been well understood. The droplets have a very high activation threshold; its frequency dependence does not comply with homogeneous nucleation theory and localized nucleation spots have been observed. Here we show that acoustic droplet vaporization is initiated by a combination of two phenomena: highly nonlinear distortion of the acoustic wave before it hits the droplet and focusing of the distorted wave by the droplet itself. At high excitation pressures, nonlinear distortion causes significant superharmonics with wavelengths of the order of the droplet size. These superharmonics strongly contribute to the focusing effect; therefore, the proposed mechanism also explains the observed pressure thresholding effect. Our interpretation is validated with experimental data captured with an ultrahigh-speed camera on the positions of the nucleation spots, where we find excellent agreement with the theoretical prediction. Moreover, the presented mechanism explains the hitherto counterintuitive dependence of the nucleation threshold on the ultrasound frequency. The physical insight allows for the optimization of acoustic droplet vaporization for therapeutic applications, in particular with respect to the acoustic pressures required for activation, thereby minimizing the negative bioeffects associated with the use of high-intensity ultrasound.

/pdf/acoustic-droplet-vaporization-is-initiated-by-superharmonic-4tgfrsgu6c.pdf

Acoustic droplet vaporization is initiated by superharmonic focusing

Fundamental issues relative to structural vibration and to scattering of sound from structures with imprecisely known internals are explored, with the master structure taken as a rectangular plate in a rigid baffle, which faces an unbounded fluid medium on the external side. On the internal side is a fuzzy structure, consisting of a random array of point-attached spring-mass systems. The theory predicts that the fuzzy internal structure can be approximated by a statistical average in which the only relevant property is a function m F (Ω) which gives a smoothed-out total mass, per unit plate area, of all those attached oscillators which have their natural frequencies less than a given value Ω. The theory also predicts that the exact value of the damping in the fuzzy structure is of little importance, because the structure, even in the limit of zero damping, actually absorbs energy with an apparent frequency-dependent damping constant proportional to dm F (ω)/dω incorporated into the dynamical description of the master structure. A small finite value of damping within the internals will cause little appreciable change to this limiting value.

/pdf/fundamental-structural-acoustic-idealizations-for-structures-217h9l0k2s.pdf

Fundamental Structural-Acoustic Idealizations for Structures with Fuzzy Internals

Numerous contributions have been made in the enhancement of the Parabolic Equation (PE) approximation method, which has been shown to be a useful tool for solving realistic problems in many different scientific fields. Evidence of its usefulness is the application of PE to solve ocean acoustic propagation problems. In early years, when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations in range-dependent environments; thus, there were certain limitations. In the recent decade, these limitations have been relaxed a great deal due to many useful contributions. The time has come to survey and report these important contributions and to discuss how these contributions enhance the capability of the PE method. This paper gives a brief review of what had been done before 1984 and highlights some important PE developments from 1984 to 1994. Also, some applications of the PE to predict ocean acoustic propagation problems will be presented. We shall call attention to a few important issues related to the PE developments and applications. Looking ahead we will discuss what more a PE can do in order to stimulate future research, development, as well as applications.

Parabolic equation development in recent decade

The partitioning of internal mass among bands of resonant frequencies is addressed for a prototype internal vibrating structure with small damping, attached via an arbitrary number N A of attachment points to an external structure. Insofar as the dynamics of the latter are concerned, the internal structure is adequately described by a frequency-dependent impedance matrix, any given column of which lists the ratios of the 3N A force components induced by one of the attachment points' velocity components when all of the other velocity components are held to zero. The properties of matrix elements and their frequency dependence are discussed in relation to principles of mechanics, especially the requirements of translational and rotational invariance of the potential energy functions. Among the deductions are that modal masses can be defined with values calculable solely from the impedance matrix measurements, and that the modal masses sum to the total mass of the internal vibrating system.

Allan D. Pierce

Papers

Fundamental Structural-Acoustic Idealizations for Structures with Fuzzy Internals

Parabolic equation development in recent decade

Resonant-Frequency-Distribution of Internal Mass Inferred From Mechanical Impedance Matrices, With Application to Fuzzy Structure Theory

A theory of aneurysm sounds.

Mathematical Theory of Wave Propagation