182 HERMANN VON HELMHOLTZ 



sound-producing agency. By means of the repeated applica- 

 tion of Green's theorem to four distinct spaces, he was enabled, 

 under certain assumptions as to the magnitudes involved, with- 

 out knowing the special form of opening or the motion of the 

 air within the opening, to deduce certain relations between 

 the plane and the hemispherical waves that spread out into 

 the remoter space ; and thus the unsolved problem of the influ- 

 ence of the open end upon plane waves was determined. 



In the first place it was found for the form of the waves 

 in the tube, that the maxima and minima of the vibrations, 

 i.e. their nodes and internodes, occurred at quarter wave- 

 lengths from each other, and that the phases of motion differed 

 by a quarter of a period at the maximal and minimal points. 

 Helmholtz termed the distance of the cross-section from a 

 point on the axis at a given definite distance from the mouth, 

 the reduced length of the tube, and found that the maxima 

 of vibration occurred throughout where the reduced length 

 was equal to an even multiple of the quarter wave-length, 

 while the surfaces of least motion, or nodal surfaces, occurred 

 on the contrary wherever the reduced length of the tube 

 equalled an uneven multiple of the quarter wave-length. 



After deducing this general law, by which the problem is 

 referred to the determination of the reduced length in the 

 different forms of tube, Helmholtz next proposes to discover 

 in what forms of tubes the aerial motion at the mouth, and the 

 reduced length, may be fully determined for sound-waves of 

 such great wave-length, that the dimensions of the opening 

 of the tube, its cross-section, and that of the part of the tube 

 that deviates from the cylindrical, vanish. 



Helmholtz contributed a supplement to these inquiries in the 

 lecture given on Feb. 27, 1863, to the Nat. Hist. Med. Verein, 

 'On the Influence of Friction in the Air upon the Motions 

 of Sound/ He returns in this to the theoretical differences 

 between real and reduced lengths in particular forms of the 

 mouth of tubes, since the theory for narrow tubes shows far 

 smaller differences than are actually found by experiment; 

 the correspondence was much closer when the friction of the 

 air is taken into consideration, which Helmholtz was able to do 

 on the basis of Stokes's investigations, as he had previously in 

 the case of fluids. 



