An intuitive justification of this approximation can be found in the article by Louisnard . Also, relevant discussions were made in Section 1 on the same fact that real part of the wavenumber is well predicted by linear theory as also reported by experimental studies. Finally, with the use of equations (2.49) and (2.50), the Helmholtz equation for the nonlinear wave propagation can be obtained as; (2.51) In Figure 2.5, the real and imaginary part of the wavenumber for the driving frequency 20 kHz, bubble volume fraction % and the distribution of bubbles β=0.005 with same radius R0=5 µm, is plotted. One can observe that the imaginary part of the wave number increases very quickly for the driving amplitudes over the Blake threshold and it can be as high as the real part for very large driving amplitudes. Figure The real and imaginary part of the wavenumber used in the nonlinear wave propagation.
An intuitive justification of this approximation can be found in the article by Louisnard . Also, relevant discussions were made in Section 1 on the same fact that real part of the wavenumber is well predicted by linear theory as also reported by experimental studies. Finally, with the use of equations (2.49) and (2.50), the Helmholtz equation for the nonlinear wave propagation can be obtained as; (2.51) In Figure 2.5, the real and imaginary part of the wavenumber for the driving frequency 20 kHz, bubble volume fraction β=0.005 % and the distribution of bubbles with same radius R0=5 µm, is plotted. One can observe that the imaginary part of the wave number increases very quickly for the driving amplitudes over the Blake threshold and it can be as high as the real part for very large driving amplitudes. Figure The real and imaginary part of the wavenumber used in the nonlinear wave propagation.
Thesis_Hakan_Dogan.docx