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A motion (wave) $\mathbf{x}: \mathcal{B}_0 \times [t_0,t_1] \to \mathcal{E}:$ such that
$q-o = \mathbf{x}(p,t)=(p-o)+\mathbf{a}_0 cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$

can propagate in an elastic fluid with constitutive equation:
$\mathbf{T}=-\pi(\rho)\mathbf{I}$

not subject to any volume force? What restrictions on polarization vector $\mathbf{a}_0$ and the vector $\mathbf{k}_0$ are needed? $\mathbf{a}_0$ and $\mathbf{k}_0$ should be parallel? What restrictions on $\pi(\rho)$ are needed?

I can write the balance equation of linear momentum, in an eulerian description I suppose: $\rho \mathbf{a} = \mathrm{div}_q \mathbf{T} = -\nabla_q \pi = - \pi' \nabla_q \rho$

But it is easier to compute the acceleration in a lagrangian description:
$\ddot{\mathbf{x}}(p,t) = -\omega_0^2 \mathbf{a}_0cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$

At this point computing $\nabla_q\rho$ is not easy I think. There is some mistake here? There is a better and simpler way to prove the results?

Thanks and Please excuse my bad english

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The answer to the question in the title is "Yes" - e.g. sound waves. –  cm2 Feb 26 '12 at 3:30
    
The reals questions are in the body. What kind of wave can propagate in an elastic fluid? What kind of elasthic fluid is needed? How can I derive these result? –  unlikely Feb 26 '12 at 8:32
1  
It is a little unclear what you are asking for. I assume p and o are two points in space; what is the significance of point q? If it is a simple translation between the two points, then the gradient operators are equal. –  cm2 Feb 27 '12 at 3:22
    
$q-o = \mathbf{x}(p,t)$, so $q$ it's the position of $p$ after a time $t$, it's the Lagrangian description of the motion. –  unlikely Jan 31 at 14:25
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