Wavelength of a Stationary Wave
What is a Stationary Wave?
Imagine shaking one end of a rope that is tied to a fixed point, like a wall. The wave you create travels down the rope, hits the wall, and reflects back. When the incoming wave and the reflected wave meet, they combine. If you shake the rope at just the right frequency, a special pattern emerges where certain points on the rope seem to be vibrating wildly, while other points don't move at all. This is a stationary wave.
Unlike a traveling wave, a stationary wave does not transfer energy from one end to the other. Instead, the energy is trapped, oscillating back and forth within the medium. The pattern is characterized by two key features:
- Nodes (N): These are points of zero amplitude. The rope does not move at these points at all. They are the "still points" in the wave.
- Antinodes (A): These are points of maximum amplitude. The rope vibrates with the greatest possible displacement at these points.
In a stationary wave, nodes and antinodes occur at fixed positions, creating the illusion of a wave that is "standing" still.
Defining the Wavelength
The wavelength ($\lambda$) is a crucial property of any wave. For a traveling wave, like a ripple on a pond, the wavelength is the distance between two consecutive crests or two consecutive troughs. For a stationary wave, we need a different, but equally valid, definition.
It is important to note that the distance between a node and the next antinode is not the wavelength. That distance is actually one-quarter of a wavelength ($\lambda/4$). Let's see why this definition makes sense by looking at the simplest stationary wave pattern.
The most basic pattern, called the fundamental mode or first harmonic, has one single "loop" between the two ends. In this case:
- The entire wave is contained between two nodes (at the fixed ends).
- There is one antinode exactly in the middle.
- The distance from one node to the next node is half of a "full wave." Therefore, the length of the rope (L) is equal to half the wavelength: $L = \lambda / 2$.
So, if the rope is 2 meters long, the wavelength of the fundamental stationary wave is 4 meters. You can see that the distance between the two nodes at the ends is 2 meters, but the defined wavelength is twice that distance.
Harmonics and Wavelength
When the frequency of vibration increases, more complex stationary wave patterns can form. These are called harmonics. The relationship between the length of the medium (L) and the wavelength ($\lambda$) changes for each harmonic.
| Harmonic | Description | Wavelength ($\lambda$) | Number of Loops |
|---|---|---|---|
| 1st (Fundamental) | One antinode between two end nodes. | $\lambda = 2L$ | 1/2 |
| 2nd | Two antinodes between three nodes. | $\lambda = L$ | 1 |
| 3rd | Three antinodes between four nodes. | $\lambda = \frac{2L}{3}$ | 1.5 |
| n-th | n antinodes between (n+1) nodes. | $\lambda = \frac{2L}{n}$ | n/2 |
In the table, 'n' is the harmonic number. Notice the general formula for wavelength: $\lambda = \frac{2L}{n}$. This shows that as the harmonic number increases, the wavelength decreases. The distance between any two adjacent nodes (or antinodes) is always $\lambda / 2$, which you can see is equal to $L / n$.
Seeing Stationary Waves in Action
Stationary waves are not just a theoretical concept; they are all around us, especially in music.
Example 1: The Guitar String
When a guitarist plucks a string, it vibrates and creates a stationary wave. The fixed ends of the string are nodes. The pitch of the note we hear is determined by the frequency, which is directly related to the wavelength. Pressing a finger on a fret shortens the effective length (L) of the vibrating string. According to the formula $\lambda = 2L$ for the fundamental wave, a shorter L means a shorter wavelength. A shorter wavelength results in a higher frequency, which is why the pitch of the note becomes higher.
Example 2: The Flute
In a wind instrument like a flute, stationary waves are formed in a column of air. However, the boundary conditions are different. The end where the player blows is approximately an antinode (air moves maximally), and the open far end is also an antinode. For a tube open at both ends, the fundamental wavelength is $\lambda = 2L$, just like the string. But the pattern has antinodes at both ends and a node in the middle. When you open or close the holes on a flute, you are changing the effective length of the air column, thus changing the wavelength and the pitch of the sound produced.
Example 3: The Microwave Oven
Microwaves are a type of electromagnetic wave. Inside a microwave oven, these waves reflect off the metal walls and create stationary wave patterns. The antinodes are points of high energy. This is why many microwaves have a rotating turntable—to move the food through these different points and ensure it heats evenly. If your food doesn't rotate, you might find some parts are very hot (at an antinode) and other parts are cold (at a node).
Common Mistakes and Important Questions
Q: Is the wavelength the distance between a node and an antinode?
No, this is a very common mistake. The distance between a node and the very next antinode is only one-quarter of a wavelength ($\lambda/4$). The wavelength is defined as the distance between two adjacent nodes OR two adjacent antinodes, which is twice that distance ($\lambda/2$).
Q: Can a stationary wave have a wavelength longer than the medium itself?
Yes, absolutely. Look at the fundamental mode on a string fixed at both ends. The length of the string is L, but the wavelength is $2L$. The entire wave pattern only shows half of the "full wave," so the actual wavelength is twice the length of the medium.
Q: Why is it important to measure from node-to-node or antinode-to-antinode?
This definition ensures consistency with the wavelength of the original traveling waves that combined to form the stationary wave. The distance between two equivalent points (two points of zero displacement or two points of maximum displacement) on any wave should be one full wavelength. This universal definition allows physicists and engineers to use the same mathematical relationships, like the wave equation $v = f\lambda$, for both traveling and stationary waves.
Footnote
[1] Standing Wave: Another term for a stationary wave, a wave pattern that remains in a constant position.
[2] Node: A point in a stationary wave where the amplitude is always zero.
[3] Antinode: A point in a stationary wave where the amplitude of vibration is at a maximum.
[4] Fundamental Frequency: The lowest frequency at which a system naturally vibrates, creating the simplest stationary wave pattern.
[5] Harmonics: Frequencies at which a system vibrates that are integer multiples of the fundamental frequency. They produce more complex stationary wave patterns.