Summary
By substituting in an allowed manner the actual waves with plane waves and considering that any segments of the trajectories between the same wave fronts are isochronic ways for the vibrations: on derives the well known travel-time formulas quite plainly without the tiresome geometrical calculations applied hitherto.
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Kilczer, J. Die theoretische Bestimmung der Laufzeit. Geofisica Pura e Applicata 19, 144–150 (1951). https://doi.org/10.1007/BF01993402
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DOI: https://doi.org/10.1007/BF01993402