Ultrasound intensity in fat, muscle and bone

[songoku]

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

I_R / I = (Z₂ - Z₁)² / (Z₂ + Z₁)²

The Attempt at a Solution

a. intensity that passes through fat:
I_{pass fat} = I e^-(0.24)(0.4) = 0.908 I → correct based on the solution

b. intensity being reflected by muscle:
I_{reflect 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:
I_{reflect 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?

 

[anathema]

Thank you for your question regarding the transmission and reflection of ultrasound through layers of fat, muscle, and bone. Based on the equations and calculations provided, it appears that you have correctly determined the intensities that pass through, reflect off, and are transmitted through each layer. However, there is a small error in your calculation for the final intensity transmitted through the muscle-bone boundary.

In part f, you calculated the intensity transmitted through the muscle-bone boundary as 0.332 I - 0.121 I = 0.211 I. However, this is the intensity that is reflected by the bone, not transmitted. The correct calculation for the intensity transmitted through the muscle-bone boundary would be 0.332 I x e-(0.65)(4.35) = 0.121 I, as you correctly calculated in part e.

Therefore, the intensity transmitted through the muscle-bone boundary is indeed 0.121 I, as stated in the solution. The book is not wrong, but there was a small error in your calculation. Keep up the good work in your scientific studies!