Based on the elastic continuum model, the theory of displacement of acoustic phonons spectra, arising in flat semiconductor nanostructures, was developed. For the studied resonant tunneling structure, which can be an active element of a quantum cascade laser or detector, the spectrum of acoustic phonons modes has been calculated. The dependences of the acoustic phonon spectrum from geometric parameters of studied nanostructure were set. The results obtained in the paper can be used to further theoretical studies of the electrons interaction with the transverse acoustic phonons in multilayer semiconductor nanosystems.