Reflection and transmission of nonlinear waves from arterial branching Hilmi Demiray * Department of Mathematics, Faculty of Arts and Science, Isik University, Bu ¨ yu ¨ kdere Caddesi, 34398 Maslak, Istanbul, Turkey Received 23 April 2005; received in revised form 24 March 2006; accepted 17 May 2006 Available online 22 September 2006 Abstract In this work, treating the arteries as a prestressed thin walled elastic tube and the blood as an inviscid fluid, we have studied the reflection and transmission of nonlinear waves from arterial branching, through the use of reductive perturba- tion method. The reflected and the transmitted waves at the bifurcation point are calculated in terms of the incident wave. The numerical results indicate that the reflected wave is comparatively small whereas the transmitted waves in branches are comparable with the incident wave. This result is quite consistent with the experimental measurements [N. Sergiopulos, M. Spiridon, F. Pythoud, J.J. Meister, On wave transmission and reflection properties of stenosis, J. Biomechanics 26 (1996) 31–38]. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Elastic tubes; Reflected and transmitted waves 1. Introduction A remarkable feature of the arterial blood flow is its pulsatile character. The intermittent ejection of blood from the left ventricle produces pressure and flow pulses in the arterial tree. Experimental studies reveal that the flow velocity in blood vessels largely depends on the elastic properties of the vessel wall and they propagate towards the periphery with a characteristic pattern [1]. Theoretical investigations for blood waves have been developed by several researchers (c.f. [2–5]), through the use of weakly nonlinear theories. It is shown that the dynamics of the blood waves governed by the KdV equation, perturbed KdV equation and their modified forms. The solitary-wave model gives a plausible explanation for the peaking and steeping of the pulsatile waves in arteries. In all these works, the effects of arterial branching on flow characteristics have been neglected. In the process of solitary wave propagation in arteries, the transmission and the reflection of waves at arte- rial branching become very important. In order to elucidate the transmission and the reflection of blood flow at the arterial branching, Duan et al. [6–8] proposed a one-dimensional fluid-filled elastic tube model with branching, and they observed that the incident wave can be transmitted into each bifurcation without 0020-7225/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2006.05.015 * Tel.: +90 216 528 7172; fax: +90 216 712 14 74. E-mail address: demiray@isikun.edu.tr International Journal of Engineering Science 44 (2006) 1164–1172 www.elsevier.com/locate/ijengsci