A method is described for measuring and examining ocean waves in a way which allows their amplitude and period to be determined with some precision. Data obtained in this way are compared with meteorological charts of the ocean in an attempt to assess the velocity of propagation of swell over long distances. A critical estimate is only possible when the meteorological conditions are sufficiently simple, but in one selected example it appears that the velocity of propagation is within 5% of the value prescribed by hydrodynamical theory. The evidence in more complicated instances does not disagree with this result, but does not permit of such an exact interpretation. The waves are measured by the fluctuating pressure which they produce upon an instrument laid on the sea bed in shallow water near the coast. The resulting curves are examined by a machine which draws the frequency spectrum of the recorded waves. The information given by these spectra is combined with the information of wind strength given by the meteorological charts to form a 'propagation diagram' whose appearance is a test of the validity of the theoretical group velocity. A suitable theoretical basis is given to the work. Three examples are discussed in detail: a depression in the North Atlantic, a tropical storm off the coast of Florida and a train of swell which appears to have originated in a storm off Cape Horn. The swell in these examples had travelled to Land's End distances of 1200, 2800 and 6000 miles respectively. The paper deals only with the velocity of propagation of the swell and does not discuss how the amplitude of the swell may depend upon the distance the swell has travelled or the wind strength and fetch in the generating area.