pressure increases, this reduces propeller efficiency and increases noise. The forces generated by the bubble collapse can cause permanent damage to the surfaces of the blade. Propeller thrust equation Single blade Taking an arbitrary radial section of a blade at r, if revolutions are N then the rotational velocity is . If the blade was a complete screw it would advance through a solid at the rate of NP, where P is the pitch of the blade. In water the advance speed, , is rather lower. The difference, or slip ratio, is: where is the advance coefficient, and is the pitch ratio, with being the diameter of the propeller. The forces of lift and drag on the blade, dA, where force normal to the surface is dL: where: These forces contribute to thrust, T, on the blade: where: As , From this total thrust can be obtained by integrating this expression along the blade. The transverse force is found in a similar manner: Substituting for and multiplying by r, gives torque as: which can be integrated as before. The total thrust power of the propeller is proportional to and the shaft power to . So efficiency is . The blade efficiency is in the ratio between thrust and torque: showing that the blade efficiency is determined by its momentum and its qualities in the form of angles and , where is the ratio of the drag and lift coefficients. This analysis is simplified and ignores a number of significant factors including interference between the blades and the influence of tip vortices. Thrust and torque The thrust, T, and torque, Q, depend on the propeller's diameter, D, revolutions, N, and rate of advance, , together with the character of the fluid in which the propeller is operating and gravity. These factors create the following non-dimensional relationship: where is a function of the advance coefficient, is a function of the Reynolds' number, and is a function of the Froude number. Both and are likely to be small in comparison to under normal operating conditions, so the expression can be reduced to: For two identical propellers the expression for both will be the same. So with the propellers , and using the same subscripts to indicate each propeller: For both Froude number and advance coefficient: where is the ratio of the linear dimensions. Thrust and velocity, at the same Froude number, give thrust