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Found in: Page 506

### Fundamentals Of Physics

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?

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

See the step by step solution

## Step 1: Given data

Frequency of diagnostic ultrasound is

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

## Step 2: Determining the concept

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

The expression for the velocity is given by,

${{\mathbit{v}}}_{\mathbf{a}\mathbf{i}\mathbf{r}}{\mathbf{=}}{{\mathbit{\lambda }}}_{\mathbf{ }\mathbf{a}}{\mathbf{×}}{\mathbit{f}}$

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

## Step 3: (a) Determining the wavelength in air of such a sound wave

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

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

Therefore, the wavelength can be calculated as,

${v}_{air}={\lambda }_{ a}×f$

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

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

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

## Step 4: (b) Determine the wavelength of the wave in tissue

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

. $f=4.50×{10}^{\text{6}}\text{Hz}$

Therefore, wavelength can be calculated as,

${v}_{t}={\lambda }_{ t}×f\phantom{\rule{0ex}{0ex}}⇒{\lambda }_{ t}=\frac{{v}_{t}}{f}$

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

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

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