Mathematical modeling of the orchestral timpani

The orchestral timpani is modeled as an ideal membrane capping a rigid kettle in an air medium. Room acoustics are ignored. Green function and boundary integral methods are used to investigate the effects of air loading and kettle shape on the vibrational spectrum. It is shown that experimental data on timpani modal vibrations may be fit quantitatively. The relationship among timpani membrane and kettle parameters which give modal frequency ratios f$\sb{11}$:f$\sb{21}$:f$\sb{31}$:f$\sb{41}$ very close to the musically desirable values 2:3:4:5 are investigated using the theoretical model