Counting open nodal lines of random waves on planar domains

We compute the asymptotic expectation of the number of {\em open} nodal lines for random waves on smooth planar domains. We find that for both the long energy window $[0,\lambda]$ and the short one $[\lambda,\lambda+1]$ the expected number of open nodal lines is proportional to $\lambda$, asymptotically as $\lambda\to\infty$. Our results are consistent with the predictions in the physics literature made by Blum, Gnutzmann and Smilansky \cite{BGS}.