GENERAL THEORY OF SOUND WAVES 203 



The real meaning of the property which differentiates the 

 present type of motion from all others is most clearly expressed 

 in terms of the "circulation" round a closed curve. If we divide 

 the curve into infinitesimal linear elements, and multiply the 

 length of each element by the tangential component of the 

 velocity, estimated always in the same direction round the 

 curve, the result is the "circulation" referred to. It may be 

 denoted by 



\( u-^- +v -^ -\-w-j-Jds, or \(udx + vdy + wdz). ...(8) 



On the present hypothesis the tangential velocity is d<f>/ds, and 

 the integral of this, taken round the circuit, is zero, the first and 

 last values of < being the same. The circulation is therefore 

 zero in every circuit which can be drawn in the region in 

 question. For a reason which may be understood by reference 

 to the case of an infinitesimal circuit, the type of motion now 

 under consideration is called "irrotational." The name has the 

 advantage of calling attention to a geometrical property rather 

 than to an analytical form of expression. 



A dynamical interpretation can also be given to the 

 velocity-potential. The equations (3), when written in the 

 forms 



p u = pfi^/dx, p v = - pdd<f>/dy, p w = pfifydz, (9) 



shew that < is the potential per unit mass of a system of 

 extraneous impulsive forces which would generate the actual 

 motion of the fluid instantaneously from rest. 



The theorem as to the persistence of the irrotational character 

 is most important; but it is necessary to observe the restrictions 

 under which it has been proved. It was implied, in the first 

 place, that the fluid was frictionless, and this is essential. 

 Again the medium has been supposed free from extraneous 

 forces, but the restriction is easily removed in the case of forces 

 which, like gravity, have a potential (per unit mass). Finally, 

 the assumption has been made that the motion is infinitely 

 small. This simplifies the proof, and covers most cases which are 

 of interest in acoustics. A more rigorous investigation would 

 shew that the circulation is (under the above condition) still 



