Conclusions

Conclusions

Three practical ways to represent the phase velocities: the 3D three axis. the 3D constant radius and the planar representations.

The 3D three axis representation allows a visualization of the unsuspected connection nature among the intersecting surfaces. In other words, sometimes there are not three velocity surfaces, there is only one. The continuity of the line through the three main planes reveals the existence of only one surface.

The 3D constant radius representation allows a real comparison of the size for an open-deformed (120 degrees) rectangular wall (90 degrees).

The 2D planar representation is very useful in the wave reflection-refraction problem solution.

The application developed in Excel allows the user to obtain a planar, a 3D constant radius and a 3D three-axis representations of the phase velocity surfaces for any material by changing the stiffnesses constants.

Although the direction cosines are evaluated numerically, there are some errors in the direction of the velocities (the magnitudes and graphs are correct). By reviewing the data, they occur near to the main axis and acoustic axis when the numbers reach high values (infinite tendency). There is a limit in which the precision is lost. So, it is necessary to use the reduced mathematical expressions for those special situations, like the assumption of the one wave is always purely transverse.

It is important to note two additional issues. The first is that the information for the graphs is taken every five degrees. The second issue is the way in which the graph is done, always it is considered the lower root inside the middle solution and inside the greater. As a consequence of this two issues, the graphs could show sometimes a different shape of the real situation (especially in the area crossings).

The 3D three axis representation allows a visualization of the unsuspected connection nature among the intersecting surfaces. In other words, sometimes there are not three velocity surfaces, there is only one. The continuity of the line through the three main planes reveals the existence of only one surface.

The 3D constant radius representation allows a real comparison of the size for an open-deformed (120 degrees) rectangular wall (90 degrees).

The 2D planar representation is very useful in the wave reflection-refraction problem solution.

The application developed in Excel allows the user to obtain a planar, a 3D constant radius and a 3D three-axis representations of the phase velocity surfaces for any material by changing the stiffnesses constants.

Although the direction cosines are evaluated numerically, there are some errors in the direction of the velocities (the magnitudes and graphs are correct). By reviewing the data, they occur near to the main axis and acoustic axis when the numbers reach high values (infinite tendency). There is a limit in which the precision is lost. So, it is necessary to use the reduced mathematical expressions for those special situations, like the assumption of the one wave is always purely transverse.

It is important to note two additional issues. The first is that the information for the graphs is taken every five degrees. The second issue is the way in which the graph is done, always it is considered the lower root inside the middle solution and inside the greater. As a consequence of this two issues, the graphs could show sometimes a different shape of the real situation (especially in the area crossings).