## 7 Dynamic Surface Distortion

We address here the motion of the reflecting surface of a mirror receiving a time-varying power flow creating a time-varying temperature and finally time-varying stresses. For the temperature field, the problem was solved in the preceding Section 6. We take the generic form

And we search for a displacement vector in the (already used) form
with, as usual, . The equilibrium equations must be modified with respect to Equation (3.91) to take into account inertial effects
with the expression of the displacement vector, and the expression of thermoelastic stresses, this gives
By taking (7.4)(7.5) , we get
The general solution of which is
where are arbitrary constants, and where the transverse wave number is defined as
. Equation (7.4) can then be written as
which allows one to find :
where and are two more arbitrary constants, is a special solution of
and the longitudinal wave number is defined as
Once is found, one obtains :
The boundary conditions on the faces allow one to determine the arbitrary constants. It is convenient to introduce the dimensionless parameter
Then, we have
with
and
are defined as in Equation (3.148). The surface distortion is given by
and we have
We have seen that the dynamic temperature field at frequencies within the GW band is practically proportional to the beam intensity profile, and consequently negligible on the edge. This is why we neglect the boundary conditions for the edge stresses here. At this point, we can specialize for the two cases of coating/bulk heating.