Comparison of the Khokhlov–Zabolotskaya–Kuznetsov equation and Fourier-continuation for modeling high-intensity focused ultrasound beams.

Abstract

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 Comparative Approach to the Solution of the Zabolotskaya-Khokhlov Equation by Iteration Methods

“…Substituting these values in (8) and making unity, we get the solution of (38) in the form of the following infinite series: Using ADM. Let = 2 / and = + ( ) 2 so that (38) can be written as…”

“…It is obvious that the convergence of the above relation depends mainly on the auxiliary parameter ℎ. With the proper choice of the initial guess 0 ( , ), the power series (8) converges to the exact solution at = 1…”

“…The ZK equation describes the propagation of a confined wave beam or sound beam in nonlinear medium without dispersion or absorption and it investigates the beam deformation associated with the properties of the nonlinear medium [3,4]. This equation has applications in many fields of life such as it is used to simulate the estimation of fish stock abundance, discrimination between fish species, the effect of excess attenuation which occurs due to the nonlinear sound propagation in water, absorption, and diffraction in the focused sound beams [5][6][7][8].…”

A Laplace substitution method of space‐time fractional order including jumbled partial derivatives

“…29,30 Also, this equation has implementations in numerous areas of life such as the appreciation of fish stock plenitude, recognition between fish types, diffraction in the concentrated sound beams, imbibition, and the effect of abundance lessening which happens because of the nonlinear sound expansion in water. 31,32 In Kumar et al, 29 the similarity transformations method was used to find new closed form solutions of the (2 + 1) Z-K equation and was also reduced to a new PDE with fewer independent variables. In Bruzon et al, 33 the no classical symmetry reductions was used to find some traveling wave solutions of the dissipative Z-K equation.…”