  We study the inverse boundary value problem for the Helmholtz equation using
the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional
Lipschitz stability estimate for the inverse problem holds in the case of
wavespeeds that are a linear combination of piecewise constant functions
(following a domain partition) and gives a framework in which the scheme
converges. The stability constant grows exponentially as the number of
subdomains in the domain partition increases. We establish an order optimal
upper bound for the stability constant. We eventually realize computational
experiments to demonstrate the stability constant evolution for three
dimensional wavespeed reconstruction.
