  Water keeps puzzling scientists because of its numerous properties which
behave oppositely to usual liquids: for instance, water expands upon cooling,
and liquid water is denser than ice. To explain this anomalous behaviour,
several theories have been proposed, with different predictions for the
properties of supercooled water (liquid at conditions where ice is more
stable). However, discriminating between those theories with experiments has
remained elusive because of spontaneous ice nucleation. Here we measure the
sound velocity in liquid water stretched to negative pressure, and derive an
experimental equation of state, which reveals compressibility anomalies. We
show by rigorous thermodynamic relations how these anomalies are intricately
linked with the density anomaly. Some features we observe are necessary
conditions for the validity of two theories of water.
