MEMBRANES AND PLATES 155 



may assume a great variety of forms, owing to the superposition 

 of different modes having the same frequency. The gravest 

 mode of a free plate is that in which the nodal lines form a 

 cross through the centre, with arms parallel to the sides. 

 The figure shews other cases in which possible forms can be 

 assigned to the nodal lines from considerations of symmetry. 



57. Vibrations of Curved Shells. 



When we proceed to the vibrations of curved plates, or 

 shells, we meet with further complications due to the fact that 

 no absolutely sharp line can be drawn between flexural and 

 extensional modes. This has been already exemplified in the 

 case of the ring ( 51). It appears, however, that as the thick- 

 ness is (in imagination) reduced the normal modes tend to fall 

 into two distinct categories. In one of these the frequencies 

 tend to definite limits, the deformations being mainly 

 extensional, and so analogous to the longitudinal vibrations 

 of a bar, where the dimensions of the cross-section were found 

 to be immaterial. In the second category the frequencies 

 diminish without limit, being ultimately proportional to the 

 thickness, as in the flexural vibrations of a bar or plate. 



It will be understood that, acoustically, the flexural vibra- 

 tions are alone of real interest. When the shape is one of 

 revolution about an axis, the nodal lines will evidently be 

 parallels of latitude and equidistant meridians. As in the case 

 of 51 these are not lines of absolute rest, the tangential motion 

 being there relatively at its greatest. This has an application 

 to bells. A theoretical calculation of the frequencies of an 

 actual bell is of course out of the question ; but it is somewhat 

 remarkable that no systematic experimental study appears to 

 have been made until the subject was taken up by Lord 

 Rayleigh in 1890. Some unexpected results were obtained. 

 To quote a typical case, the normal modes of a particular bell, 

 when arranged in ascending order of frequency, were found to 

 have the following numbers of nodal meridians and parallels, 

 and the pitches indicated : 



(4,0) (4,1) (6,?) (6,?) (8,?) 

 e c" f" + &"(, d'". 



