Suppose I have an elastic body described by a vector field $\pmb{\Delta} \left( \pmb{x}, t \right) $ which gives the displacement of a point from equilibrium. This field can be separeted in a component with null divergence and a component with null curl. I am going to focus on the first one: one can prove it solves a wave equation of the kind:
$\left( \frac{\partial^2}{\partial t^2} - v_L^2 \nabla^2 \right) \pmb{\Delta}_L = 0$.
I have read one can prove that the longitudinal and transversal part of the wave each has an energy conservation law. I have also read this conserved energy for the case of a longitudinal wave should have the following form:
$E_L = \frac{1}{2} \int d \pmb{x} \left[ \lvert \partial_t \pmb{\Delta}_L \rvert^2 + v_L^2 (div\pmb{\Delta}_L)^2 \right] $.
I have tried proving this statement but I get a different result, here is my attempt:
Multiplying both sides of the wave equation for $\frac{\partial \pmb{\Delta}_L}{\partial t} $ I get:
$\frac{\partial \pmb{\Delta}_L}{\partial t} \cdot \frac{\partial^2 \pmb{\Delta}_L}{\partial t^2} - v_L^2 \frac{\partial \pmb{\Delta}_L}{\partial t} \cdot \nabla^2 \pmb{\Delta}_L= 0$, therefore:
$\frac{\partial }{\partial t} \left[ \frac{1}{2} \lvert \frac{\partial \Delta_L}{\partial t} \rvert^2 \right] - v_L^2 \frac{\partial \Delta_L^{i}}{\partial t} \nabla^2 \Delta_L^i= 0$, where I have used an improper Einstein convention to express the dot product.
The last term is the real trouble here. I have tried to deal with it using Green's first identity. By integrating over the entire volume (and therefore neglecting the integral over the boundary) this identity reduces to:
$\int \left[ \psi \nabla^2 \phi +\nabla \phi \cdot \nabla \psi \right] dV = 0$.
By applying it to the second term of the equation I get: $\int d^3 \pmb{x} \frac{\partial \Delta_L^{i}}{\partial t} \nabla^2 \Delta_L^i = - \int d^3 \pmb{x} (\partial_j \Delta_L^i) \partial_t \left( \partial_j \Delta_L^i \right) = \frac{\partial}{\partial t} \left[ -\frac{1}{2} \sum_{i j} \left( \partial_j \Delta_L^i \right)^2 \right] $
which gives me the following conservation law:
$ E_L = \frac{1}{2} \int d^3 \pmb{x} \left[ \lvert \partial_t \pmb{\Delta_L} \rvert^2 +v_L^2 \sum_{i j} \left( \partial_j \Delta_L^i \right)^2 \right] = constant $
which is similar to what I was expecting, except the last term isn't quite the divergence squared. Did I do the math wrong?
