Ultrasound intensity in fat, muscle and bone

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Homework Statement


The specific impedance of fat, muscle and bone are 1.4 x 106 kg m-2 s-1 , 1.6 x 106 kg m-2 s-1 and 6.5 x 106 kg m-2 s-1 respectively. The linear absorption coefficients in fat and in muscle are 0.24 cm-1 and 0.23 cm-1 respectively.

A parallel beam of ultrasound of intensity I is incident on the layer of fat. Discuss quantitatively, in terms of I, the reflection and the transmission of the beam of ultrasound as it passes through the layer of fat of thickness 4.0 mm, into the muscle of thickness 43.5 mm and finally into the bone


Homework Equations


I = I0 e-kx

IR / I = (Z2 - Z1)2 / (Z2 + Z1)2


The Attempt at a Solution


a. intensity that passes through fat:
Ipass fat = I e-(0.24)(0.4) = 0.908 I → correct based on the solution

b. intensity being reflected by muscle:
Ireflect muscle = (0.2)2 / 9 I = 1/225 I

c. intensity passing through fat-muscle boundary = 0.908 I - 1/225 I = 0.904 I → correct based on the solution

d. intensity transmitted through muscle = 0.904 I x e-(0.23)(4.35) = 0.332 I → correct based on the solution

e. intensity being reflected by bone:
Ireflect bone = 0.332 I x (4.9)2 / (8.1)2 = 0.121 I

f. intensity transmitted through muscle - bone boundary = 0.332 I - 0.121 I = 0.211 I → wrong


The solution said 0.121 I transmitted into bone. I think 0.121 I is the intensity reflected, not transmitted, by the bone. Am I, or is the book, wrong?
 

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