Abstract
The coordinate-free one-way wave equation is transferred in spherical coordinates. Therefore it is necessary to achieve consistency between gradient, divergence and Laplace operators and to establish, beside the conventional radial Nabla operator ∂Φ/∂r, a new variant ∂rΦ/r∂r. The two Nabla operator variants differ in the near field term Φ/r whereas in the far field r≫0 there is asymptotic approximation. Surprisingly, the more complicated gradient ∂rΦ/r∂r results in unexpected simplifications for – and only for – spherical waves with the 1/r amplitude decrease. Thus the calculation always remains elementary without the wattless imaginary near fields, and the spherical Bessel functions are obsolete.
Highlights
Seismics, sonar, sodar, room and machine acoustics, ultrasound diagnosis and tomography depend on the calculation of sound paths in solid, liquid or gaseous media with heterogenous physical properties and complex geometry
Calculations are based on the well-known equations of motion listed by Augustin-Louis Cauchy 200 years ago: In a continuum with stationary cartesian coordinates x = {x ex, y ey, z ez} and modulus of elasticity E [Pa] a vectorial, elastic deflection s = s(x, t) [m] causes the stress tensor T = E∇s [Pa]
From the quadratic velocity term c2 = (+c)2 = (−c)2 can be seen that there are two waves travelling in opposite directions +c and −c, results the designation “Two-way wave equation”
Summary
Sonar, sodar, room and machine acoustics, ultrasound diagnosis and tomography depend on the calculation of sound paths in solid, liquid or gaseous media with heterogenous physical properties and complex geometry. From the quadratic velocity term c2 = (+c)2 = (−c) can be seen that there are two waves travelling in opposite directions +c and −c, results the designation “Two-way wave equation”. Regardless of this ambiguity, irregular phantom effects occur in numerical seismic FE or FD wave calculations. To eleminate these unwanted effects a great number of auxiliary equations have been developed, but no specific approach has been able to prevail. Because of the economic importance of the “One-Way”/“Two-Way” problem there exist numerous patent applications beside the scientific literature [1,2,3]
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