Intermediate Physics for Medicine and Biology  Russell K. Hobbie & Bradley J. Roth
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Sound (or acoustics) plays two important roles in our 
x 
x + dx 

study of physics in medicine and biology. First, ani 



mals hear sound and thereby sense what is happening in 



their environment. Second, physicians use highfrequency 



sound waves (ultrasound ) to image structures inside the 

(a) 

body. This chapter provides a brief introduction to the 



physics of sound and the medical uses of ultrasonic imag 
ξ(x,t) 
ξ (x + dx,t) 

ing. A classic textbook by Morse and Ingard (1968) pro 



vides a more thorough coverage of theoretical acoustics, 



and books such as Hendee and Ritenour (2002) describe 
sn(x,t) 
sn(x + dx,t) 

the medical uses of ultrasound in more detail. 

(b) 

In Sec. 13.1 we derive the fundamental equation gov 





erning the propagation of sound: the wave equation. Sec 
FIGURE 13.1. An elastic rod. (a) The rod in its equilibrium 

tion 13.2 discusses some properties of the wave equation, 
position. (b) Each point on the rod has been displaced from its 

including the relationship between frequency, wavelength, 
equilibrium position by an amount ξ which depends on x and 

and the speed of sound. The acoustic impedance and 
t. As a result there is a normal stress sn which also depends 

its relevance to the reﬂection of sound waves are intro 
on x and t. 





duced in Sec. 13.3. Section 13.4 describes the intensity 



of a sound wave and develops the decibel intensity scale. 
rium position. In this section we consider sound waves 

The ear and hearing are described in Sec. 13.5. Section 
propagating along the x axis. The results can be general 

13.6 discusses attenuation of sound waves. Physicians use 
ized to three dimensions [See Morse and Ingard (1968)]. 

ultrasound imaging for medical diagnosis, as described in 
We ﬁrst consider an elastic rod, and then a ﬂuid in which 

Section 13.7. Ultrasonic imaging can provide information 
viscous e ects are not important. 

about the ﬂow of blood in the body by using the Doppler 



e ect, as shown in Sec. 13.8. 
13.1.1 Plane Waves in an Elastic Rod 




13.1 
The Wave Equation 
The simplest case to consider is an elastic rod which is 

forced to move longitudinally at one end. This results in 

In Chapter 1, we assumed that solids and liquids are in 
the propagation of a sound wave along the rod.1 We set up 

a coordinate system where x measures distance along the 

compressible. If a long rod were truly incompressible, a 
rod from a ﬁxed origin when no sound wave is traveling 

displacement of one end would instantly result in an iden 
along the rod. We also assume that the disturbance of 

tical displacement of the other end. In fact, the displace 
the rod depends only on the position along the rod, x, 

ment does not propagate instantaneously. It travels at the 



speed of sound in the rod. 
1This simple geometry assures that all motion is parallel to the x 

The propagation of sound involves small displacements 
axis. In general, motion in an elastic solid involves both longitudinal 

of each volume element of the medium from its equilib 
waves and transverse waves. 
