# Wave Speed

Wave speed is the velocity of a progressive wave, which is a disturbance in the form of an oscillation that travels from one location to another and transports energy.

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The velocity of the wave depends on its frequency f’ and wavelength ‘λ’. The speed of a wave is an important parameter, as it allows us to calculate how fast a wave spreads in the medium, which is the substance or material that carries the wave. In the case of ocean waves, this is the water, while in the case of sound waves, it is the air. The velocity of a wave also depends on the type of wave and the physical characteristics of the medium in which it is moving.

Figure 1. A sinusoid (sine function signal) propagates from left to right (A to B). The speed at which the sinusoid oscillation travels is known as wave speed.

## How to calculate wave speed

To calculate wave speed, we need to know the wavelength as well as the frequency of the wave. See the formula below, where the frequency is measured Hertz, and the wavelength is measured in metres.

$v = f \cdot \lambda$

The wavelength ‘λ’ is the total length from one crest to the next, as shown in figure 2. The frequency ‘f’ is the inverse of the time it takes for a crest to move to the position of the next one.

Figure 2. The wave period is the time it takes for a wave crest to reach the position of the next crest. In this case, the first crest has a time $$T_a$$ and moves to the position where the crest $$X_b$$ was before at time $$T_a$$.

Another way to calculate wave speed is by using the wave period Τ’, which is defined as the inverse of the frequency and provided in seconds.

$T = \frac{1}{f}$

This gives us another calculation for wave speed, as shown below:

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

The period of a wave is 0.80 seconds. What is its frequency?

$$T = \frac{1}{f} \Leftrightarrow \frac{1}{T} = \frac{1}{0.80 s} = 1.25 Hz$$

Wave speed can vary, depending on several factors, not including the period, frequency, or wavelength. Waves move differently in the sea, the air (sound), or in a vacuum (light).

### Measuring the speed of sound

The speed of sound is the velocity of mechanical waves in a medium. Remember that sound also travels through fluids and even solids. The speed of sound decreases as the density of the medium is lower, allowing sound to travel faster in metals and water than in the air.

The speed of sound in gases such as the air depends on the temperature and density, and even humidity can affect its speed. In average conditions such as an air temperature of 20°C and at sea level, the speed of sound is 340.3 m/s.

In the air, the speed can be calculated by dividing the time it takes for sound to travel between two points.

$v = \frac{d}{\Delta t}$

Here, ‘d’ is the distance travelled in metres, while Δt’ is the time difference.

The speed of sound in the air at average conditions is used as a reference for objects moving at high speeds by using the Mach number. The Mach number is the object speed u’ divided by ‘v’, the speed of sound in the air at average conditions.

$M = \frac{u}{v}$

As we said, the speed of sound also depends on the air temperature. Thermodynamics tells us that heat in a gas is the average value of the energy in the air molecules, in this case, its kinetic energy.

As the temperature increases, the molecules that make up the air gain velocity. Faster movements allow the molecules to vibrate faster, transmitting sound more easily, which means that sound takes less time to travel from one place to another.

As an example, the speed of sound at 0°C at sea level is around 331 m/s, which is a decrease of about 3%.

Figure 3. The speed of sound in fluids is affected by their temperature. Larger kinetic energy due to higher temperatures makes molecules and atoms vibrate faster with sound. Source: Manuel R. Camacho, StudySmarter.

### Measuring the speed of water waves

Wave speed in water waves is different from that of sound waves. In this case, the speed depends on the depth of the ocean where the wave propagates. If the water depth is more than twice the wavelength, the speed will depend on the gravity g’ and the wave period, as shown below.

$$v = \frac{g}{2 \pi}T$$

In this case, g = 9.81 m/s at sea level. This can also be approximated as:

$$v = 1.56 \cdot T$$

If waves move to shallower water and the wavelength is larger than twice the depth ‘h’ (λ > 2h), then wave speed is calculated as follows:

$$v = \sqrt{g \cdot h}$$

As with sound, water waves with larger wavelengths travel faster than smaller waves. This is the reason why large waves caused by hurricanes arrive at the coast before the hurricane does.

Here is an example of how the speed of waves differs depending on the depth of the water.

A wave with a period of 12s

In the open ocean, the wave is not affected by the water depth, and its velocity is approximately equal to v = 1.56 · T. The wave then moves to shallower waters with a depth of 10 metres. Calculate by how much its speed has changed.

Wave speed ‘Vd’ in the open ocean is equal to the wave period multiplied by 1.56. If we substitute the values in the wave speed equation, we get:

$$Vd = 1.56 m/s^2 \cdot 12 s = 18.72 m/s$$

The wave then propagates to the coast and enters the beach, where its wavelength is larger than the depth of the beach. In this case, its speed ‘Vs’ is affected by the beach depth.

$$Vs = \sqrt{9.81 m/s^2 \cdot 10 m} = 9.90 m/s$$

The difference in speed is equal to the subtraction of Vs from Vd.

$$\text{Speed difference} = 18.72 m/s - 9.90 m/s = 8.82 m/s$$

As you can see, the speed of the wave decreases when it enters shallower waters.

As we said, the speed of waves depends on the depth of the water and the wave period. Larger periods correspond to larger wavelengths and shorter frequencies.

Very large waves with wavelengths reaching more than a hundred meters are produced by large storm systems or continuous winds in the open ocean. Waves of different lengths are mixed in the storm systems that produce them. However, as the larger waves move faster, they leave the storm systems first, reaching the coast before the shorter waves. When these waves reach the coast, they are known as swells.

Figure 4. Swells are long waves with high speed that can travel across whole oceans.

### The speed of electromagnetic waves

Electromagnetic waves are different from sound waves and water waves, as they do not require a medium of propagation and thus can move in the vacuum of space. This is why sunlight can reach the earth or why satellites can transmit communications from space to earth base stations.

Electromagnetic waves move in a vacuum at the speed of light, i.e., at approximately 300,000 km/s. However, their speed depends on the density of the material they are passing through. For instance, in diamonds, light travels at a speed of 124,000 km/s, which is only 41% of the speed of light.

The dependence of the speed of electromagnetic waves on the medium they travel in is known as the refractive index, which is calculated as follows:

$n = \frac{c}{v}$

Here, ‘n’ is the index of refraction of the material, c’ is the speed of light, and v’ is the speed of light in the medium. If we solve this for the speed in the material, we get the formula for calculating the speed of electromagnetic waves in any material if we know the refractive index n.

$v = \frac{c}{n}$

The following table shows the light velocity in different materials, the refractive index, and the material’s average density.

 Material Speed [m/s] Density [kg/m3] Refractive index Vacuum of space 300,000,000 1 atom 1 Air 299,702,547 1.2041 1,00029 Water 225,000,000 9998.23 1.333 Glass 200,000,000 2.5 1.52 Diamond 124,000,000 3520 2,418

The values for air and water are given at standard pressure 1 [atm] and a temperature of 20°C.

As we said and is illustrated in the table above, the speed of light depends on the density of the material. The effect is caused by the light impacting atoms in the materials.

Figure 5. Light is absorbed by the atoms when passing through a medium. Source: Manuel R. Camacho, StudySmarter.

Figure 6. Once the light has been absorbed, it will be released again by other atoms. Source: Manuel R. Camacho, StudySmarter.

As the density increases, the light encounters more atoms in its way, absorbing the photons and releasing them again. Each collision creates a small time delay, and the more atoms there are, the greater the delay.

## Wave Speed - Key takeaways

• Wave speed is the speed at which a wave propagates in a medium. The medium can be the vacuum of space, a liquid, a gas, or even a solid. Wave speed depends on the wave frequency ‘f’, which is the inverse of the wave period ‘T’.
• In the sea, lower frequencies correspond to faster waves.
• Electromagnetic waves normally move at the speed of light, but their speed depends on the medium in which they move. Denser mediums cause electromagnetic waves to move more slowly.
• The speed of ocean waves depends on their period, although in shallow water, it only depends on the depth of the water.
• The speed of sound travelling through the air depends on the air temperature, as colder temperatures make sound waves slower.

#### Flashcards in Wave Speed 14

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What speed do electromagnetic waves travel at?

Electromagnetic waves travel at the speed of light, which is approximately 300,000 km/s.

How do we calculate wave speed?

Generally, the speed of any wave can be calculated by multiplying the wave frequency by its wavelength. However, the speed can also depend on the density of the medium as in electromagnetic waves, the depth of the fluid as in ocean waves, and the temperature of the medium as in sound waves.

What is wave speed measured in?

Wave speed is measured in units of velocity. In the SI system, these are metres over second.

What is wave speed?

It is the speed at which a wave propagates.

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