  We design extremely-thin acoustic metasurfaces, providing a versatile
platform for the manipulation of reflected pressure fields, that are
constructed from mass loads and stretched membranes fixed to a periodic rigid
framework. These metasurfaces demonstrate deeply subwavelength control and can
have thicknesses an order of magnitude less than those based around Helmholtz
resonators. Each sub-unit of the metasurface is resonant at a frequency tuned
geometrically, this tunability provides phase control and using a set of finely
tuned membrane resonators we create a phase-grating metasurface. This surface
is designed to exhibit all-angle negative reflections with the ratio of
wavelength, $\lambda$, to thickness, $h$, of $\lambda/h\approx 23.1$, and to
create a flat mirror using the phase profile of an elliptic reflecting mirror.
A further important acoustic application is to sound diffusers and we proceed
to design a deeply subwavelength membrane-based meta-diffuser that can be two
orders of magnitude thinner than the operating wavelength, i.e. thickness
$\approx\lambda/102$. This paves the way for developing advanced acoustic
metasurfaces with applicability to functional acoustic devices in sound-related
industries.
