  Acoustic transparency is the capability of a medium to transmit mechanical
waves to adjacent media, without scattering. This characteristic can be
achieved by carefully engineering the acoustic impedance of the medium -- a
combination of wave speed and density, to match that of the surroundings. Owing
to the strong correlation between acoustic wave speed and static stiffness, it
is challenging to design acoustically transparent materials in a fluid, while
maintaining their high structural rigidity. In this work, we propose a method
to design architected lattices with independent control of the elastic wave
speed at a chosen frequency, the mass density, and the static stiffness, along
a chosen loading direction. We provide a sensitivity analysis to optimize these
properties with respect to design parameters of the structure, that include
localized masses at specific positions. We demonstrate the method on five
different periodic, three dimensional lattices, to calculate bounds on the
longitudinal wave speed as a function of their density and stiffness. We then
perform experiments on 3-D printed structures, to validate our numerical
simulations. The tools developed in this work can be used to design lightweight
and stiff materials with optimized acoustic impedance for a plethora of
applications, including ultrasound imaging, wave filtering and waveguiding.
