The phenomena occurring when an uncased explosive charge is detonated in a fluid medium are examined by hydrodynamical methods. Attention is focused chiefly on the pressure and velocity distributions in the gaseous products of the explosion, which expand laterally behind the detonation wave as it travels down the charge, the results being shown in graphical form. To simplify the problem, the charge, and the gas and fluid fields, were treated as two-dimensional. The hydrodynamical equations were solved numerically using the method of characteristics. This dates back to Monge, but it is only recently that it has been applied to the numerical solution of hyperbolic equations. The methods of numerical integration used in this paper are similar to those developed early in the war by the Research Section of the External Ballistics Department, Ordnance Board, for determining the velocity distributions around projectiles moving at supersonic speeds. The nature of the boundary conditions made it necessary to find explicit theoretical formulae for the gas field near the charge, and the analysis involved is given at length. For the problem in which the surrounding medium is air, the shape and position of the shock waves set up by the explosion are calculated. The shock waves are found to be straight to the nominal accuracy of the calculations (1 in 5000) for six charge widths from their intersections with the block of explosive.