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Standing Waves and Normal Modes

Last updated date: 09th Apr 2024
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Standing Wave Definition

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.

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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

1. What is the speed of a standing wave?

Let's consider a standing wave freely reversing in an isolated environment. In that particular case, the energy which transmits along a standing wave is completely zero. But the whole game changes if we treat the standing wave as two separate traveling waves in opposite directions. Then it applies as if the velocity of the standing wave is measured to be zero. This separation of standing waves can supply us the right energies of the standing waves, which are clamped or fastened with stretching strings (the length is L). The speed of the standing wave can be measured with this method and can be tabulated for future use.

2. Why do only specific frequencies produce standing waves?

A standing wave pattern is a particular vibrational pattern created within a medium as soon as the vibrational frequency of the source causes reflected waves to move from one end of the medium to interfere with incident waves that are generated from the source. Such patterns are only created, ranging within the medium at particular frequencies of vibration. These particular frequencies are noticed, tabulated for experimental purposes of experimenting with the standing waves or its nodal or antinodal forms. If we can reproduce those frequencies with the help of instruments, we can see the occurrence of the standing waves again. So, in turn, we can repeatedly experiment with the standing waves too.

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