Elastic tubes are widely used nowadays in fluid flow, and their importance no longer needs to be demonstrated. From the biological field up to the industrial domain, different types of elastic tubes are encountered and while many studies have been done, there still exist unresolved questions.

The first studies on elastic tubes presented a linear approach of the fluid flow therein with differences arising only in the consideration of the tube wall contribution [

1,

2]. The nonlinear studies that came after brought out the importance of the fluid convective term and particularly took into account the dynamical properties of the tube wall where the nonlinear stress–strain relationship is considered [

3–

6]. Most of the works that followed contributed to improve the modeling, in a more realistic way, of elastic tubes [

7–

9] and took into account viscosity, tapering aspect of the tubes [

5,

10], pathological cases of vessels and prostheses [

11].

Some studies on blood waves focused on the study of the behaviour of waves in the regions of discontinuities [

12,

13]. In general, any discontinuity zone present in an artery gives rise to the phenomena of wave transmission and reflection. These discontinuities are in general due to the change in the elastic properties of the arterial wall, diseases, bifurcations and prostheses.

The objective of this paper is to determine the expression of the transmission and reflection coefficients in the prosthesis using a nonlinear model of blood flow. This theory is based on the separation of pressure waves into their forward and backward running components proximal and distal to the prosthesis. We then use the same approach to investigate the characteristics of a reflectionless prosthesis.

In the study, we consider that the fluid in the vessel is Newtonian, viscous, homogeneous and incompressible. Furthermore, we take into account the variation of the radius and elasticity of the wall. The rest of the paper is therefore organized as follows: in Section

2, the physical and mathematical problems are formulated. In Section

3, the nonlinear model of the flow is used to determine the transmission and reflection coefficients of the prosthesis, and the conditions for a reflectionless prosthesis are derived accordingly. The last section is devoted to concluding remarks.