  This work is concerned with the study of fundamental models from nonlinear
acoustics. In Part~I, a hierarchy of nonlinear damped wave equations arising in
the description of sound propagation in thermoviscous fluids is deduced. In
particular, a rigorous justification of two classical models, the Kuznetsov and
Westervelt equations, retained as limiting systems for consistent initial data,
is given. Numerical comparisons that confirm and complement the theoretical
results are provided in Part~II.
