KAHIBARO
Table of Contents
What wave speed means
Wave speed tells us how fast a disturbance travels through a medium. When a wave moves along a rope, through air, or across water, the material itself usually does not travel along with the wave. Instead, the disturbance, the pattern of motion, moves from one place to another.
For example, if you shake one end of a rope, a pulse travels along the rope. The rope pieces move mainly up and down, but the pulse moves sideways along the rope. The speed of that pulse is the wave speed.
Wave speed is usually written as $v$.
The basic wave speed relation
A wave repeats itself after one wavelength $\lambda$. If the wave travels one wavelength in one period $T$, then its speed is
$$
v = \frac{\lambda}{T}
$$
Since frequency and period are related by
$$
f = \frac{1}{T}
$$
we also get the very important formula
$$
v = f\lambda
$$
This is one of the most useful equations for waves.
Important wave speed formula:
$$
v = f\lambda
$$
where $v$ is wave speed, $f$ is frequency, and $\lambda$ is wavelength.
Understanding the formula
The equation $v = f\lambda$ says that wave speed equals how many wave cycles pass each second, multiplied by the length of each cycle.
If a wave has a large wavelength and many cycles pass each second, it moves quickly. If either the wavelength or the frequency is smaller, the speed is smaller.
A simple unit check helps confirm the formula:
$$
[f\lambda] = \text{Hz} \cdot \text{m} = \frac{1}{\text{s}} \cdot \text{m} = \text{m/s}
$$
So the unit of wave speed is meters per second.
Wave speed depends on the medium
A very important idea is that wave speed is usually determined by the properties of the medium. For mechanical waves, the medium is the material through which the wave travels, such as a string, air, or water.
In many situations, the source sets the frequency, but the medium sets the speed. If the speed changes because the wave enters a different medium, then the wavelength changes too.
This means that for a given wave entering a new medium, frequency usually stays the same, while wavelength changes so that
$$
v = f\lambda
$$
still remains true.
For a wave moving from one medium to another, the frequency usually stays the same, but the wavelength changes because the wave speed changes.
Examples in common media
Sound waves travel at different speeds in different materials. They usually travel faster in solids than in liquids, and faster in liquids than in gases. A stretched string also carries waves, and those waves travel faster when the string is under greater tension.
Water waves are more complicated because their speed can depend on depth and wavelength, but the main idea remains the same, wave speed describes how fast the disturbance travels.
The table below gives some familiar examples.
| Wave type | Typical medium | What mainly affects speed |
|---|---|---|
| Wave on a string | Stretched string | Tension and mass per length |
| Sound wave | Air, water, solid | Elastic properties and density |
| Water surface wave | Water | Depth and wave properties |
| Seismic wave | Earth materials | Material stiffness and density |
Example calculation
Suppose a wave on a rope has frequency
$$
f = 4\,\text{Hz}
$$
and wavelength
$$
\lambda = 2\,\text{m}
$$
Then the wave speed is
$$
v = f\lambda = (4\,\text{Hz})(2\,\text{m}) = 8\,\text{m/s}
$$
So the wave travels along the rope at $8\,\text{m/s}$.
Now suppose the same source produces waves of frequency $4\,\text{Hz}$ in a different rope where the wave speed is $12\,\text{m/s}$. Then the wavelength becomes
$$
\lambda = \frac{v}{f} = \frac{12}{4} = 3\,\text{m}
$$
The frequency stays $4\,\text{Hz}$, but the wavelength changes.
Visual picture of wave speed
A wavelength is the distance between two matching points on the wave, such as crest to crest. If one crest moves forward by one full wavelength during one period, then the speed is the distance traveled divided by the time taken.
Rearranging the formula
The relation $v = f\lambda$ can be rearranged depending on what quantity is unknown.
If you want wavelength,
$$
\lambda = \frac{v}{f}
$$
If you want frequency,
$$
f = \frac{v}{\lambda}
$$
These forms are very useful in solving problems.
Useful rearrangements:
$$
v = f\lambda, \qquad \lambda = \frac{v}{f}, \qquad f = \frac{v}{\lambda}
$$
Distinguishing wave speed from particle motion
It is easy to confuse the speed of the wave with the speed of the particles in the medium. They are not the same thing.
In a transverse wave on a string, the string elements move up and down, while the wave travels along the string. In a sound wave, air molecules oscillate back and forth, while the disturbance moves through the air.
So wave speed describes propagation of the disturbance, not the travel of the material itself.
Common mistakes
A frequent mistake is to assume that increasing frequency always increases wave speed. That is not generally true for a given medium. If the medium stays the same, the wave speed is often fixed by that medium. Changing the frequency then changes the wavelength instead.
Another common mistake is mixing up period and frequency. Remember that they are inverses:
$$
f = \frac{1}{T}
$$
If you are given period, first convert it if needed, then use
$$
v = \frac{\lambda}{T}
$$
Summary
Wave speed is the speed at which a wave disturbance travels through a medium. Its basic relation to frequency and wavelength is
$$
v = f\lambda
$$
This equation is central to wave physics. In mechanical waves, the medium usually determines the speed. If the speed changes, the wavelength changes, while the frequency usually remains the same. Understanding this idea makes it much easier to analyze all kinds of waves.