Causal impulse response for circular sources in viscous media
Piston and coordinate geometry. The piston, centered at the origin and surrounded by an infinite rigid baffle in the plane, has a radius . The radial and axial observation coordinates are denoted by and , respectively. The distance between origin and observer is given by and the angle between the radial and axial coordinates is .
Comparison of the asymptotic form of the Green’s function for the Stokes wave equation (Eq. (3)) and the reference Green’s function computed numerically via the MIRF approach. The Green’s function is shown for (a) and , (b) and , and and (d) and .
The RMS error for the asymptotic Green’s function relative to the reference MIRF result. The RMS error is displayed on a log-log plot relative to the dimensionless parameter . The slope on the log-log plot is approximately 0.5, indicating that RMS error is proportional to .
On-axis velocity potential for a circular piston of radius in a viscous medium. The piston is excited by a Hanning-weighted toneburst with center frequency and pulse length for four combinations of axial distance and relaxation time : (a) and , (b) and , (c) and , (d) and . The velocity potential for each combination of and is computed via both the lossy impulse response approach and a frequency-synthesis approach for verification.
The RMS error for the on-axis lossy impulse response relative to the reference MIRF result for a piston with radius . The RMS error is plotted versus the normalized axial distance .
Snapshots of the lossless impulse response generated by a circular piston with radius at and are displayed in panels (a) and (b). The constant amplitude component within the paraxial region represents the direct wave. The remaining component represents the edge wave generated by the discontinuity in particle velocity at .
Snapshots of the lossy impulse response generated by a circular piston with radius with for and are displayed in panels (a) and (b). Unlike the lossless impulse response depicted in Fig. 6, the direct wave is attenuated due to viscous diffusion. The edge wave also experiences additional attenuation relative to Fig. 6.
Normalized velocity potential field produced by a circular piston of radius excited by a Hanning-weighted toneburst in a viscous medium with relaxation time . Snapshots of the normalized velocity potential for and are displayed in panels (a) and (b).
Lossy impulse response for a circular piston and . In panel (a), the near field impulse response and the far field impulse response were evaluated at both on axis and off axis ( and ). Panel (b) shows the near field and far field impulse responses evaluated at on axis and off axis ( and ).
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