Q68P - University Physics with Modern Physics

Question

BIO Ultrasound in Medicine. A 2.00-MHz soundwave travels through a pregnant woman’s abdomen and is reflected from the fatal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 72 beats per second are detected. The speed of sound in body tissue is 1500 m/s. Calculate the speed of the fatal heart wall at the instant this measurement is made.

Step-by-Step Solution

Answer

a) $\lambda$=0.625mm 

b) the corresponding wavelengths are 0.075 mm to 1.5 mm.

STEP 1: Given Data

Frequency emitted is $2.00\text{\hspace{0.17em}MHz}$ 

Sound’s speed is $1500\text{\hspace{0.17em}m}/\text{s}$

Number of beats per second are  $72$

STEP 2: Concept of the frequency of sound wave

The frequency of the sound wave is defined as the number of oscillation produced per unit time. It is given as-

 $\mathbf{f}\mathbf{=}\frac{v}{\lambda }$

where, v is the velocity , $\mathit{\lambda }$ is the wavelength and  $\mathit{f}$ is the frequency of sound wave.

STEP 3: Calculate the wavelength of the wave

The frequency of the ultrasound waves is by using equation $f=2.40\times {10}^6\text{\hspace{0.17em}Hz}$.The wavelength of the wave is $\lambda =\frac{v}{f}$

$\begin{array}{c}\lambda =\frac{1.5\times {10}^3\text{\hspace{0.17em}m/s}}{2.40\times {10}^6\text{\hspace{0.17em}Hz}}\\ =0.625\text{\hspace{0.17em}mm}\\ =0.625\times {10}^{-3}\text{\hspace{0.17em}m}\end{array}$

STEP 4: Calculation of wavelength

Range of the frequencies is $f_1=1.00\text{\hspace{0.17em}MHz}$  to $f_2=20\text{\hspace{0.17em}MHz}$ . The corresponding wavelength  $f_2$ to is smaller than the corresponding wavelength to $f_1$ , according the equation  $\mathrm{v}=\mathrm{\lambda f}$  as v does not change in the same medium. So,

${\lambda }_2=\frac{1.5\times {10}^3\text{\hspace{0.17em}m/s}}{\begin{array}{c}20.0\times {10}^6\text{\hspace{0.17em}Hz}\\ =0.075\times {10}^{-3}\text{\hspace{0.17em}m}\\ =0.075\text{\hspace{0.17em}mm}\end{array}}$

${\lambda }_1=\frac{1.5\times {10}^{3}m/s}{\begin{array}{c}1.00\times {10}^{6}Hz\\ =1.5\times {10}^{-3}m\\ =1.5mm\end{array}}$

 

Therefore, the corresponding wavelengths are $0.75 mm$ to  $1.5mm$.