  The purpose of this paper is twofold. Firstly, we study the dynamical density
fluctuations around the critical point(CP) of Quantum Chromodynamics(QCD) using
dissipative relativistic fluid dynamics in which the coupling of the density
fluctuations to those of other conserved quantities is taken into account. We
show that the sound mode which is directly coupled to the mechanical density
fluctuation is attenuated and in turn the thermal mode becomes the genuine soft
mode at the QCD CP. We give a speculation on the possible fate of a Mach cone
in the vicinity of the QCD CP as a signal of the existence of the CP on the
basis of the above findings. Secondly, we clarify that the so called
first-order relativistic fluid dynamic equations have generically no problem to
describe fluid dynamic phenomena with long wave lengths contrary to a naive
suspect whereas even Israel-Stewart equation, a popular second-order equation,
may not describe the hydrodynamic mode in general depending on the value of the
relaxation time.
