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The condition for the validity of your derivation [for
compression waves] was Hooke's law, the restoring force
must be proportional to displacement. Is this not
true for an ideal transverse wave (small amplitude)?
What prevents me from applying the same to a transverse
displacement and from using k in Hooke's law (rather than s)?
I am assuming that there is no tension in a long string of tiny
masses connected by tiny relaxed springs.