Question

(a) Audible wavelengths. The range of audible frequencies is from about 20 Hz to 20,000 Hz. What is the range of the wavelengths of audible sound in air? (b) Visible light. The range of visible light extends from 380 nm to 750 nm. What is the range of visible frequencies of light? (c) Brain surgery. Surgeons can remove brain tumors by using a cavitron ultrasonic surgical aspirator, which produces sound waves of frequency 23 kHz. What is the wavelength of these waves in air? (d) Sound in the body. What would be the wavelength of the sound in part (c) in bodily fluids in which the speed of sound is 1480 m>s but the frequency is unchanged?

Answer 1

(a) 0.017 m - 17.2 m

The wavelength of a sound wave is given by

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

where

v = 343 m/s is the sound speed in air

f is the frequency

For the wave of frequency f = 20 Hz, we have

$\lambda=\frac{343 m/s}{20 Hz}=17.2 m$

while for the wave of frequency f = 20,000 Hz, we have

$\lambda=\frac{343 m/s}{20,000 Hz}=0.017 m$

So, the range of wavelength is

0.017 m - 17.2 m

(b) $4.0 \cdot 10^{14} Hz-7.9 \cdot 10^{14} Hz$

The frequency of an electromagnetic wave is given by

$f=\frac{c}{\lambda}$

where

$c=3\cdot 10^8 m/s$ is the speed of light

$\lambda$ is the wavelength

For the visible light with wavelength $\lambda=380 nm=3.8\cdot 10^{-7} m$, the frequency is

$f=\frac{3\cdot 10^8 m/s}{3.8\cdot 10^{-7} m}=7.9 \cdot 10^{14} Hz$

For the visible light with wavelength $\lambda=750 nm=7.5\cdot 10^{-7} m$, the frequency is

$f=\frac{3\cdot 10^8 m/s}{7.5\cdot 10^{-7} m}=4.0 \cdot 10^{14} Hz$

So, the range in frequencies is

$4.0 \cdot 10^{14} Hz-7.9 \cdot 10^{14} Hz$

(c) 0.015 m

As said in part (a), the wavelength of a sound wave is given by

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

where

v = 343 m/s is the sound speed in air

f is the frequency

For the wave in this problem,

$f=23 kHz = 23,000 Hz$

so the wavelength is

$\lambda=\frac{343 m/s}{23,000 Hz}=0.015 m$

(d) 0.064 m

In solid, the speed of sound is

$v=1480 m/s$

This means that if we use the formula to find the wavelength of the wave

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

we should now use v=1480 m/s.

Since the frequency of the wave is f = 23,000 Hz, the wavelength in the body is:

$\lambda=\frac{1480 m/s}{23,000 Hz}=0.064 m$