  Scalar field should have no spin angular momentum according to conventional
understandings in classical field theory. Yet, recent studies demonstrate the
undoubted existence of wave spin endowed by acoustic and elastic longitudinal
waves, which are of irrotational curl-free nature without vorticity and can be
described by scalar fields. Here, to solve this seeming discrepancy, we uncover
the origin of wave spin in scalar fields beyond traditional formalism by
clarifying that the presence of higher order derivatives in scalar field
Lagrangians can give rise to non-vanishing spin. For scalar fields with only
first order derivative, we can make the hidden wave spin emerge, by
constructing a latent field that leads to the original field through a time
derivative so that is of second order. We exemplify the wave spin for elastic
and acoustic fields, as well as for dissipative media, following Noether's
theorem in higher-order derivative Lagrangian. The results would prompt people
to build more comprehensive and fundamental understandings of structural wave
spin in classical fields.
