# How to Calculate Wavelength

What do light, your home’s Wi-Fi, an earthquake, and the music in Adele’s latest hit “Easy on me” have in common? They all consist of waves. You have probably already heard this word in different contexts, but it turns out waves are used in many fields of physics, such as mechanics, acoustics, electromagnetism, and even quantum mechanics!

One of the most important characteristics of waves is their wavelength, which relates to both its speed and its frequency. This property allows us to understand how a certain type of wave manifests itself, for example through the specific color of a ray of light or the pitch of a musical note.

Let’s discover what wavelength is about and how to calculate it in a few simple steps.

## How to calculate a wave’s wavelength

To calculate a wave’s wavelength simply divide its speed (in m/s) by its frequency (in 1/s):

The resulting wavelength will have units of length (m).

## What is the wavelength?

A cool device to work out your upper body are battle ropes. They consist of heavy ropes attached to a wall on one end and free on the other. The idea is then to take the free ends and shake them up and down with all your strength. When doing so, waves are generated along the ropes, as shown in this image.

This interesting effect arises due to the transmission of movement between adjacent sections of the rope and can be explained if we make a simple thought experiment.

Imagine you break down the rope into several independent rigid sections held together by flexible joints. Now, if you hold the rope and then move your hand upwards, the movement will be transmitted away from you, from one section to the next one. Since every section has a certain mass, it will oppose this motion at first but will eventually displace upwards. The rope’s inertia ends up delaying the transmission of movement along its length. Now, if you immediately move your hand downwards, each section will pull their neighbor down and this new motion will be also transmitted, as the following image shows.

In real life, the transmission of movement from one rope section to the next one happens extremely fast. Therefore, it might be hard to see the wave formation as shown on the previous image with the naked eye. Check out this slow motion video of wave formation on a rope to see how this effect takes place.

Let’s look at another example. An earthquake is caused by a sudden release of pressure in the Earth’s tectonic plates, which triggers a violent movement that is transmitted across the ground. In this case, since said plates form a solid medium, particles inside it initially displace but will immediately try to bounce back to their original position due to the plates’ elasticity. When doing so, they surpass their initial resting position and move away from it in the opposite direction. This way, the tectonic plates’ particles end up oscillating around what is called their equilibrium position

So far, we have discussed examples of one specific type of waves: mechanical waves. These refer to the oscillation of matter, which constituent particles form what is called the medium in which the wave propagates. These media are made of atoms, which have mass and are joined together by bonds with a certain elasticity. Nevertheless, mechanical waves can also propagate on other types of media, like liquids or gases, where each particle or molecule is not necessarily bonded to the next, but in which collective phenomena take place. Let’s look at one example:

In gases, pressure arises due to the constant collision of its constituent particles. A local pressure change can be generated, for example, by a speaker membrane, called the source. When it vibrates, it pushes the air molecules next to its surface and forces them to get closer to other particles right next to them. When this happens, the increasing number and intensity of collisions will push particles away from each other. Nevertheless, this again forces them to get closer to the next layer of particles in the air, creating a new local pressure change. This repeated process explains how pressure waves propagate, which we usually refer to as sound. The following image explains this phenomenon:

We have discussed waves as the propagation of movement. Nevertheless, they can also be expressed as the transfer of energy, since motion is directly related to the potential and the kinetic energy of the particles that oscillate in a certain medium. Furthermore, some types of waves do not even require a physical medium to transport energy from one place to another, which is the case of electromagnetic waves

This type of waves is created by the correlation of electric and magnetic phenomena, into what is called the electromagnetic induction. Basically, when a magnetic field varies in time, it generates a time-varying electric field perpendicular to it, and vice versa. Since both fields oscillate in time and move through space, they behave as waves, which can propagate together without the need of a physical medium, even in vacuum. This is because both fields sustain each other.

Light, radio waves, your home’s Wi-Fi, 5G mobile networks, and heat are all made of electromagnetic waves with different characteristics. Nevertheless, they all propagate at the same speed when travelling through the same medium. This is known as the speed of light, which has the incredible value of 299.792.458 meters per second in vacuum.

Since waves are involved in so many physical phenomena, it is only reasonable that physicists try to describe them mathematically. This can be achieved by assigning measurable values to its characteristics. Let’s discover the most important ones!

Waves are usually represented as shown in the following image, where the y axis is time, and the x axis is the measurable quantity that oscillates. In the case of a sound wave, this would be the pressure, while in the case of a particle’s motion during an earthquake it would be its position.

The distance between the highest (peak) and lowest points (trough) in a wave is called the peak-to-peak amplitude. This value directly relates to its intensity, which tells us, for example, how loud a sound is, or how bright a source of light can be. The wave’s period is the time it takes for one complete oscillation to occur, and it is the reciprocal of frequency, another important characteristic of waves. These quantities offer valuable information like the specific pitch of a sound wave or the color of a light beam, and they relate as shown in the following equation:

Where T is the period, which has units of time (s), and f the wave’s frequency, which has units of 1/s, also called hertz. Finally, since waves propagate through space, another way to represent them is by plotting the oscillating variable against space instead of time, which can look something like this:

Here, we can measure another important characteristic of waves: their wavelength. This value equals the distance over which one entire oscillation occurs and has therefore units of length (m). Since its definition is very similar to that of the period, wavelength is often referred to as the spatial period of a wave.

Waves of different wavelength have different properties. For example, those that have a long wavelength can travel great distances without interfering with objects standing in their way, which is useful to transmit information, for example, in the form of television or radio signals. On the other hand, ultra-short wavelengths are useful to interfere with very small objects, for example, human cells in a radiotherapy as cancer treatment. The following image shows two waves, the upper one with a longer wavelength than the lower one.

## The wavelength equation

So far, we have established that the period of a wave represents the time it takes for one entire oscillation to occur, while the wavelength is the distance covered by the wave in the same amount of time. It is therefore possible to express the speed at which the wave propagates in a certain medium by dividing both quantities:

Since the wave’s period relates to its frequency as shown in equation 2, we can write the previous relation as:

When rearranged, we obtain equation 1, which allows us to calculate a wave’s wavelength using its other main characteristics. Keep in mind this definition is valid for the medium where you measure the wave’s speed and frequency. If the wave propagates slower inside another medium, its resulting wavelength will be shorter.

The experimental procedure to measure the characteristics of a wave in order to calculate its wavelength depends on the specific type of wave. Nevertheless, wavelength is mostly used to describe electromagnetic waves, since the difference between all the different types (light, wireless signals, heat, etc.) is precisely this value.

The range of all possible wavelengths forms the electromagnetic spectrum. The following image by Encyclopædia Britannica shows the most common uses of electromagnetic waves in the different parts of the spectrum, ranging from radio waves (λ between 1 mm and hundreds of kilometers) to gamma rays (λ in the order of picometers).