The dynamics of amorphous water ice structures of different densities have been studied by high-resolution neutron time-of-flight and backscattering spectroscopy. An accurate determination of the vibrational density of states G(omega) in the energy range of phonons h omega less than or similar to 40 meV of a many fold of structures comprising the low-density amorphous (LDA, rho approximate to 31 molecules/nm(3)), high-density amorphous (HDA, rho approximate to 39 molecules/nm(3)), very-high-density amorphous (vHDA, rho approximate to 41 molecules/nm(3)) and modifications of intermediate density in respect to HDA and vHDA has been achieved. Unlike the G(omega) of high-density crystalline phases IX, V, and XII, which have been measured as reference systems, the G(omega) of all high-density amorphous counterparts proves to be textureless except for a predominant peak at low energies. In vHDA this peak is centered at about 10 meV and redshifted upon density decrease to 7 meV in LDA. A concomitant upshift of the low-energy librational band edge from 34 to 45 meV is detected in deuterated samples. Mean-square displacement and Debye temperatures T-D for vHDA, HDA'-a structure obtained as a transient product of the temperature induced vHDA to LDA transformation-and LDA are extracted from the highest resolution backscattering experiments. T-D values indicate the absence of a dominant excess of low-energy modes in G(omega), referred to as boson peak in the literature, being in agreement with the G(omega) properties directly monitored by the time-of-flight technique. Having applied deuterated sample material we are able to display phase coherence effects within the phonon system in the second and third pseudo-Brillouin zone (1 angstrom(-1) <= Q <= 5 angstrom(-1)) of the amorphous samples. A phase coherent signal from acoustic phonons can be followed up to energies of at least 15 meV.