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Derivation of Poisson's ratio

Figure E-5. Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities ( and ) and density .

Figure E-5 in Sheriff’s Encyclopedic Dictionary of Applied Geophysics contains basic information on elastic constants in isotropic media expressed in terms of each other and compressional and shear wave velocities and , respectively. The following are derivations of in terms of and Poisson’s ratio s, in terms of and s, and s in terms of / where and are initially defined in terms of density , shear modulus and Lame’s constant . [1]


By definition, Poisson's ratio:


By definition:


Dividing equation 1 by equation 2,




By definition:

By definition:

By definition, Poisson's ratio:


Equation 6 is the same as equation 3


  1. Sheriff, Robert E. (2002). Encyclopedic Dictionary of Exploration Geophysics (4th ed.). Society of Exploration Geophysicists, SEG Geophysical Reference Series No. 13. ISBN 978-1-56080-118-4. DOI:

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