<p>A long-wavelength approximation of Fluid-Structure-Interaction (FSI) waves in vascular networks is developed, the relevance of which is first discussed. Long-wavelength FSI-waves continuity condition within networks are derived. The resulting coupled wave system is solved within any network generalizing a recently proposed quantum-graph eigen-mode decomposition approach. The spectral condition associated with the secular matrix evaluation is derived, providing the intrinsic pulsating modes of arterial pulse waves. The theory is applied to the circle of Willis arterial network for which the spectrum is explicitly computed and compared with clinical observations related to the intra-cranial pressure dynamics. Retrieving the cardiac frequency modes from the observed pulse signal produce consistent results between the new modes predictions and observations. This leads to a new understanding of intra-cranial pressure frequency content, related to pressure pulse propagation within the arterial network, containing relevant clinical information about the arterial’s physiological and mechanical response.</p>