The speed of a wave is fixed by the type of wave and the physical properties of the medium in which it travels (except for electromagnetic waves which need no medium to travel). Examples:

- For sound waves in a fluid (for example air or water) the speed is determined by
*v = (B/ρ)*where B is the bulk modulus or compressibility of the fluid in Newtons per meter squared and^{1/2}*ρ*is the density in kilograms per cubic meter. - For sound waves in a solid the speed is determined by
*v = (Y/ρ)*where^{1/2}*Y*is Young’s modulus or stiffness in Newtons per meter squared and*ρ*is the density in kilograms per meter cubed. - For waves on a string the speed is determined by
*v = (T/μ)*where T is the tension in the string in Newtons and^{1/2}*μ*is the mass per length in kilograms per meter. - Although electromagnetic waves do not need a medium to travel (they can travel through a vacuum) their speed in a vacuum,
*c = (1/**μ*_{o}*ε*_{o})^{1/2}= 3.0×10^{8}m/s is governed by two physical constants, the permeability*μ*_{o}and the permittivity,*ε*_{o}of free space.

- A more comprehensive list of the speed of sound in various materials.

As we saw in the previous chapter, there is a relationship between the period, wavelength and speed of the wave. The period of a cork floating in the water is affected by how fast the wave passes (wave speed) and the distance between peaks (wavelength). The relationship between speed, period and wavelength of a sine wave is given by *v = λ /T *where wavelength and period for a sine wave were defined previously. This can also be written as *v = λ f* since frequency is the inverse of period. Notice that, since wave speed is normally a fixed quantity the frequency and wavelength will be inversely proportion; higher frequencies mean shorter wavelengths.

Often it is easier to write *ω = 2π f* where *ω* is the angular frequency in radians per second instead of having to write *2π f* everywhere. Likewise it is easier to write *k = 2π/λ* where *k* is the wave number in radians per meter rather than having to write *2π/λ* a lot. Using these new definitions the speed of a wave can also be written as *v = fλ = ω/k*.

If the medium is uniform the speed of a wave is fixed and does not change. There are circumstances where the speed of a particular wave does change, however. Notice that the speed of sound in air depends on the density of the air (mass per volume). But the density of air changes with temperature and humidity. So the speed of sound can be different on different days and in different locations. The temperature dependence of the speed of sound in air is given by *v = 344m/s + 0.6 (T – 20)* where *T* is the temperature in Celsius. Notice that at room temperature (20 C) sound travels at 344 m/s.

The speed of sound can also be affected by the movement of the medium in which it travels. For example, wind can carry sound waves further if the sound is traveling in the same direction or it can slow the sound down if the sound is traveling in a direction opposite to the wind direction.

Electromagnetic waves travel at c = 3.0×10^{8} m/s in a vacuum but slow down just a little when they pass through a medium (for example light passing from air to glass). This occurs because the material has a different value for the permittivity and/or permeability due to the interaction of the wave with the atoms of the material. The amount the speed changes is given by the index of refraction *n = c/v* where c is the speed of light in a vacuum and v is the speed in the medium. The frequency of the wave does not change when it slows down so, since *v = λ f*, the wavelength of electromagnetic waves in a medium must be slightly smaller.

In this chapter we have assumed that the speed of a wave does not depend on its frequency or wavelength. This is generally true; for example all the sounds of the instruments in an orchestra reach your ear at the same time, no matter what frequencies they are playing. However, it is the case that under some circumstances speed can depend on the frequency of the wave, a phenomenon known as dispersion. For electrical signals in a cable this means the signal gradually deteriorates in quality because high frequency components travel at a different speed compared to lower frequency components. Different colors of light travel at slightly different speeds through glass which is how a prism separates out the different frequencies of white light.

**Video/audio examples:**

**What is the speed of sound in a vacuum? Why is there no sound when the air is removed from the jar?**

**The speed of sound in several gasses.**

**The Zube Tube is a toy that has a spring attached to two plastic cups on either end. Vibrations in the spring travel at different speeds so a sound starting at one end (for example a click when you shake the tube and the spring hits the cup) ends up changing pitch at the other end as the various frequencies arrive. See if you can figure out from the video which frequencies travel faster, high frequencies or low.**

**These two videos demonstrate the Allasonic effect. The speed of sound is different in a liquid with air bubbles because the density is different. As the bubbles burst, the speed of sound changes, causing the frequency of sound waves in the liquid column to change, thus changing the pitch. Example: one, two. What do you hear in each case?**