Energy Ratio Explorer

Quick Links


You will need to have the Java runtime environment (JRE) installed on your computer in order to use this software. As of Java Version 7 Update 51, you will also need to add to the Java Exception Site List. This has been tested in Windows 10 using Microsoft Internet Explorer, Microsoft Edge and Mozilla Firefox.

To start the Energy Ratio Explorer click on the 'Launch' button above.

  • To use the explorer, select whether the incident wave is P or S, and whether it originates in the upper or lower level. These choices may be changed at any time.
  • Also select which components you wish to have plotted. These may also be changed at any time. Only those for one type of incident wave may be displayed simultaneously.

Version History

The Energy Ratio Explorer was created at CREWES in February 2002 by Chuck Ursenbach.

The Energy Ratio Explorer was updated September 2, 2005 to correct errors in the calculation of Bortfeld approximations to Rps and Tps. Note that there is an error in equation 22 of Bortfeld (1961) which was incorporated in the earlier version.

Technical Notes

  • The plot shows how the energy component ratios change with angle of incidence. To see how they change with properties of each medium, use the six scroll bars in the control panel to change the density and velocities of each layer. These may be fixed to particular values in the text fields, or interactively scanned over a range of values using the slider bars. Only four of these six variables are independent, one for the densities and three for the velocities. Accordingly one can use the drop down menus to select up to four density and velocity ratios as well. The slider bars generate ratios between 0 and 2, but other values can be accessed through the text fields. Note that you are not prevented from selecting properties corresponding to a negative Poisson's ratio.
  • The location of critical angles is indicated by vertical lines, which are annotated with the value of the critical angle, and the relevant velocity conditions.
  • The color code for the various components is given on the control panel. The scales may be adjusted using the control panel. Angles may only be adjusted to integer numbers of degrees, and the incident angle must be between 0 and 90.
  • The Aki-Richards and Bortfeld approximations can be accessed using the appropriate checkbox on the panel. These were originally defined only for sub-critical angles, but are extended beyond the critical point in this applet.

Comments on the Energy Ratios

The energy ratios show what fraction of an incident wave’s energy is carried away from the interface by various types of waves. Because of energy conservation, the energy ratios for all waves corresponding to one of the three possible incident wave types (P, SV, or SH) sum to unity. The energy ratios do not describe however the partitioning of energy flux parallel to the interface. Thus, after a critical point, the energy ratio of one wave will always drop to zero, even though that wave will continue to have a non-zero Zoeppritz coefficient. This indicates an evanescent wave that travels along the interface, but decays exponentially away from it.


  1. Aki & Richards, 1980, "Quantitative Seismology", vol. I, sec. 5.2.
  2. Bortfeld, R., 1961, "Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves," Geophys. Prosp., 9, 485-502.

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