# Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures
## 1 Introduction  
Many conventional techniques applied in nondestructive testing are based on an active mode, in which testing loadings are applied during testing to deliver signals or energy from the outside to the test body. In contrast, the acoustic emission (AE) technique is a passive method that does not require the application of external energy to the test structure, as AE is generated by a material as a result of a sudden release of energy (other than heat) from localized sources within the solid, in turn due to a failure of lattice vibrations in materials. AE sources are usually classified as primary or secondary. Primary AE sources include material degradations related to deformation and fracture development, whereas leak, flow, and the fabrication process are secondary AE sources. Due to this unique characteristic, AE techniques are uniquely applicable to structure health monitoring (SHM) [1, 2, 3, 4]. Among the primary AE sources, crack formation and growth are the most important practical nondestructive testing (NDT) because the detecting and monitoring of these failures can prevent or slow further damage. Thus, in SHM, the point source (PS) is adapted as an AE excitation source; for example in seismic displacements, crack fracture and cleavage, and concentrated vertical step forces [5, 6, 7, 8, 9, 10]. The displacements generated by a PS were first introduced by [PERSON] [11]. Later, [PERSON] presented mathematical formulas for displacements in spherical geometry by defining it as a concentrated force (CF) loaded at a point [12]. In mathematical formulas, Helmholtz potentials for the displacements were derived in terms of scalar potentials generated by PS excitation in an infinite domain [12, 13, 14]. Although the AE generated by the PS is important for characterizing real signals observedin practical SHM, theoretical modeling was limited to spherical geometries in an infinite domain. To our knowledge, no theoretical work on AE excited by an internal PS in cylindrical geometries was reported in the literature. Most theoretical works on elastic wave properties in cylindrical geometries focused only on wave propagation and scattering situations with or without external perturbations [15; 16; 17; 18; 19; 20; 21; 22; 23]. In linear elastodynamics, the Navier-Lame (NL) equation is the most popular and effective method for solving the displacement fields in elastic media. In cylindrical coordinates, the NL equation was solved by combining three potentials responsible for one compression (P) wave and two shear waves (horizontally and vertically polarized; SH and SV, respectively), using the models proposed by [PERSON] [24] and [PERSON] [25], respectively. The basic difference between the two models is that the compression and shear parts are separated in one of them, but not the other. The two models were examined comprehensively for the case with no body forces by [PERSON] et al. [15; 18] and [PERSON] et al. [20; 21]. Additionally, [PERSON] et al. developed a method to find exact solutions for axisymmetric wave propagation in functional cylinders by matching continuity and boundary conditions at layer junctions [26; 27]. Previously, we constructed the NL equation specifically for displacements generated by PS excitation in a transversely isotropic cylinder (TIC) [28] and a two-dimensional plate [29]. As an internal defect, the CF acting in a preferent direction had both spatial and temporal properties, represented by the delta function and time-dependent CF vector. As the first task, Green's function for the delta function was determined in a given geometry. Green's function contributes the CF distribution in a given domain. For TIC, the CF vector contributes to the temporal Helmholtz potentials for P, SV, and SH waves based on the [PERSON] and [PERSON] model. Contrary to ordinary potentials, these potentials are characterized by the nature of the material defect.