X. ULTRASONIC VIBRATIONS 309 



ultrasound in liquids and solids while loudspeakers and sirens are 

 used for airborne ultrasound. 



5. Reflection and Transmission at Boundaries 



Whenever a plane sound wave strikes a boundary between two 

 media, part of the energy is reflected to the first medium and part is 

 transmitted through the second. If pi and Vi are the density and 

 velocity of propagation of the first medium and P2 and V2 those of the 

 second, then, if the wavelength of the sound is small compared to the 

 dimensions of the boundary, we have for the ratio of reflected to inci- 

 dent energy (1, p. 271): 



/. jr . T/ \ s 



(9) 



PiVi + P2V2/ 



This gives, for the percentage of sound energy reflected from the 

 boundary, water-glass 66%, oil-glass 70%, water-steel 85%, and 

 oil-steel 88%. For sound energy incident on an air-water interface 

 or vice versa, the above relation indicates that 99.9% would be re- 

 flected. 



Consider now a plate of thickness d, density p2 and velocity V2 im- 

 mersed in a liquid of density pi and velocity Vi. If a sound wave 

 present in the liquid strikes the plate at right angles, the ratio of re- 

 flected to transmitted energy becomes : 



'piVi 



R = 



\P2 



V-1 



4 . +2 '^ -J- ('^^ Pz^sV 



A2 \P2^ 2 pJ^l/ . 



(10) 



where X2 is the wavelength of the sound in the plate. It is easily 

 seen that, if the plate is infinitely thick, the above formula reduces to 

 the one previously given for two media. On the other hand, if d 

 is small compared to X2, virtually all the sound energy is transmitted 

 through the plate. Note also that, if the product pV for the plate 

 is the same as that for the medium, complete transmission results. 

 This criterion has been used in many instances in the selection of ma- 

 terials for underwater sound equipment. 



For intermediate thicknesses almost complete transmission will 

 result for d = 71X2/2, where n is an integer. Practically speaking, n 

 must be confined to small numbers (1 through about 10). This fact 

 is extremely important in biological work in which the ultrasonic 

 energy must traverse the glass wall of a beaker or flask in order to 



