Efficient source coding is desired for any data storage and transmission. It could be enabled by adopting a transform inspired by natural phenomena. Based on the mechanical vibration models, a family of bases applicable to data compression is constructed. The eigenvectors of vibrating thin square plates, which are composed of sinusoidal and hyperbolic functions, are used to construct these real-orthonormal bases. Our analyses show that their attributes and performance are comparable to those of the widely used Discrete Cosine Transform. Since compressive sensing is a way to build the data compression directly into the acquisition, we propose to apply the set of low-frequency atoms of these bases that carry the main portion of the signal energy as sensing patterns. Thus, the measurement ensemble contains the high-energy Sinusoidal-Hyperbolic Transform's (SHT's) coefficients of the scene under-view. This sampling method leads to high fidelity reconstruction along with good efficiency in encoding and decoding. The dictionary matrix that is made by the SHTs and a non-trigonometric sparsifying basis like Hadamard is well-conditioned for the pursuit algorithm. Both subjective and objective evaluation of the reconstruction results validates the effectiveness of our method. Compared to the random Gaussian sensing patterns, for the same Compression Ratio (CR), the proposed sampling method results in images with significantly higher fidelity. The sensing patterns are also shown to be robust against Gaussian and Poisson noise. Application of our scheme to image compression is also discussed.