  Elastic wave velocity in a soft fiber that varies depending on material
constitution and axial stress level is an essential measure of mechanical
signals in many technical applications. In this work, based on the
small-on-large theory, we establish a model of linear elastic wave propagation
in a finitely pre-stretched soft fiber. The formulas of longitudinal (Primary,
P-) and transverse (Secondary, S-) wave velocities are provided and validated
by numerical simulations as well as by experimental data on spider silk. The
influences of material constitution, compressibility, and pre-stress on the
wave propagation are investigated. We found that with increasing pre-stress,
the variation of P-wave velocity highly relies on the concavity of the
stress-strain curve. In contrast, an increase of S-wave velocity exhibits
regardless of any constitutive model. For both P- and S-waves, the variation of
the velocities is more significant in a compressible fiber than that in a
nearly-incompressible one. Moreover, for minuscule pre-stress, we propose a
modified formula for S-wave velocity based on the Rayleigh beam theory, which
reveals the competition mechanism between "string vibration" and "beam
vibration." This may provide a reliable theoretical basis for precise
mechanical characterization of soft fibers and open a route for lightweight,
tunable wave manipulation devices.
