  High-$T_{\rm{c}}$ cuprate superconductors host spin, charge and lattice
instabilities. In particular, in the antiferromagnetic glass phase, over a
large doping range, lanthanum based cuprates display a glass-like spin freezing
with antiferromagnetic correlations. Previously, sound velocity anomalies in
La$_{2-x}$Sr$_{x}$CuO$_4$ (LSCO) for hole doping $p\geq 0.145$ were reported
and interpreted as arising from a coupling of the lattice to the magnetic glass
[Frachet, Vinograd et al., Nat. Phys. 16, 1064-1068 (2020)]. Here we report
both sound velocity and attenuation in LSCO $p=0.12$, i.e. at a doping level
for which the spin freezing temperature is the highest. Using high magnetic
fields and comparing with nuclear magnetic resonance (NMR) measurements, we
confirm that the anomalies in the low temperature ultrasound properties of LSCO
are produced by a coupling between the lattice and the spin glass. Moreover, we
show that both sound velocity and attenuation can be simultaneously accounted
for by a simple phenomenological model originally developed for canonical spin
glasses. Our results point towards a strong competition between
superconductivity and spin freezing, tuned by the magnetic field. A comparison
of different acoustic modes suggests that the slow spin fluctuations have a
nematic character.
