Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

156 Stiffness matrix method for layered media
ωh / π
15
10
5
0
-5
5
10
0
5
15
10
15
Re kh / π Im kh / π
0
Figure 10.3: Wave spectrum for Mindlin
plate, ν = 0.31.
Top: symmetric modes. Bottom: antisymmetric
modes.
ωh / π
15
10
5
0
-5
5
10
0
5
15
10
15
Re kh / π Im kh / π
0
By the way, observe that = ps − q 2 is the Rayleigh function, which is useful for
showing that at high frequencies, the phase velocity of some of these modes converges to
the Rayleigh wave velocity. After finding the modes, the displacements for the line load
could be obtained by modal superposition, very much along the lines of what was done in
the example in Section 9.1. In principle, such superposition must include both the propagating
modes and the evanescent modes (real vs. complex eigenvalues or characteristic
wavenumbers k). The exception is when we are interested only in the response at some
distance from the source, in which case the propagating modes may suffice. The great
advantage of modal solutions is that the inverse transforms back into the space domain
can be carried out analytically.