How Standing Waves Form: Principles and Applications
We can consider a system in which an elongated string is bounded at both ends. We tend to send a continuous sinusoidal wave without a break of a specific frequency towards the positive x-direction. When the wave generally reaches the right fixed end, it reflects and traverses to the left end. In this particular procedure, the left traversing wave overlaps the wave traveling to the right. The same cycle continues, as the left proceeding wave reflects the left part and begins to travel right; it again overlaps the left-going wave. This ultimately results in the production of many overlapping waves that always interfere with each other. The two types of waves that are longitudinal waves (e.g., sound) and transverse waves (e.g., water) can together constitute the standing waves.
Image will be uploaded soon
Standing Wave Equation
We can consider that, at any point in time, you and time t, there are generally two waves, one which moves to the left-hand side and the other which moves to the right-hand side. The wave when keeps traveling in the positive direction of the x-axis is given as,
y1(u, t) = a sin (Ku – ωt),
and that when traveling in the negative direction of the x-axis is given as,
y2(u, t) = a sin (Ku + ωt),
By the principle of superposition, the combined wave is stated as,
y (u, t) = y1(u, t) + y2(u, t)
= a (sin) of (ku – ωt) + a (sin) of (ku + ωt)
= (2a sin ku) cos ωt
This is the standing wave equation.
How Are Standing Waves Formed?
Here, the term, which is stated as (2a sin Ku), provides us the information about the amplitude of the oscillation from all the elements of the wave at a position 'u.' We can also notice that the wave, which is stated by this type of equation, can not be described as the same as a traveling wave or as a waveform that can not move to either side. This expression surely represents the standing waveform, where the point of every minimum, maximum, and the null remains stagnant at one position only throughout its propagation.
The amplitude is called to be as zero for all the given values of Ku that give sin Ku = 0. Those values that are given by the Ku = nπ, for the given values n = 0, 1, 2, 3, …
By substituting the values of k = 2π/λ into the expressions like for the amplitude, we will get u=(nλ)/2, for the values of n = 0, 1, 2, 3, …
The particular positions of the zero amplitudes are called nodes. The distance covered by 2λ, or we can consider a half wavelength, can separate both consecutive nodes. The amplitude is undoubtedly said to have a maximum value of 2a, which automatically occurs for all Ku's values that give ⎢sin Ku ⎢= 1.
The values are Ku = (n + ½) π for all the values of n = 0, 1, 2, 3, …therefore, by substituting the values for k = 2π/λ in the given expression, we can automatically get the value for u=(n+12)λ2 for the values of n = 0, 1, 2, 3, … as that of the positions selected for maximum amplitude. These are known as the Anti-nodes. These antinodes are said to be divided into two beams by λ/2 and located about half away between the pairs of nodes. Now, we can consider a string with length L whose ends are fixed. These two ends of this string are known as nodes. This is how standing waves formed, and along with it, the formation of antinodes also takes place.
Nodes and Antinodes
A node is a particular point along with a standing wave where the wave certainly has minimum amplitude.
The different elementary subject of a node is an antinode, a particular point where one standing wave's amplitude is a maximum. These take place on the midway between the nodes.
Solved Examples
What is Normal Mode?
Let a mass be situated on a spring having one natural frequency at which it can freely oscillate up and down. A stretched string having fixed ends can oscillate back and forth with a holistic spectrum of frequencies and varying patterns of vibration. These unique "Modes of Vibration" of a single string are known as standing waves or normal modes.
Did You Know?
Michael Faraday was the inventor of the standing waves. It was a Mid-19th-Century Discovery which resulted in the whole world to see the future of wave mechanics.
This particular phenomenon was first seen and studied in 1831 by Michael Faraday. He was an English scientist who keenly observed the formation of standing waves on the surface of a liquid in a container that would be vibrating.
FAQs on Standing Waves and Normal Modes Explained
1. What is a standing wave and how is it formed?
A standing wave, also known as a stationary wave, is a wave pattern that remains in a constant position. It is formed by the superposition of two identical waves (having the same frequency, wavelength, and amplitude) travelling in opposite directions through the same medium. This interference results in specific points of no vibration (nodes) and maximum vibration (antinodes), creating a wave that does not appear to move or propagate energy through the medium.
2. What is the primary difference between a standing wave and a progressive (travelling) wave?
The main difference lies in their energy transfer and amplitude. Here's a quick comparison:
- Energy Transfer: A progressive wave transfers energy from one point to another. A standing wave does not transfer energy; the energy is confined between its nodes.
- Amplitude: In a progressive wave, all particles oscillate with the same amplitude. In a standing wave, the amplitude varies with position, being zero at the nodes and maximum at the antinodes.
- Waveform: The waveform of a progressive wave moves through the medium, while the waveform of a standing wave is stationary.
3. Can you explain the concepts of nodes and antinodes in a standing wave?
Nodes and antinodes are fundamental features of a standing wave pattern.
- Nodes: These are points along the standing wave that have zero amplitude. At these points, the two interfering waves always cancel each other out due to destructive interference. They remain stationary.
- Antinodes: These are points of maximum amplitude, located midway between two consecutive nodes. At these points, the interfering waves always reinforce each other due to constructive interference.
4. What are normal modes of vibration in the context of physics?
A normal mode of an oscillating system is a specific pattern of motion where all parts of the system move sinusoidally with the same frequency and with a fixed phase relationship. These are the natural frequencies at which an object or system will vibrate if disturbed. For a stretched string fixed at both ends, the normal modes are the different standing wave patterns (like the fundamental, first overtone, second overtone, etc.) that can be sustained on the string.
5. How are standing waves and normal modes related? Are they the same thing?
They are closely related but not identical concepts. A standing wave is the physical wave pattern itself. A normal mode refers to a specific, allowable standing wave pattern that corresponds to one of the natural resonant frequencies of the system. In simple terms, a system like a guitar string can only support standing waves at certain frequencies. Each of these allowed standing wave patterns is called a normal mode. So, all normal modes are standing waves, but a standing wave is only considered a normal mode if its frequency matches one of the system's resonant frequencies.
6. Why is energy not transferred along a standing wave, even though it's formed by two energy-carrying waves?
This is a key characteristic of standing waves. While it's true that the two component travelling waves are carrying energy, they do so in opposite directions. In an ideal standing wave, the rate of energy flow from the first wave is exactly equal and opposite to the rate of energy flow from the second wave at every point. This results in zero net transfer of energy across any point in the medium. The energy is effectively 'trapped' or localized in segments between the nodes, oscillating between kinetic and potential energy within each segment.
7. What is the mathematical equation for a standing wave and what do its components represent?
The equation for a standing wave formed on a string along the x-axis is typically written as:
y(x, t) = [2A sin(kx)] cos(ωt)
Here's what each part means:
- y(x, t): The transverse displacement of the string at position 'x' and time 't'.
- [2A sin(kx)]: This term represents the amplitude of oscillation at any position 'x'. It is not constant; it depends on the position. 'A' is the amplitude of the individual interfering waves and 'k' is the angular wave number (2π/λ).
- cos(ωt): This term describes the simple harmonic motion of all particles in the wave over time. 'ω' is the angular frequency (2πf).
8. What are some common real-world examples where standing waves occur?
Standing waves are very common in nature and technology. Some key examples include:
- The vibration of strings on musical instruments like guitars, violins, and pianos.
- The column of air vibrating inside wind instruments like flutes or organ pipes.
- Microwaves inside a microwave oven, which create a standing wave pattern to heat food.
- The vibrations on the surface of a drumhead after it is struck.
- In lasers, where light forms a standing wave between two mirrors to amplify the beam.
