Variational Principles For Surface Wave Propagation On A Laterally Heterogeneous Earth‐Iii. Potential Representation

Jeroen Tromp, F. A. Dahlen

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We present a frequency‐domain analysis of travelling and standing surface waves on a smooth, laterally heterogeneous Earth model, using a potential representation that is valid everywhere, including in the neighbourhood of surface wave caustics. the Love and Rayleigh wave displacement fields are written in the form uL=k‐1LW (‐řX▿1L and uR=UřχR+k‐1RV▿1χR; the quantities U, V and W are the local radial eigenfunctions, kL and kR are the local Love and Rayleigh wavenumbers, and χL and χR are surface wave potentials that vary rapidly on the surface of the unit sphere. the natural normalization condition for the local radial eigenfunctions is cCI1= 1, where c is the local phase velocity, C is the local group velocity and I1 is the local radial kinetic energy integral; with this normalization, the Love and Rayleigh wave potentials satisfy the spherical Helmholtz equations ▿21χL + k2LχL = 0 and ▿21χR + k2RχR = 0. Standing wave eigenfunctions χL and χR can be determined either by solving a truncated matrix eigenvalue problem or by using the EBK semi‐classical method; the results incorporate multiplet coupling along a single dispersion branch but ignore cross‐branch coupling. the theory allows for slowly varying topography of the Earth's surface and the core‐mantle boundary, and incorporates the effect of self‐gravitation.

Original languageEnglish (US)
Pages (from-to)195-209
Number of pages15
JournalGeophysical Journal International
Issue number2
StatePublished - Feb 1993

All Science Journal Classification (ASJC) codes

  • Geophysics
  • Geochemistry and Petrology


  • free oscillations
  • lateral heterogeneity
  • potential representation
  • surface waves
  • variational principles.


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