From some quick searches (so not a definitive answer, but a place to start), it seems that sound waves are most likely longitudinal which doesn’t cause shear. However, shear forces can be created by sound waves when they hit a surface.
From that information, I don’t think the shear energy imparted by a sound wave is very large. Since non-newtonian fluids only thicken under shear, they may not actually behave very differently than a regular fluid in these conditions. Preventing sound waves from traveling is usually accomplished by causing lots of scattering (open cell foam) or by absorbing the energy in a viscoelastic material (usually polymers).