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A standing wave, unlike a moving wave, does not appear to move. Each point on the axis of the standing wave will oscillate around that point. Adjacent points are in phase with one another (parts of the wave flap up and down at the same time), therefore points of a given phase stay in the same place as time passes. Adjacent locations have distinct amplitude oscillations. (In a travelling wave, adjacent places have the same amplitude but the phase shifts along the wave.) Unlike a travelling wave, no energy is transferred from one location to the next since adjacent points are in phase.
Standing waves are created by the superposition of two travelling waves of the same frequency moving in opposite directions. This is commonly accomplished by the use of a travelling wave and its reflection, which ensures that the frequency remains constant.
Antinodes are the highest-amplitude oscillating spots on a stationary wave. Nodes are zero-amplitude spots that appear to be fixed.
Standing wave in String and Pipes
If a string has fixed ends, these cannot oscillate, and so any stationary wave established on it must have nodes at both ends. The length of the string L, is related to the wavelength of possible standing waves λ by
L= nλ/2
where,n=1,2,3…
The frequencies of the standing waves which can be supported in this example will be f
f= c/λ =n c/2L
The 1st harmonic (fundamental frequency) is the lowest frequency of standing waves that can be supported on the system. The second, third, fourth and so on harmonics are simple multiples of this fundamental frequency.
If one of the string’s endpoints is free, however, it must be an antinode. As a result, the wavelength and the length of the string are connected.
L = (2n-1)λ/2
where n = 1, 2, 3,
The frequencies of standing waves that can be maintained in the medium in this situation are f,
f= c/λ=(2n-1)c/4L
As a result, if the fundamental frequency is f0,, the higher harmonics supported are 3f0,, 5f0, 7f0, and so on. The odd harmonics are present, but the even harmonics are absent.
Acoustic Waves
Acoustic waves are mechanical and longitudinal waves (with the same vibration direction as the propagation direction) that emerge from a pressure oscillation travelling in a wave pattern through a solid, liquid, or gas. Wavelength, frequency, duration, and amplitude are only a few of the features of these waves. Sound is experienced as acoustic waves by the ear.
Derivation Of Standing Waves
A wave travelling positive x direction can be written as
y1 =Acos(ωt−kx)
where ω=2πf is the angular frequency of the wave, t is time,
k= λ/ 2π is the wavenumber and x is position.
The reflected wave of the same amplitude, travelling in the −x direction can then be written as
y2 =Acos(ωt+kx)
The resultant wave formed by superposition of these two is y=y1+y2
y=A(cos(ωt−kx)+cos(ωt+kx))
using the trigonometric formula for the addition of two cosines:
y=2Acos(ωt)cos(kx)
The time and position terms become independent, so that for a fixed time the displacements at each point y(x) are modulated by the cos(kx) term and for a fixed position y varies with time. However the effects on y of these two variables are independent – x determines the amplitude of the oscillation at a point and x determines the phase of the entire wave – there is no point of given phase that varies with x and t. This means that the wave no longer travels, but has instead a maximum displacement which depends on position.
y (x)=2Acos(kx)
Conclusion
Finally, standing wave patterns are created by the repetitive interference of two waves of the same frequency travelling in opposite directions across the same medium. There are nodes and antinodes in every standing wave pattern. Acoustic waves are mechanical and longitudinal waves (with the same vibration direction as the propagation direction) that emerge from a pressure oscillation travelling in a wave pattern through a solid, liquid, or gas.
