Question

The speed of sound in dry air at \(20^{\circ} \mathrm{C}\) is 343 \(\mathrm{m} / \mathrm{s}\) and the lowest frequency sound wave that the human ear can detect is approximately 20 \(\mathrm{Hz}\) (a) What is the wavelength of such a sound wave? (b) What would be the frequency of electromagnetic radiation with the same wavelength? (c) What type of electromagnetic radiation would that correspond to? [Section 6.1]

Short answer

The wavelength of a sound wave with a frequency of 20 Hz and a speed of 343 m/s is 17.15 meters. Electromagnetic radiation with the same wavelength has a frequency of \(1.75 \times 10^7\ Hz\) and corresponds to radio waves.

Step-by-step solution

Find the wavelength of the sound wave

The relationship between speed, wavelength, and frequency of a wave can be described by the formula: \(v = \lambda f\) Where \(v\) is the speed of the wave, \(\lambda\) is the wavelength, and \(f\) is the frequency. Since we have the values of speed and frequency of the sound wave, we can solve for the wavelength. \(343\ m/s = \lambda (20\ Hz)\) Now, we'll isolate the wavelength, \(\lambda\), by dividing both sides by the frequency (20 Hz): \(\lambda = \frac{343\ m/s}{20\ Hz}\)

Calculate the wavelength of the sound wave

Now we will calculate the value of the wavelength using the formula obtained in step 1: \(\lambda = \frac{343\ m/s}{20\ Hz} = 17.15\ m\) The wavelength of the sound wave is 17.15 meters.

Find the frequency of electromagnetic radiation with the same wavelength

The relationship between the speed of light, wavelength, and frequency of electromagnetic radiation can be described by the formula: \(c = \lambda f\) Where \(c\) is the speed of light \((3 \times 10^{8}\ m/s)\), \(\lambda\) is the wavelength, and \(f\) is the frequency. To find the frequency of electromagnetic radiation with the same wavelength, we'll plug in the values into the equation. \((3 \times 10^{8}\ m/s) = (17.15\ m) f\) Now, we'll isolate the frequency, \(f\), by dividing both sides by the wavelength (17.15 m): \(f = \frac{(3 \times 10^{8}\ m/s)}{(17.15\ m)}\)

Calculate the frequency of electromagnetic radiation

Now we will calculate the value of the frequency using the formula obtained in step 3: \(f = \frac{(3 \times 10^{8}\ m/s)}{(17.15\ m)} = 1.75 \times 10^7\ Hz\) The frequency of electromagnetic radiation with the same wavelength is \(1.75 \times 10^7\ Hz\).

Identify the type of electromagnetic radiation

To identify the type of electromagnetic radiation, we can use the electromagnetic spectrum and its range of frequencies. The frequency of the radiation is \(1.75 \times 10^7\ Hz\), which falls into the range of radio waves which have frequencies ranging from \(3 \times 10^4\ Hz\) to \(3 \times 10^{11}\ Hz\). Therefore, the type of electromagnetic radiation that corresponds to the same wavelength as the sound wave is a radio wave.

Key concepts

Wavelength Calculation

Understanding the calculation of wavelength is fundamental when studying waves, be it in sound, water, or electromagnetic waves. Wavelength, symbolized by the Greek letter lambda \( \lambda \), is the distance between two consecutive peaks or troughs of a wave. In the context of sound waves, which are longitudinal waves traveling through a medium such as air, the calculation of wavelength can be intuitively understood through the formula: \[ v = \lambda f \].

The speed of sound at standard temperature and pressure (20°C in dry air) is approximately 343 meters per second. By knowing both the speed (v) and the frequency (f) of a wave, you can algebraically solve for the wavelength (\lambda) by rearranging the formula: \[ \lambda = \frac{v}{f} \]. When applying this to a 20 Hz sound wave, which is at the lower end of the human hearing range, you calculate a significant wavelength of 17.15 meters, showing the inversely proportional relationship between frequency and wavelength; the lower the frequency, the longer the wavelength.

This concept is not exclusive to sound and applies across the board to all wave phenomena, including the vast electromagnetic spectrum, which brings us to electrically charged particles generating oscillating electric and magnetic fields that propagate as electromagnetic radiation across space.

Frequency of Electromagnetic Radiation

Electromagnetic radiation encompasses a wide range of frequencies and is a vital concept in physics and everyday life, influencing everything from communication technologies to medical imaging. The frequency of electromagnetic radiation, represented by \( f \), measures the number of wave cycles that pass a point per unit of time, typically expressed in hertz (Hz).

For electromagnetic waves, the speed of light in a vacuum, denoted \( c \), is a universal constant at about \( 3 \times 10^8 \ m/s \). Given the invariance of \( c \), and knowing the wavelength, we can determine the frequency using the rearranged version of the same fundamental wave equation \( c = \lambda f \): \[ f = \frac{c}{\lambda} \].

By observing how a 17.15-meter wavelength translates to a frequency of approximately 17.5 million Hz (or 17.5 MHz), students can learn about the relationship between wavelength and frequency, noting how an increase in one results in a reduction of the other due to the constant speed of light. This helps in understanding that all electromagnetic waves share this fundamental characteristic, although the implications differ across the varying types of radiation.

Electromagnetic Spectrum

The electromagnetic spectrum is a continuum of all electromagnetic waves arranged according to frequency or wavelength. It includes, in order of increasing frequency, radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each category has its unique characteristics and practical applications.

A radio wave, which has the longest wavelength in the electromagnetic spectrum, can have wavelengths that range from a few millimeters to kilometers. The frequency calculated from a 17.15-meter wavelength places this particular electromagnetic wave within the radio wave range, confirming its identity as a radio wave. Radio waves are utilized in broadcasting, communication, and other technologies like MRI in medicine.

Understanding where a given frequency or wavelength falls within the electromagnetic spectrum is critical to grasping both the nature of electromagnetic radiation and its interactions with matter. This categorization based on frequency helps in identifying the applications and potential hazards of electromagnetic radiation, reflecting its importance in both scientific and daily contexts.