  In this work we propose and analyze an extension of the approximate component
mode synthesis (ACMS) method to the heterogeneous Helmholtz equation. The ACMS
method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale
method to solve elliptic partial differential equations. The ACMS method uses a
domain decomposition to separate the numerical approximation by splitting the
variational problem into two independent parts: local Helmholtz problems and a
global interface problem. While the former are naturally local and decoupled
such that they can be easily solved in parallel, the latter requires the
construction of suitable local basis functions relying on local eigenmodes and
suitable extensions. We carry out a full error analysis of this approach
focusing on the case where the domain decomposition is kept fixed, but the
number of eigenfunctions is increased. The theoretical results in this work are
supported by numerical experiments verifying algebraic convergence for the
method. In certain, practically relevant cases, even exponential convergence
for the local Helmholtz problems can be achieved without oversampling.
