Figure 13.5

In Chapter 13 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss imaging using ultrasound. One of the key concepts in this chapter is the reflection of an ultrasonic wave at the boundary between tissues having different acoustic impedances. The acoustic impedance is equal to the square root of ρ/κ, where ρ is the density and κ is the compressibility. An ultrasonic image is formed by recording echoes of an ultrasonic wave and using the timing of those echoes to determine the distance from tissue boundaries.

Figure 13.5 from Intermediate Physics for Medicine and Biology.

Figure 13.5 shows schematically how part of a pressure wave approaching a surface (the incident wave) passes through the surface (the transmitted wave) and part travels back toward the wave’s source (the reflected wave, or echo). When you look closely, however, there is something odd about this figure: the transmitted wave has a larger amplitude than the incident wave. How can this be? Doesn’t this violate some conservation law? If we consider particles rather than waves, we would expect that if 100 particles were incident on a surface, perhaps 75 of them would be transmitted and 25 reflected. Figure 13.5 seems to imply that 100 incident particles result in 133 transmitted and 33 reflected! How?

The figure is not wrong; the problem is with our intuition. Pressure is not a conserved quantity. There is no reason to expect the sum of the pressures of the transmitted and reflected waves to equal the pressure of the incident wave. The amplitudes are consistent with equations 13.26 and 13.27 in IPMB relating the three waves. Yet there is a conserved quantity, one we all know: energy.

The intensity of a wave is the energy per unit area per unit time. The intensity I, pressure p, and acoustic impedance Z are related by equation 13.29: I = p²/2Z. The transmitted wave in Fig. 13.5 has a pressure amplitude that is 1.33 times the amplitude of the incident wave, but it is moving through a tissue that has twice the acoustic impedance (the caption says that for this figure, Z₂ = 2 Z₁). For simplicity, take the acoustic impedance on the left (the incident side of the boundary, region 1) to be Z₁ = 0.5 and the amplitude of the incident wave to be p_i = 1 (let’s not worry about units for now, because our goal is to compare the relative intensities of the three waves). In this case, the intensity of the incident wave is equal to one. If the transmitted pressure is 1.33 and the acoustic impedance on the right (region 2) is Z₂ = 1 (twice Z₁), then the transmitted intensity is (1.33)²/2 = 0.89. The reflected wave has amplitude 0.33, and is propagating through the tissue on the left, so its intensity is (0.33)²/1 = 0.11. The sum of the intensities of the transmitted and reflected waves, 0.89 + 0.11, is equal to the intensity of the incident wave. Energy is conserved! The figure is correct after all.

Here is another way to think about intensity: it’s one half of the product of the pressure times the tissue speed. When I say “tissue speed” I don’t mean the propagation speed of the wave, but the speed of the tissue itself as it oscillates back and forth. The acoustic impedance relates the pressure and tissue speed. The large pressure of the transmitted wave is associated with a small tissue speed. The transmitted wave in Fig. 13.5 looks “big” only because we plot the pressure. Had we plotted tissue speed instead, we would not be wondering why the transmitted wave has such a large amplitude. We would, however, be scratching our head about a funny phase shift in the reflected wave, which the version of the figure showing the pressure hides.

So, Fig. 13.5 is correct. Does that mean it looks exactly the same in the 5th edition of IPMB (due out this summer)? No, we did change the figure; not to correct an error, but to emphasize another point. Figure 13.5, as presently drawn, shows the wavelength of the transmitted wave to be the same as the wavelength of the incident wave. The wavelength does not depend on the acoustic impedance, but it does depend on the wave speed (the propagation speed of the wave itself, usually given the symbol c, which is not the same as the speed of the oscillating tissue). The wave speed is equal to the square root of the reciprocal of the product of the density and compressibility. One can cook up examples where two tissues have the same wave speed but different acoustic impedances. For instance (again, not worrying about units and only comparing relative sizes), if the tissue on the left had twice the compressibility and half the density of the tissue on the right, then the left would have half the acoustic impedance and the same wave speed as the right, just as shown in Fig. 13.5. But tissues usually differ in compressibility by a greater factor than they differ in density. If we assume the two regions have the same density but the right has one-fourth the compressibility, then Z₂ = 2 Z₁ as before but also c₂ = 2 c₁, so the wavelength is longer on the right. In the 5th edition, the figure now shows the wavelength longer on the right.

What is the moral to this story? Readers (and authors) need to think carefully about illustrations such as Fig. 13.5. They tell a physical story that is often richer and more complicated than we may initially realize.