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.

7.1 Dynamic surface distortion caused by coating absorption

7.1.1 Mean displacement

7.1.2 Under cutoff regime

7.1.1 Mean displacement

7.1.2 Under cutoff regime

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