Phase relation between strain and particle velocity

[emq]

I'm trying to understand the following derivation. Starting with the one-dimensional equation for a traveling wave $u = u_0 \exp{[j(\omega t - \beta z)]}$ the goal is to derive the phase relation between strain and velocity. The author first derives the relationship between strain and particle displacement, $S = \frac{\partial u}{\partial z} = -j\beta u$. Then to find the phase relationship between S and the particle velocity, $v = \frac{\partial u}{\partial t}$, the author takes the time derivative of S: $$ \frac{\partial S}{\partial t} = -j\beta \frac{\partial u}{\partial t} $$ and uses that to get: $$ S = -\frac{\beta}{\omega} \frac{du}{dt} = -\frac{\beta}{\omega} v$$

I don't see how the author gets from the time derivative of the strain to the last equation.

 

[haruspex]

the author takes the time derivative of S:

I cannot say why the author did that, but I don't think it was on the path to finding that last equation.
That is quite simply done by expanding $\frac{\partial u}{\partial t}$ and comparing with the expression for S.