On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events.

High-intensity focused ultrasound (HIFU) employs acoustic amplitudes that are high enough that nonlinear propagation effects are important in the evolution of the sound field. A common model for HIFU beams is the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation which accounts for nonlinearity and absorption of sound beams. The KZK equation models diffraction using the parabolic or paraxial approximation. For many HIFU sources the source aperture is similar to the focal length and the paraxial approximation may not be appropriate. Here we compare results obtained using the “Texas code” a time-domain numerical solution to the KZK equation to a new method of solving nonlinear differential equations: the Fourier-continuation (FC) method. Here the FC method solves the underlying fluid dynamics equations and does not invoke the paraxial approximation. Results were obtained for a 1.1-MHz focused HIFU source. The medium parameters were taken to either be hyperviscous water or to match human liver. For a low focusing gain transducer (focal length 50λ and radius 10λ) the KZK and FC models showed excellent agreement through the focal region. As the source radius was increased, discrepancies started to appear and for a radius of 30λ, the KZK model did not capture diffraction of the HIFU source accurately. [Work supported by NSF 0835795.]

A model is presented for the linear pulsation of a small bubble in a tube with locally reactive walls but with otherwise arbitrary wall impedance. The model is based on the normal mode expansion of the Green's function presented by Morse and Ingard [Theoretical Acoustics (McGraw-Hill, 1968), Eq. (9.2.10)]. The specific case of a cylindrical tube is considered. For a bubble that is located in the center of the tube and that is small compared with both tube diameter and wavelength, the radiation impedance on the bubble is given by a summation of the normal modes evaluated in the center of the tube. From the radiation impedance, the frequency response of the bubble to an applied sound field is obtained. For tube walls that are either rigid of pressure release, the solution agrees with the frequency response calculated using the method of images for a square tube having the same cross-sectional area. In tubes with hard walls the resonance frequency decreases as tube radius decreases because of radiation damping. In tubes with very soft walls the radiation damping is negligible below the cutoff frequency of the lowest mode, and the resonance frequency increases slightly as tube radius is decreased.

A recently developed Fourier-continuation (FC) method is used to develop a numerical algorithm, which can accurately solve the fully nonlinear acoustic vector wave equation for the type of hundred-wavelength domains arising in the field of focused medical ultrasound. The FC nonlinear-ultrasound solver is used to outline the domain of applicability of the widely used Khokhlov–Zabolotskaya–Kuznetsov (KZK) approximation in focused ultrasound particularly in configurations arising in the field of high intensity focused ultrasound (HIFU). Good agreement was found for small-radius transducers; but as the source radius was increased to dimensions equivalent to practical HIFU sources the KZK approximation leads to errors in the location of the predicted focus. Examples for media with multiple scattering centers also show the utility of the FC approach in modeling complex media. In general the FC method produces solutions with significantly reduced discretization requirements over those associated with finite-difference and finite-element methods, and they lead to improvements in computing times by factors of hundreds and to thousands over competing approaches. [Work supported by NSF Grant No. 0835795.]