52 MATTER WAVES: SOUND AND ULTRASOUND 



expansion or contraction increases with increasing applied voltage. Quartz 

 and barium titanate are currently in wide use. If the applied voltage is 

 varied, the crystal shape varies accordingly, or vibrates, and the matter 

 wave so established is transmitted by contact with the medium. The ampli- 

 tude of the vibration is higher the higher the vibrating voltage applied. The 

 frequency of vibration follows that of the electrical signal, if the crystal is not 

 too big. Figure 3-2 illustrates these points. 



Apparatus with output which ranges from a few to a million cycles per 

 second, and from next to nothing up to a few hundred watts per square 

 centimeter of crystal, has been built and used. 



Constructed with a concave radiating surface (Figure 3-2 (d)), an array of 

 piezoelectric crystals, if properly oriented, can be made to focus an intense 

 beam of matter waves at a point a few centimeters from the radiating sur- 

 face. For example, in recent therapeutic work beams of 1 Mc (1,000,000 

 cps) were focused on a small target, and delivered energy at a rate (inten- 

 sity) of 8 kw/cm 2 of cross-section of the target ! 



Absorption 



If waves are diverging, or being dissipated or scattered, the important gen- 

 eral rule, called the "inverse square law," is obeyed. It says simply that the 

 intensity, /, decreases as the distance from the source gets larger, in such a 

 manner that if, for example, the distance between source and receiver is 

 doubled, the intensity at the receiver falls to only one quarter. Quantita- 

 tively, 



I(x) oc \/x 2 



where I(x) is the intensity at any distance, x, away from the source. See 

 Figure 3-3. 



If a parallel beam of matter waves is absorbed by the medium, the rate of 

 absorption at a point is proportional to the intensity at that point; or 



dl/dx = -kl 



which integrates (see Chapter 1) to 



/ = I e-* 



if / is the value of / where x = 0. 



For the case in which the waves are diverging and also being absorbed, a 

 linear combination of the inverse square law and the absorption law applies. 

 The energy absorbed from the matter-wave beam by the medium contri- 

 butes to the thermal motion of the molecules of the medium. The absorp- 

 tion coefficient, k, is intimately related to several physical properties of the 

 medium. 



