To determine which transition will result in the largest change in the wavelength of the ultrasound wave, we need to consider the relationship between speed, frequency, and wavelength. The wavelength (\(\lambda\)) of a wave is given by the equation:
\[ \lambda = \frac{v}{f} \]
where \(v\) is the speed of the wave and \(f\) is its frequency. When the ultrasound wave moves from one tissue to another, its speed changes, but the frequency remains constant. Thus, the change in wavelength can be expressed as:
\[ \Delta \lambda = \lambda_2 - \lambda_1 = \frac{v_2}{f} - \frac{v_1}{f} = \frac{1}{f}(v_2 - v_1) \]
This shows that the change in wavelength is proportional to the change in speed of the ultrasound wave.
We can calculate the change in speed for each of the transitions:
-
Fat to Liver:
- Speed in Fat = 1,450 m/s
- Speed in Liver = 1,550 m/s
- Change in speed = 1,550 - 1,450 = 100 m/s
-
Muscle to Kidney:
- Speed in Muscle = 1,580 m/s
- Speed in Kidney = 1,560 m/s
- Change in speed = 1,560 - 1,580 = -20 m/s
-
Blood to Fat:
- Speed in Blood = 1,570 m/s
- Speed in Fat = 1,450 m/s
- Change in speed = 1,450 - 1,570 = -120 m/s
-
Liver to Muscle:
- Speed in Liver = 1,550 m/s
- Speed in Muscle = 1,580 m/s
- Change in speed = 1,580 - 1,550 = 30 m/s
Now, we can summarize the changes in speed:
- Fat to Liver: 100 m/s
- Muscle to Kidney: -20 m/s
- Blood to Fat: -120 m/s
- Liver to Muscle: 30 m/s
The largest change in speed occurs when transitioning from Blood to Fat (with a change of -120 m/s).
Therefore, the transition that results in the largest change in the wavelength of the ultrasound wave is:
Blood to Fat.
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