I will give you mathematical background that explains where sound waves, as well as shear, and other waves come from in continuum approximations and why viscosity has no influence on such waves: In Eulerian form, the vector equations of motion for a fluid or solid continuum can be written as $$\frac{\partial \mathbf{q}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{x}} = \vec{\nabla} \cdot \underline{\mathbf{S}}$$ where $\mathbf{q}$ is a vector of size $n$ and $\underline{\mathbf{S}} = \underline{\mathbf{S}}(\frac{\partial\mathbf{q}}{\partial\mathbf{x}})$ is a matrix of size $n \times 3$. For fluids you would typically have $\mathbf{q}=(\rho,\rho u,\rho v,\rho E)$ and solids $\mathbf{q}=(\rho,\rho u,\rho v,\rho E, \epsilon_x,\epsilon_y)$. The term $\mathbf{F}(\mathbf{q})$ represents conservative flux. It is important to note that $\mathbf{F}$ only depends on $\mathbf{q}$ directly and not on any of its spatial derivatives. Alternatively, $\underline{\mathbf{S}}$ depends only on spatial derivatives of $\mathbf{q}$. While this equation is a very complicated non-linear partial differential equation, for small disturbances it is entirely analogous to the simple advection-diffusion equation $$\frac{\partial q}{\partial t} + \lambda\frac{\partial q}{\partial x} = \kappa\frac{\partial^2 q}{\partial x^2}$$ with wave speed $\lambda$ and diffusion coefficient $\kappa$. To put the equations of motion for solids and fluids into a similar form simply apply the chain rule to yield $$\frac{\partial \mathbf{q}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{q}}\frac{\partial \mathbf{q}}{\partial\mathbf{x}} = \frac{\partial \underline{\mathbf{S}}}{\partial(\partial\mathbf{q}/\partial \mathbf{x})}\vec{\nabla} \frac{\partial\mathbf{q}}{\partial\mathbf{x}} $$ Comparing this to the simple linear advection-diffusion equation we can extract all of the state dependent wave speeds from the Jacobian $\mathbf{\underline{J}}=\frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{q}}$ by noting that if $\mathbf{F}(\mathbf{q})$ were constant then a transformation of variables according to it's eigenvectors will produce $n$ independent advection-diffusion equations with wave speeds corresponding to the eigenvalues of the Jacobian. Likewise, the right hand side corresponds strictly to diffusion like processes.
