264 AUDITORY BIOPHYSICS 



On the hypothesis that adjoining modifications in tissue structure 

 reflect the usage to which they are put, the circles of maximum strength 

 in the eardrum indicate the regions of maximum deflection, and inter- 

 mediate regions nearly devoid of reinforcing material indicate regions 

 of minimum deflection. The question then arises: does the tympanic 

 membrane resemble a diaphragm subjected to partitioning into charac- 

 teristic nodes and antinodes following the distribution of the reinforce- 

 ments in the circular and radial fibrous layer? An answer may be 

 obtained by considering the eardrum as if it were a shallow conical shell. 



Modes of Vibration of a Conical Shell 



Owing to the difficulties involved in a mathematical analysis of a 

 vibrating shallow conical shell, the analysis of its vibrations will be pre- 

 sented with the aid of a model. 



An ideally rigid disk inserted in an infinite flat wall may be driven by 

 an alternating force of constant magnitude at all audible frequencies. 

 Under these circumstances it is found that the driving force is opposed 

 by (a) the inertia of the disk and a certain quantity of air set in motion by 

 the disk, and (b) the radiation of acoustic energy. The first acts like a 

 mass reactance, and the second like a mechanical resistance. The mass 

 of the disk is constant, but the mass of the air set in motion decreases 

 with rise in frequency and at high frequencies becomes negligibly small. 



If a disk is used as a model the resistance due to sound radiation is 

 negligibly small. The acoustic output of a 20-cm disk, for example, is 

 found to be constant up to 1000 cycles, after which it diminishes to 

 1/100 of this value at 8000 cycles. It is, therefore, unsuited for repro- 

 duction of speech or music because, in the upper register, the acoustic 

 output falls away too rapidly on account of the relatively large mass of 

 the disk with its corresponding reduction of velocity. For constant 

 acoustic output its velocity should be the same for all frequencies, but 

 unfortunately its velocity decreases inversely as its frequency. Since 

 acoustic power varies as the square of the velocity, its acoustic output 

 thus decreases as the square of the frequency. At high frequency its 

 acoustic output is therefore very small. 



Modifying the flat disks to a shallow cone structure improves the 

 output at high frequencies. A 20-cm (diameter), free-edged paper cone 

 driven as an electrodynamic loud speaker exhibits radial modes of vibra- 

 tion which start below 100 cycles (McLachlan [1936]). These may be 

 suppressed by crimping and reinforcing the edge. 



In commercial loud-speaker cones the radial modes of vibration are 

 suppressed as much as possible, leaving the concentric nodal circles 



