From material derivatives to partial derivatives in the wave equation

Question

Consider the Cauchy momentum equation:

$$\rho \frac{d^2 \mathbf{u}}{d t^2} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$$

where $\rho(\mathbf{x},t)$ is the density, $\mathbf{u}(\mathbf{x},t)$ is the displacement vector, $\boldsymbol{\sigma}(\mathbf{x},t)$ is the Cauchy stress tensor, and $\mathbf{f}(\mathbf{x},t)$ represents the body force per unit mass.

Let us focus on the left-hand side of the equation. The material derivative of $\mathbf{u}$ is:

$$ \frac{d \mathbf{u}}{d t} = \frac{\partial \mathbf{u}}{\partial t} + \frac{d\mathbf{x}}{dt} \cdot \nabla \mathbf{u}, $$

if you proceed to take the second derivative of this expression you get something nasty looking. However, if $d\mathbf{x}/dt=0$, you get great simplification:

$$ \frac{d^2\mathbf{u}}{dt^2}=\frac{\partial^2\mathbf{u}}{\partial t^2} $$

What I am confused with is that for the wave equation this is pretty much what is assumed, but for Navier-Stokes equation we need to keep the $d\mathbf{x}/dt$ term.

I do not see why this difference arrises.

The problem I have at hand is to derive that for an elastic string which is held fixed at endpoints, the governing equation of motion is $\partial^2 \mathbf{u}/\partial t^2=c^2\partial^2 \mathbf{u}/\partial x^2$. For the right hand side it is relatively easy: assume $\mathbf{f}=0$ and use Hooke's law to calculate $\mathbf{\sigma}$. Again, the lack of terms containg $d\mathbf{x}/dt$ on the left hand side is surpriding to me.

Answer 1

It happens that the convective term $\mathbf{\dot x}\cdot \nabla \mathbf u$ describes the transport of the field $\mathbf u$ over the flow, but in waves there is no transport of matter.

Another way to see it is that in the traditional wave equation, there is no macroscopic flow velocity, i.e., $\mathbf{\dot x} =0$.

Of course we could study another wave equations, but this is an hypothesis in the construction of this specific model.

For Navier-Stokes equation the situation is different, since very often the flow is exactly what is interesting to study in the problem.

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If you keep such convective term, you could study a convective wave equation, a model that takes into account the flow of its media.