Helmholtz Equation Research Paper
Sound pressure p, due to the sound wave propagation in a media satisfies the Helmholtz equation that is written as where is the Laplacian operator, is the free field wave number, is the angular frequency, and is the speed of sound in the media. But, in real media, sound wave propagation causes energy loss. So, the sound pressure satisfies the dissipative wave equation as [17]: where is called “relaxation time” in which, η is the shear viscosity coefficient, ηB is the bulk viscosity coefficient, and is the equilibrium density of the media. If we assume a mono-frequency sound pressure wave, p, with the time dependency of , Eq. (3) gives that is called “modified complex Helmholtz equation”. This equation describes the sound pressure …show more content…
3(a) and 3(b) show the real part and the imaginary part of the complex wave number, respectively. The inviscid–liquid (i.e. ) curves in the mentioned figures are obtained from the Helmholtz equation (i.e. ). The viscous–liquid (i.e. ) curves are obtained from the modified complex Helmholtz equation (i.e. ).
Fig. 3 shows the wave number of the Helmholtz equation (i.e. ) and the modified complex Helmholtz equation (i.e. ) in comparison with each other, clearly. Fig. 3(a) shows that the real part of wave number in the modified complex Helmholtz equation is much less than the wave number of the Helmholtz equation. Also, the real part of wave number in the modified complex Helmholtz equation is approximately insensitive to the increasing of frequency, especially by increasing the characteristic impedance of the liquids. The wave number of Helmholtz equation decreases by increasing the characteristic impedance of the liquids. As it is shown in Fig. 3(b), the imaginary part of wave number in the Helmholtz equation is zero. However, the imaginary part of wave number in the modified complex Helmholtz equation increases by increasing the frequency in THz range. This increase is due to the effect of term in Eq. (6). Also, Fig. 3(b) shows the imaginary part of the wave number in the modified complex Helmholtz equation decreases by increasing the characteristic impedance (i.e. ) of the viscous
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