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]

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.

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.

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.