KAHIBARO
Table of Contents
Spatial Repetition in a Wave
Wavelength is the distance over which a wave repeats its shape in space. If you look at a wave at one instant of time, you will see a pattern of crests, troughs, or compressions that repeats again and again. The wavelength tells you how far you must move along the wave before the pattern looks the same again.
For a transverse wave, such as a wave on a string, the wavelength is the distance from one crest to the next crest, or from one trough to the next trough. For a longitudinal wave, such as sound in air, the wavelength is the distance from one compression to the next compression, or from one rarefaction to the next rarefaction.
The wavelength, usually written as $\lambda$, is a length.
Its SI unit is the meter, $\text{m}$.
How to Recognize Wavelength
A wavelength is always measured between two nearest points that are in the same phase, meaning they are at the same stage of oscillation. For example, two neighboring crests are in the same phase, and so are two neighboring troughs.
If you measure from a crest to the next trough, that is not a full wavelength. It is only half a wavelength:
$$
\text{crest to next trough} = \frac{\lambda}{2}
$$
Similarly, the distance covering two full repeats is $2\lambda$, and so on.
Wavelength in Different Types of Waves
The meaning of wavelength depends slightly on the kind of wave, but the basic idea is always spatial repetition.
| Type of wave | One wavelength is the distance between |
|---|---|
| Transverse wave | two neighboring crests or two neighboring troughs |
| Longitudinal wave | two neighboring compressions or two neighboring rarefactions |
| Water surface wave | repeating surface shapes |
| Light wave | repeating positions of the electromagnetic oscillation |
For beginners, it is useful to remember that wavelength is about how stretched out the wave is in space.
Symbol and Basic Relation
The standard symbol for wavelength is $\lambda$, the Greek letter lambda.
Wavelength is closely related to wave speed and frequency. If a wave moves with speed $v$ and has frequency $f$, then
$$
v = f\lambda
$$
This can also be written as
$$
\lambda = \frac{v}{f}
$$
This means that for a fixed wave speed, a higher frequency gives a shorter wavelength, and a lower frequency gives a longer wavelength.
Important wave relation:
$$
v = f\lambda
$$
So,
$$
\lambda = \frac{v}{f}
$$
Use this only when $v$, $f$, and $\lambda$ refer to the same wave.
Physical Meaning
Wavelength tells us how much distance the wave pattern occupies per cycle. If the source makes one complete oscillation, the disturbance moves forward by one wavelength during that cycle.
This gives a simple picture of the formula $v = f\lambda$. If the wave completes $f$ cycles every second, and each cycle occupies a length $\lambda$, then in one second the wave travels a distance $f\lambda$.
Examples
Suppose a sound wave travels in air with speed $340\ \text{m/s}$ and frequency $170\ \text{Hz}$. Its wavelength is
$$
\lambda = \frac{v}{f} = \frac{340}{170} = 2\ \text{m}
$$
So the distance from one compression to the next is $2\ \text{m}$.
If a wave on a string travels at $8\ \text{m/s}$ and has frequency $4\ \text{Hz}$, then
$$
\lambda = \frac{8}{4} = 2\ \text{m}
$$
Again, the pattern repeats every $2\ \text{m}$.
Reading Wavelength from a Picture
When a wave is drawn, wavelength is measured horizontally along the direction of propagation, not up and down. The vertical height of the wave is related to amplitude, not wavelength.
In this drawing, the red marked distance between two neighboring crests is one wavelength.
Wavelength in a Longitudinal Wave
In a longitudinal wave, the repeating pattern is harder to see because the particles move back and forth along the same line the wave travels. Still, wavelength is measured from one compression to the next.
The dense regions represent compressions. The distance between neighboring compressions is one wavelength.
Common Mistakes
A common mistake is to confuse wavelength with amplitude. Wavelength measures distance along the wave, while amplitude measures maximum displacement from equilibrium.
Another common mistake is to count the distance between unlike points, such as a crest and the next trough, as one wavelength. That distance is only half of $\lambda$.
Students also sometimes think wavelength stays the same when a wave enters a new medium. In general, the frequency is set by the source, while the wave speed may change in a new medium. Because
$$
\lambda = \frac{v}{f},
$$
a change in speed usually causes a change in wavelength.
Do not confuse these quantities:
Wavelength $\lambda$, measures spacing in space.
Period $T$, measures repetition in time.
Frequency $f$, measures cycles per second.
Amplitude, measures size of oscillation.
Summary
Wavelength is the spatial length of one complete wave cycle. It is written as $\lambda$ and measured in meters. It can be found by measuring the distance between repeating points, such as crest to crest or compression to compression. It is connected to speed and frequency by
$$
v = f\lambda
$$
and this makes wavelength one of the most important quantities for describing waves.