# Frequency To Wavelength Calculator

Created by Davide Borchia
Reviewed by Komal Rafay
Last updated: Feb 02, 2023

Our frequency to wavelength calculator will help you understand the intimate relationship between these two quantities in a wave.

Here you will learn:

• What is the wavelength?;
• What is the relationship between wavelength and frequency?;
• How to calculate the wavelength given a frequency?;
• Some examples of how to use our frequency to wavelength calculator.

## What is a wave?

A wave is a periodic motion where the equilibrium state of a system is perturbated, creating a characteristic oscillating behavior.

You can find waves just about everywhere: sound, earthquakes, electromagnetic radiation. If there is a propagation of energy, it's likely that with a better look, you will observe a wave-like motion.

## The sinusoidal wave equation

There are many types of waves: let's focus only on the most common and easy-to-understand one: the sinusoidal wave.

A sinusoidal wave is described by... you guessed it, a sine function. It follows the equation:

$\footnotesize u(x,t)=A\times\sin{(\frac{2\pi}{\lambda}x - 2\pi f t + \phi)}$

where:

• $u$ - the amplitude of the wave at a certain position $x$ and time $t$;
• $A$ - the maximum amplitude of the wave ($1/2$ of the distance from the highest to the lowest point of the wave);
• $\lambda$ - the wavelength;
• $f$ - the frequency; and
• $\phi$ - the phase constant.

## What are frequency and wavelength?

Frequency and wavelength define the shape of a wave minus a vertical stretch (that's the purpose of the amplitude $A$) and the definition of a "starting point" (the phase constant tells us the initial position at position $x=0$ and time $t=0$).

• The frequency $f$ tells us how many times the wave completes a period of its oscillation in a given amount of time (if that interval is a second, the frequency is measured in Hertz: $1\ \text{Hz} = 1/\text{s}$). Note that the frequency is the inverse of the period $T$; $f=1/T$.
• The wavelength $\lambda$ measures the horizontal distance between two peaks on the same side of the baseline.

## How to calculate the wavelength from the frequency

What is the relationship between frequency and wavelength? You will be pleased to discover that it all boils down to a simple formula that relates the two quantities to the speed of propagation of the wave, $v$.

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

You can intuitively understand this formula by noting that the speed of the wave corresponds to the distance covered in a given unit of time. At the same time, the wavelength is the distance to complete a period of the oscillation. Dividing the wavelength by the speed returns the time necessary to complete an oscillation: the period!

## Calculate the wavelength given the frequency

Let's look at some examples of waves to better understand the relationship between frequency and wavelength.

Light is nothing but electromagnetic radiation. It propagates at the ludicrous speed of almost $300,000\ \text{km}/\text{s}$. As humans, we are most familiar with visible light, a portion of the electromagnetic spectrum associated with light we perceive as colors. Each color has a specific frequency.

From the lowest to the highest frequency, we selected some colors we liked:

• Red with a frequency of $4.62\times10^{14}\ \text{Hz}$;
• Green with a frequency of $5.45\times10^{14}\ \text{Hz}$; and
• Blue with a frequency of $6.66\times10^{14}\ \text{Hz}$.

Try our frequency to wavelength calculator to find out the value of the wavelengths: choose the desired wave velocity, in this case, light in a vacuum.

The calculator will calculate the wavelengths given the frequencies. To input the frequencies, select teraHertz from the units menu of the frequency variable, then input the values above as $462$, $545$, and $666$.

For the red:

$\lambda = \frac{299,792,458 \text{m}/\text{s}}{4.62\times10^{14}\ \text{Hz}} \simeq 6.49 \times10^{-7}\ \text{m} = 649\ \text{nm}$

🙋 $\text{nm}$ stands for nanometer, a billionth of a meter. It's a measurement unit good for molecules and nanotechnologies! Think of something a thousand time smaller than a hair.

For the color green, we find that:

$\lambda= \frac{299,792,458 \text{m}/\text{s}}{5.45\times10^{14}\ \text{Hz}} \simeq 5.50\times10^{-7}\ \text{m} = 550\ \text{nm}$

And finally, for the blue, we calculate the wavelength from the frequency with:

$\lambda= \frac{299,792,458 \text{m}/\text{s}}{6.66\times10^{14}\ \text{Hz}} \simeq 4.50\times10^{-7}\ \text{m} = 450\ \text{nm}$

As you can see, the higher the frequency, the shorter the wavelength (if the speed remains constant). This happens because the time required to complete a period gets smaller, and at a constant speed, this corresponds to a reduced distance between peaks.

🔎 Do you know that the distance traveled by light in a nanosecond roughly corresponds to a foot? Physicist David Mermin proposed a new measurement unit, the "light nanosecond", jokingly called phoot: a portmanteau of photos, light in Latin, and foot.

## Beyond our frequency to wavelength calculator!

Apart from the frequency to wavelength calculator, we made other tools that can help you with any kind of oscillating problem! Try our:

Apart from beign the descriptive quantities of a wave, frequency and wavelength appears in multiple instances across physics. An equation as simple as the one we've just met allows you to calculate, for example, the energy of a photon: check our photon energy calculator to discover the Planck's relationship. And not far from there, with the knowledge you've acquired, you can understand the experiment that awarded Einstein of his Nobel prize: the photoelectric effect. Our photoelectric effect calculator will guide you in this exploration of the quantum world!

## FAQ

### What is the wavelength?

The wavelength is a quantity that measures the distance of two peaks on the same side of a wave. You can think of the wavelength as the distance covered by a wave in the period of the oscillation.

### How to calculate the wavelength from the frequency?

To calculate the wavelength of a periodic oscillating motion given the frequency, you first need to know the speed of the wave. Then, apply the formula:
λ = v/f
where:

• λ - the wavelength;
• v - the propagation speed of the wave; and
• f - the frequency.

### What is the wavelength of microwaves?

Your microwave oven works thanks to a particular kind of electromagnetic radiation that excites the water molecules in your food. Microwave ovens typically operate with waves with frequencies of about 2.45 GHz. This is also the frequency used by wifi!

The corresponding wavelength is, considering the speed of light in air:
λ = 299,702,547 m/s / 2.45 × 10⁹ Hz = 12.23 cm

### Where can I see wavelength?

Drop a stone in a pond! You will see the surface of the water getting covered in concentric ripples. The distance between two peaks (above the surface) is the wavelength of the transverse wave.

That wavelength increases with time because the energy transported by the water is damped, and it progressively reduces to zero.

Davide Borchia
Preset (optional)
light in vacuum
Wave velocity
m/s
Wave frequency
Hz
Wavelength (λ)
m
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