The equation for the relationship between the speed and other characteristics of a wave can be derived from our basic understanding of motion. The average speed of anything that is moving is:

average speed $\displaystyle =\frac{\text{distance}}{\text{time}}$

(So, for example, a CAR on the highway traveling at a speed of 100 km/h covers 100 km during the time of 1 h.) For an electromagnetic wave to travel the distance of one of its wavelengths, $\lambda$ , at the speed of light, $c$ , we have $c=\frac{\lambda}{t}$ . The frequency of a wave is the number of cycles per second. If a wave has a frequency of a million cycles per second, then the time for each cycle to go by is a millionth of a second. So, in general, $t=\frac{1}{f}$ .

Substituting into our wave equation, we get $c=\lambda \times f$ . Now let's use this to calculate an example. What is the wavelength of visible light that has a frequency of $5.66\times10^{14}\,$ Hz?

Solution

Solving the wave equation for wavelength, we find:

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

Substituting our values gives:

$\displaystyle \lambda=\frac{3.00\times10^{8}\,\text{m/s}}{5.66\times10^{14}\,\text{Hz}}=5.30\times10^{-7}\,\text{m}$

This answer can also be written as 530 nm, which is in the yellow-green part of the visible spectrum (nm stands for nanometers, where the term “nano” means “billionths”).

Check Your Learning

“Tidal waves,” or tsunamis, are waves caused by earthquakes that travel rapidly through the ocean. If a tsunami travels at the speed of 600 km/h and approaches a shore at a rate of one wave crest every 15 min (4 waves/h), what would be the distance between those wave crests at sea?

Answer:

$\displaystyle \lambda=\frac{\text{600 km/h}}{\text{4 waves/h}}=150\,\text{km}$

Bibliography

This example is a derivative work of the Creative Commons source in [1].

[1] Andrew Fraknoi, David Morrison, and Sidney Wolff, “Example 5.1: Deriving and Using the Wave Equation,” in “5.1 The Behavior of Light,” Astronomy 2e. Houston, Texas: OpenStax, 2022. Licensed CC BY 4.0 in the original 2022 release (https://creativecommons.org/licenses/by/4.0/). Changes: converted to LaTeX for PhysicsLibrary and separated from section 5.1 as a standalone example. Access for free at https://openstax.org/books/astronomy-2e/pages/1-introduction. Section source: https://openstax.org/books/astronomy-2e/pages/5-1-the-behavior-of-light.

"example of behavior of light: deriving and using the wave equation" is owned by bloftin.
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