LET R N be an N-dimensional Euclidean space, N ⩾ 3, 0 ⩽ τ ⩽ 1, Ω τ a family of open bounded domains in R N , \\ ̄ gW T the closure of the domain Ω τ , Ω 1 τ a connected component of the set R NβΩ τ , containing the point at infinity, Ω 2 τ = R N β (Ω duτ ∪ Ω 1 τ) . Let B τ be an operator defined on the boundary of the set Ω τ and having the form ( B τ u) (x) = u(x)¦ xεΓp \\ ̄ gW τ , ( B τ u) (x) = ( δu δn + σ(x)u) (x)¦ xεΓp \\ ̄ gW τ . Let Δ be the Laplace operator, H (τ) the extension in ( B τ u) ( x) = 0 of the operator (−Δ) corresponding to the boundary conditions l 2( R N βω τ ) E( λ, H ( τ)) the spectral function of the operator H( τ). In l 2( R N βω τ ) we consider the abstract Cauchy problem δ 2(τ¦t) δt 2 = −H(τ)v , v(τ¦ + 0) = f 1 , v(τ¦ + 0) = f 2 . (1) It is easy to see that the solution of the problem (1) (which in this case is the same as the generalization of it by S. L. Sobolev) is given by the formula 2 v(τ|t)= ⨍ o ∞ cos VλtdE(λ,H(τ))ƒ2+ ⨍ o ∞ λ − 1 2 sin Vλtde(λ,H(τ))ƒ1 It follows from (2) that v(τ¦t) behaves essentially differently depending on whether or not the operator H( τ) has a discrete spectrum. The purpose of our paper is to estimate the variation of the function v(τ¦t) for so small a variation of the domain Ω τ as transforms it from a simply connected into a non-simply connected domain, and changes the nature of the spectrum of the operator v(τ¦t). The variation of the discrete spectrum of a Laplace operator when the domain is varied was previously studied in [1]. We have formulated the main results in Lemma 2.3 and Theorem 2.1. We use the method developed for solving similar problems for Schrödinger's equation in [2–4]. This method consists of the following. Let G(τ¦x, y, t) be the kernel of the operator exp (− t 0 H( τ)), redefined in such a way that G(τ¦x, y, t 0)= 0 , if either xεΩ τ, or yεΩ τ . It happens that the kernel is so well behaved that it permits the properties of the spectrum of the operator exp (− t 0 H( τ)), to be studied in detail, and corresponding conclusions about the spectrum of the operator H( τ) follow directly from this. It was shown in [5] that instead of the function exp (− t 0 H( τ)) it would be possible to consider any other function F ( H( τ)) (for example, (1 + H( τ)) −1), representable in the form F(λ)= ⨍ o ∞ exp (−λt)dμ(t) where μ( t) is a measure satisfying certain conditions. The paper consists of two sections. In section 1 we present some auxiliary estimates and lemmas. The proofs of these statements differ only negligibly from the proofs of the corresponding statements for the Schrödinger operator, and hence we will not discuss them in detail. The main results of the paper are discussed in section 2.