  We propose to study the fluid dynamic propagation of fluctuations in
relativistic heavy ion collisions differentially with respect to their
azimuthal, radial and longitudinal wavelength. To this end, we introduce a
background-fluctuation splitting and a Bessel-Fourier decomposition of the
fluctuating modes. We demonstrate how the fluid dynamic evolution of realistic
events can be build up from the propagation of individual modes. We describe
the main elements of this mode-by-mode fluid dynamics, and we discuss its use
in the fluid dynamic analysis of heavy ion collisions. As a first illustration,
we quantify to what extent only fluctuations of sufficiently large radial wave
length contribute to harmonic flow coefficients. We find that fluctuations of
short wave length are suppressed not only due to larger dissipative effects,
but also due to a geometrical averaging over the freeze-out hyper surface. In
this way, our study further substantiates the picture that harmonic flow
coefficients give access to a coarse-grained version of the initial conditions
for heavy ion collisions, only.
