# 5.2: Wavelength and Frequency Calculations

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### Do you enjoy going to the beach?

During the summer, almost everyone enjoys going to the beach. Beach-goers can swim, have picnics, and work on their tans. But if a beach-goer gets too much sun, they can burn. A particular set of solar wavelengths are especially harmful to the skin. This portion of the solar spectrum is known as UV B, with wavelengths of $$280$$-$$320 \: \text{nm}$$. Sunscreens are effective in protecting skin against both the immediate skin damage and the long-term possibility of skin cancer.

## Waves

Waves are characterized by their repetitive motion. Imagine a toy boat riding the waves in a wave pool. As the water wave passes under the boat, it moves up and down in a regular and repetitive fashion. While the wave travels horizontally, the boat only travels vertically up and down. The figure below shows two examples of waves. Figure $$\PageIndex{2}$$: (A) A wave consists of alternating crests and troughs. The wavelength $$\left( \lambda \right)$$ is defined as the distance between any two consecutive identical points on the waveform. The amplitude is the height of the wave. (B) A wave with a short wavelength (top) has a high frequency because more waves pass a given point in a certain amount of time. A wave with a longer wavelength (bottom) has a lower frequency. (Credit: Christopher Auyeung; Source: CK-12 Foundation; License: CC BY-NC 3.0(opens in new window))

A wave cycle consists of one complete wave—starting at the zero point, going up to a wave crest, going back down to a wave trough, and back to the zero point again. The wavelength of a wave is the distance between any two corresponding points on adjacent waves. It is easiest to visualize the wavelength of a wave as the distance from one wave crest to the next. In an equation, wavelength is represented by the Greek letter lambda $$\left( \lambda \right)$$. Depending on the type of wave, wavelength can be measured in meters, centimeters, or nanometers $$\left( 1 \: \text{m} = 10^9 \: \text{nm} \right)$$. The frequency, represented by the Greek letter nu $$\left( \nu \right)$$, is the number of waves that pass a certain point in a specified amount of time. Typically, frequency is measured in units of cycles per second or waves per second. One wave per second is also called a Hertz $$\left( \text{Hz} \right)$$ and in ​​​​​​​SI units is a reciprocal second $$\left( \text{s}^{-1} \right)$$.

Figure B above shows an important relationship between the wavelength and frequency of a wave. The top wave clearly has a shorter wavelength than the second wave. However, if you picture yourself at a stationary point watching these waves pass by, more waves of the first kind would pass by in a given amount of time. Thus the frequency of the first wave is greater than that of the second wave. Wavelength and frequency are therefore inversely related. As the wavelength of a wave increases, its frequency decreases. The equation that relates the two is:

$c = \lambda \nu\nonumber$

The variable $$c$$ is the speed of light. For the relationship to hold mathematically, if the speed of light is used in $$\text{m/s}$$, the wavelength must be in meters and the frequency in Hertz.

##### Example $$\PageIndex{1}$$

The color orange within the visible light spectrum has a wavelength of about $$620 \: \text{nm}$$. What is the frequency of orange light?

###### Known
• Wavelength $$\left( \lambda \right) = 620 \: \text{nm}$$
• Speed of light $$\left( c \right) = 3.00 \times 10^8 \: \text{m/s}$$
• Conversion factor $$1 \: \text{m} = 10^9 \: \text{nm}$$
###### Unknown
• Frequency

Convert the wavelength to $$\text{m}$$, then apply the equation $$c = \lambda \nu$$ and solve for frequency. Dividing both sides of the equation by $$\lambda$$ yields:

$\nu = \frac{c}{\lambda}\nonumber$

###### Step 2: Calculate.

$620 \: \text{nm} \times \left( \frac{1 \: \text{m}}{10^9 \: \text{nm}} \right) = 6.20 \times 10^{-7} \: \text{m}\nonumber$

$\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \: \text{m/s}}{6.20 \times 10^{-7}} = 4.8 \times 10^{14} \: \text{Hz}\nonumber$

The value for the frequency falls within the range for visible light.

## Summary

• All waves can be defined in terms of their frequency and intensity.
• $$c = \lambda \nu$$ expresses the relationship between wavelength and frequency.

## Review

1. Define wavelength.
2. Define frequency.
3. What is the relationship between wavelength and frequency?

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