Question

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?

Short answer

Answer

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

Step-by-step solution

Given data

Frequency of diagnostic ultrasound is

.$$\begin{gathered} f=4.5\text{0MHz} \\ =4.50\times {10}^{\text{6}}\text{Hz} \end{gathered}$$

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,

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

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

(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

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

Therefore, the wavelength can be calculated as,

$v_{air}={\lambda }_{ a}\times f$

Substitute$343 m/sfor{V}_{air,}4.50\times {10}^{\text{6}}\text{Hzforf}$ into the above equation,

$$\begin{gathered} 343={\lambda }_{ a}\times 4.50\times {10}^{\text{6}} \\ {\lambda }_{ a}=\frac{343}{4.50\times {10}^6} \\ {\lambda }_{ a}=76.22\times {\text{10}}^{\text{- 6}}\text{m} \\ {\lambda }_{ a}=76.22\mu m \end{gathered}$$

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

(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

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

Therefore, wavelength can be calculated as,

$$\begin{gathered} v_t={\lambda }_{ t}\times f \\ \Rightarrow {\lambda }_{ t}=\frac{v_t}{f} \end{gathered}$$

Substitute $150{\text{0m/sforv}}_{t}4.50\times {10}^{\text{6}}\text{Hzforv}t$into the above equation,

$$\begin{gathered} {\lambda }_{ t}=\frac{1500}{4.50\times {10}^6} \\ =333.33\times {10}^{-6}\text{m} \\ =0.\text{333mm} \end{gathered}$$

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