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Love-waves cannot exist in a half-space. Layering must be present and there also must be accompanied impedance contrasts associated with the layering.

Because layering naturally induces seismic dispersion (in certain frequency ranges) and layering must be present for Love-wave generation, does that - as a corollary - indicate that Love-waves are naturally and always dispersive?

Note that the motivation for this question primarily comes from the idea that Rayleigh-waves - which can exist in a half-space - are not dispersive when layering is not present.

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Your assertion is correct! See e.g. Stein & Wysession An Introduction to Seismology, Earthquakes, and Earth Structure (2003) p. 90 ,section 2.7.3: Love waves in a layer over a halfspace (https://books.google.ch/books?id=-z80yrwFsqoC&lpg=PP1&hl=de&pg=PA90). In particular, for any particular frequency/period, Love waves can only have certain horizontal apparent velocities/wavenumbers. Hence, different frequencies have different apparent velocities, which is what we refer to as dispersion. See for example also the figure 2.8-2 from the book, https://books.google.ch/books?id=-z80yrwFsqoC&lpg=PP1&hl=de&pg=PA96, which shows the dispersion curve for a layer over a halfspace.

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