Cerebral vasospasm (CVS) is the progressive narrowing of cerebral arteries following a haemorrhage in the brain. The primary diagnostic method (Transcranial Doppler) quantifies alterations of blood velocities in the affected vessels, but has low sensitivity when the condition affects the peripheral vessels. By exploiting the properties of the pulse waves that propagate through our system, we have used a 1D modelling approach to represent and analyse this physics and characterize the effects of different types of vasospasm on waveform features at several locations within a typical cerebral network. A sensitivity analysis empowered by the use of a Gaussian process statistical emulator was used to identify features of these waveforms that may have strong correlations with the condition. In particular, we have found that the maximum rate of pressure change can be much more effective than blood velocity for stratifying typical manifestations of the condition and its progression. The results and methodology of this study have the potential not only to improve the diagnosis and monitoring of vasospasm, but also to be used in the treatment of many other cardiovascular diseases where these cardiovascular pressure waves carry hidden data that can be decoded to provide disease characterisation.