2/ Sound and the Ear 



Fundamental 



1 st Overtone v- ^^ -^ 2 nd Harmonic 



2 nd Overtone 



3 rd Overtone 



Fundamental 



3 r Harmonic 



4 Harmonic 



Figure 3. Resonances of strings. The characteristic or reso- 

 nant frequency is w n and the characteristic or eigenfunction is 

 ip n . The displacement can be described by 



y = ZMn e±iw " 

 where the A n 's are the amplitudes. The wave velocity 



where T is the tension and e is the mass per unit length. 



V) n 



Fundamental Fundamental Al 4r 



+n 



sing 



^XT"^ ft Overtone 3 rd Harmonic fZ Jf sin 



37TX 



21 



XX] 



2 nd Overtone 5 th Harmonic fl ff sin 5 ^ 



5 41 ^t 



Figure 4. Particle velocity for various overtones of a closed- 

 end organ pipe. See Figure 3 for definition of the symbols. 

 When the particle velocity has a node, the acoustic pressure 

 has an anti-node, and conversely. The external ear canal re- 

 sembles a closed-end organ pipe with a fundamental around 3 kc. 



