Mathematical investigation of periodic acoustical waveguides of an arbitrary shape (Q1106395)

The author considers the Helmholtz equation with the wave number function being periodic in one space coordinate \(x_ 1\). For the solutions of Floquet type \(u(x)=v(x)\cdot \exp (-\xi \cdot \phi (x))\), where \(v(x_ 1+2\pi,x_ 2,x_ 3)=v(x)\), the spectrum of the \(\xi\) 's is proved to be discrete with no nontrivial solutions u(x). The Green function for the Neumann problem is constructed being exponentially decreasing iff the spectrum consists of no purely imaginary points.