When waves move through different materials, their speed changes based on the properties of those materials. To understand how waves behave, it’s important to know this relationship.

We often talk about three main types of waves:

• Sound waves
• Light waves
• Water waves

Each type of wave follows specific rules depending on where it is traveling.

The speed of a wave in any material can be shown using a simple formula:

$v = f \lambda$

Here’s what the letters mean:

• $v$ is the wave speed
• $f$ is the frequency
• $\lambda$ is the wavelength

This formula tells us how speed, frequency, and wavelength are linked. When a wave moves from one material to another, its frequency stays the same. However, its speed and wavelength can change.

Let’s first look at sound waves.

In air, sound travels at about 343 meters per second (m/s) at room temperature (20°C). But when sound travels through water, it goes much faster, around 1482 m/s. This happens because water is denser and more elastic than air. We can say:

$v_{sound\_in\_water} > v_{sound\_in\_air}$

This shows that sound moves faster in water than in air. The change from gas (air) to liquid (water) makes a big difference in the way sound travels.

Now, let’s think about light waves.

Light travels fastest in a vacuum at about 300 million meters per second ($3.00 \times 10^8$ m/s). But when light goes into materials like glass or water, it slows down. How much it slows down depends on the material and is measured by something called the refractive index ($n$):

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

In this equation:

• $c$ is the speed of light in a vacuum
• $v$ is the speed of light in the material

When light enters a material with a higher refractive index, its speed decreases. For example, the refractive index of water is about 1.33, and for glass, it can be between 1.5 and over 1.9 depending on the type.

So when light moves from a vacuum (where $n=1$) into glass, we can find its speed using the refractive index. For glass, it looks like this:

$v_{light} = \frac{c}{n}$

This shows us that light waves slow down when they go from a vacuum to a denser material, which affects how they travel.

Wavelength is also connected to speed and frequency. When a wave moves into a new material and its speed changes, the wavelength changes to keep the relationship steady. If the frequency stays the same, we can find the new wavelength like this:

$\lambda_{new} = \frac{v_{new}}{f}$

If the wave enters a medium where the speed increases, the wavelength gets longer. If the speed decreases, the wavelength gets shorter. Here’s a summary of what happens when waves change materials:

1. Sound Waves: Speed increases in denser materials (like liquid compared to gas).

   • Example: Sound travels faster in water than in air.

2. Light Waves: Speed decreases in denser materials (like glass compared to air).

   • Example: Light goes slower in glass than in a vacuum.

3. Wavelength Changes: When speed changes but frequency stays the same, the wavelength changes too.

   • Example: If a sound wave speeds up in water, it has a longer wavelength than it does in air.

To help you visualize, here’s how sound and light behave in different materials:

• Sound Going from Air to Water:

  • Speed: Increases
  • Wavelength: Increases
  • Frequency: Stays the same

• Light Moving from Vacuum to Glass:

  • Speed: Decreases
  • Wavelength: Decreases
  • Frequency: Stays the same

Also, we can talk about how wave speed changes in solid materials, depending on their properties like Young's modulus (how stretchy something is) and density. For example, the speed of sound waves in solids can be expressed using this formula:

$v = \sqrt{\frac{E}{\rho}}$

In this formula:

• $E$ is the modulus of elasticity
• $\rho$ is the density

This shows how the physical properties of solids affect how fast waves can travel through them.

To sum it up, changing materials has a big impact on wave speed. This is true for all types of waves like sound and light. Although the frequency stays constant through different materials, changes in speed lead to wavelength adjustments.

Understanding these relationships is important for learning about wave behavior and is useful in areas like acoustics (sound) and optics (light). This knowledge is key for anyone studying waves in school.

In short, knowing how waves behave in different materials helps us in many scientific fields and technologies that involve waves.