11.4: Wave Speed

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Assume that you move one end of a rope up and down just once to generate a wave in the rope. How long will take the wave to travel down the rope to the other end? It depends on the speed of the wave.

The Speed of a Wave

Wave speed is the distance a wave travels in a given amount of time, such as the number of meters it travels per second. Wave speed (and speed in general) can be represented by the equation:

Speed = Distance/Time

Wave Speed, Wavelength, and Wave Frequency

Wave speed is related to both wavelength and wave frequency. Wavelength is the distance between two corresponding points on adjacent waves. Wave frequency is the number of waves that pass a fixed point in a given amount of time. This equation shows how the three factors are related:

Speed = Wavelength x Wave Frequency

In this equation, wavelength is measured in meters and frequency is measured in hertz (Hz), or number of waves per second. Therefore, wave speed is given in meters per second, which is the SI unit for speed.

Q: If you increase the wavelength of a wave, does the speed of the wave increase as well?

A: Increasing the wavelength of a wave doesn’t change its speed. That’s because when wavelength increases, wave frequency decreases. As a result, the product of wavelength and wave frequency is still the same speed.

Calculating Wave Speed from Wavelength and Wave Frequency

The equation for wave speed can be used to calculate the speed of a wave when both wavelength and wave frequency are known. Consider an ocean wave with a wavelength of 3 meters and a frequency of 1 hertz. The speed of the wave is:

Speed = 3 m x 1 wave/s = 3 m/s

Q: Kim made a wave in a spring by pushing and pulling on one end. The wavelength is 0.1 m, and the wave frequency is 2 hertz. What is the speed of the wave?

A: Substitute these values into the equation for speed:

Speed = 0.1 m x 2 waves/s = 0.2 m/s

Calculating Wave Frequency or Wavelength from Wave Speed

The equation for wave speed (above) can be rewritten as:

Frequency = Speed/Wavelength or Wavelength = Speed/Frequency

Therefore, if you know the speed of a wave and either the wavelength or wave frequency, you can calculate the missing value. For example, suppose that a wave is traveling at a speed of 2 meters per second and has a wavelength of 1 meter. Then the frequency of the wave is:

Frequency = (2m/s)/1m=2 waves/s, or 2 Hz

Q: A wave is traveling at a speed of 2 m/s and has a frequency of 2 Hz. What is its wavelength?

A: Substitute these values into the equation for wavelength:

Wavelength = (2m/s)/2waves/s=1 m

The Medium Matters

The speed of most waves depends on the medium, or the matter through which the waves are traveling. Generally, waves travel fastest through solids and slowest through gases. That’s because particles are closest together in solids and farthest apart in gases. When particles are farther apart, it takes longer for the energy of the disturbance to pass from particle to particle through the medium.

Launch the Doppler Ducks simulation below to visualize how wave speed is related to both wavelength and wave frequency. Adjust the duck velocity slider to change how quickly the duck moves through the water. Remember that positive velocities are rightward and negative velocities are leftward. Try playing around with having the duck go faster or slower than the wave speed, or towards or away from the boat.

Interactive Element

Summary

Wave speed is the distance a wave travels in a given amount of time, such as the number of meters it travels per second.
Wave speed is related to wavelength and wave frequency by the equation: Speed = Wavelength x Frequency. This equation can be used to calculate wave speed when wavelength and frequency are known.
The equation for wave speed can be written to solve for wavelength or frequency if the speed and the other value are known.
The speed of most waves depends on the medium, or the matter through which they are traveling. Generally, waves travel fastest through solids and slowest through gases.

Review

What is wave speed?
What is the speed of a wave that has a wavelength of 2 m and a frequency of 1.5 Hz?
Calculate the frequency of a wave that is traveling at a speed of 3.0 m/s and has a wavelength of 1.2 m.
Sound energy travels through matter in waves. Do sound waves travel faster through air or water? Explain your answer.

Additional Resources

Study Guide: Waves Study Guide

Video: Speed of Mechanical Waves - Overview

Real World Application: Seismic Shakeup