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Q11P

Expert-verifiedFound in: Page 506

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**Question: Diagnostic ultrasound of frequency is 4.50 MHz 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 , what is the wavelength of this wave in tissue?**

**Answer**

** **

- The wavelength of ultrasound in air is ${\lambda}_{a}=76.2\mu m$.
- The wavelength of ultrasound in tissue is ${\lambda}_{t}=0.333nm$.

** **

Frequency of diagnostic ultrasound is

.$f=4.5\text{0MHz}\phantom{\rule{0ex}{0ex}}=4.50\times {10}^{\text{6}}\text{Hz}\phantom{\rule{0ex}{0ex}}$

** **

By using the relation between velocity and frequency, calculate the wavelength in air and tissue.

The expression for the velocity is given by,

${{\mathit{v}}}_{\mathbf{a}\mathbf{i}\mathbf{r}}{\mathbf{=}}{{\mathit{\lambda}}}_{\mathbf{\u200a}\mathbf{a}}{\mathbf{\times}}{\mathit{f}}$

Here, v* * is the velocity of the sound in air,* f *is the frequency and ${{\lambda}}_{\u200aa}$is the wavelength.

** **

Velocity of sound in air is ${v}_{air}=3\text{43m/s}$ and frequency of diagnostic ultrasound is

. $f=4.50\times 1{\text{0}}^{\text{6}}\text{Hz}$

Therefore, the wavelength can be calculated as,

${v}_{air}={\lambda}_{\u200aa}\times f$

Substitute$343\u200am/sfor{V}_{air,}4.50\times {10}^{\text{6}}\text{Hz for f}$ into the above equation,

$343={\lambda}_{\u200aa}\times 4.50\times {10}^{\text{6}}\phantom{\rule{0ex}{0ex}}{\lambda}_{\u200aa}=\frac{343}{4.50\times {10}^{6}}\phantom{\rule{0ex}{0ex}}{\lambda}_{\u200aa}=76.22\times {\text{10}}^{\text{- 6}}\text{m}\phantom{\rule{0ex}{0ex}}{\lambda}_{\u200aa}=76.22\mu m\phantom{\rule{0ex}{0ex}}$

Hence, the wavelength of ultrasound in air is${\lambda}_{\u200aa}=76.22\mu m$ .

Velocity of sound in tissue is ${v}_{t}=150\text{0m/s}$ and frequency of diagnostic ultrasound is

. $f=4.50\times {10}^{\text{6}}\text{Hz}$

Therefore, wavelength can be calculated as,

${v}_{t}={\lambda}_{\u200at}\times f\phantom{\rule{0ex}{0ex}}\Rightarrow {\lambda}_{\u200at}=\frac{{v}_{t}}{f}$

Substitute $150{\text{0m/s for v}}_{t}4.50\times {10}^{\text{6}}\text{Hz for v}t$into the above equation,

${\lambda}_{\u200at}=\frac{1500}{4.50\times {10}^{6}}\phantom{\rule{0ex}{0ex}}=333.33\times {10}^{-6}\text{m}\phantom{\rule{0ex}{0ex}}=0.\text{333mm}$

Hence, the ultrasonic wavelength on tissue is $\lambda =$0.333mm

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