By transformation optics, we construct and investigate three-dimensional acoustic cloaking devices for which the device itself and the cloaked region can be mapped into two nested $l^p$-balls ($1\leq p<\infty$). Finite energy solutions for these cloaking devices are studied in weighted Sobolev spaces with singular weights. We show that the problems in the cloaking layer and the cloaked region can be decoupled. The problem in the cloaking layer has the same exterior Cauchy data as the one in the background medium, making the cloaking device together with the cloaked region invisible to exterior measurements. The cloaking works at any fixed frequency and can deal with sources/sinks as well. Finally, a conforming finite element method incorporating the Heaviside step function is proposed for simulating the devices. Numerical experiments illustrate the effectiveness of this discretization.