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Chapter 19: Problem 3

Diagnostic ultrasound of frequency \(4.50 \mathrm{MHz}\) is used to examine tumors in soft tissue. \((a)\) What is the wavelength in air of such a sound wave? \((b)\) If the speed of sound in tissue is \(1500 \mathrm{~m} / \mathrm{s}\), what is the wavelength of this wave in tissue?

Short Answer

Expert verified
The wavelength of the ultrasound in air is approximately \(0.076 \, \mathrm{m} \) and in tissue, it is approximately \(0.33 \, \mathrm{m}\).

Step by step solution

01

Calculate The Wavelength in Air

First, calculate the wavelength of the ultrasound in air. To do this, use the formula that relates the speed of sound, frequency, and wavelength: \( v = f \lambda \). The speed of sound in air is approximately \( v = 343 \, \mathrm{m/s} \) and the frequency \( f \) is given as \( 4.5 \, \mathrm{MHz} = 4.5 \times 10^{6} \, \mathrm{Hz} \). Rearrange the equation for \( \lambda \) to get \( \lambda = v / f \). Substitute the known values into this equation to calculate the wavelength.
02

Calculate The Wavelength in Tissue

Second, calculate the wavelength of sound in the tissue. Again, use the wave formula \( v = f \lambda \). This time the speed of sound in tissue (as provided) is \( v = 1500 \, \mathrm{m/s} \), and the frequency is still \( 4.5 \, \mathrm{MHz} = 4.5 \times 10^{6} \, \mathrm{Hz} \). Rearrange the equation for \( \lambda \) to get \( \lambda = v / f \). Substitute the known values into this equation.
03

Interpret the Results

The results will show that the wavelength is different for the ultrasound in air and tissue. The wave travels faster and thus has a shorter wavelength in tissue. This is crucially important in medical applications, as the difference in wavelengths allows for the production of clear ultrasound images.

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Most popular questions from this chapter

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