## 1 Introduction  
The phenomenon of quantum entanglement [1] has generated great scientific interest and value right from the beginning of quantum mechanics. The paradox suggested by [PERSON], [PERSON], and [PERSON] was codified into a verifiable experimental inequality by [PERSON] [2]. Quantum entanglement can be considered as combining two characteristics of quantum system, namely nonlocality and nonseparability. The realization of nonlocality is uniquely the province of quantum phenomena; however, recently, classical systems able to capture the characteristic of nonseparability [3; 4; 5; 6; 7] between different degrees of freedom of the same physical manifestation. In particular, in the field of optics, degrees of freedom of photon states that span different Hilbert spaces can be made to interact in a way that leads to local correlations [5; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. In contrast, little attention has been paid to the nonseparability of sound waves; yet, remarkable new behaviors of sound, analogous to quantum physics, such as the notions of elastic pseudospin [18; 19; 20; 21; 22; 23; 24; 25; 26; 27] and Zak/Berry phase [28; 29; 30; 31; 32; 33; 34; 35; 36], are emerging. The concept of pseudospin, when one exists, is useful for understanding many fundamental quantum-like phenomena, such as the anomalous quantum Hall effect [21]. In 1932, [PERSON] originally introduced the concept of pseudospin when he described the structure of the atomic nucleus as composed of neutrons and protons and modeled as two states of the same particle [37]; hence, a pseudospin is a coherent superposition of states. The pseudospin was traditionally considered as an unmeasurable quantity. Recently, however, it has been suggested that pseudospin is a real angular momentum that might manifest itself as an observable quantity. Unlike the quantum spin, which is an intrinsic property of a particle, an elastic pseudospin is a property induced by the material that supports the elastic wave; hence, the state of superposition of the pseudospin is observable and measurable without wave function collapse. The concept of pseudospin has recently been introduced in various topological systems [38]. Building on the similar principles of photonic [39; 40] and plasmonic [41] systems, we proposed theoretically the concept of a one-dimensional (1D) elastic pseudospin. We have shown an analogy between the propagation of elastic waves on an elastic pseudospin (composed of elastically coupled 1D waveguides) and quantum phenomena [18; 23; 24; 25; 26]. More specifically, the projection on the direction of propagation of elastic waves in this pseudospin is isomorphic to the spin of a quantum particle. The pseudospin states of elastic waves in these systems can be described via a Dirac-like equation and possess spinor amplitudes. In addition, for an elastic pseudospin that is constituted of parallel arrays of elastically coupled 1D waveguides, the amplitude also spans an \(N\) dimensional Hilbert subspace, where \(N\) is the number of waveguides, and represents spatial mode degrees of freedom across the array of waveguides [42; 43; 44; 45]. These two degrees of freedom can be used to create classical nonseparability, i.e., the nonseparable superposition of elastic states in the form of linear combinations of tensor products of spatial modes and spinor amplitudes, so-called elastic Bell states. Ref. [44] reported the experimental preparation, measurement and tunability of elastic Bell states in a single system composed of three coupled waveguides. The current study expands significantly that report by analyzing elastic waves in \(N=1\), \(2\) and \(3\) waveguide systems. Particular attention is paid to the pseudospin character of elastic waves (via their dispersion character) in the two and three coupled waveguides systems. Furthermore, we report here an extensive theoretical and numerical analysis of pseudospin and nonseparable states in coupled waveguides to shed light on the experimental findings.