The standing waves They are waves that propagate in a limited medium, going and coming in a part of space, unlike traveling waves, which when propagating move away from the source that originated them and do not return to it.
They are the basis of the sounds produced in musical instruments, since they easily arise on fixed strings, either at one end or both. They are also created in tight membranes such as drums or inside tubes and structures such as bridges and buildings..
When you have a fixed string at both ends, like that of a guitar, for example, waves with identical amplitude and frequency are created, which travel in opposite directions and combine to produce a phenomenon called interference.
If the waves are in phase, the peaks and valleys are aligned and result in a wave with twice the amplitude. In that case we speak of constructive interference.
But if the interfering waves are out of phase, the peaks of one meet the valleys of others and the resulting amplitude is zero. It is then about destructive interference.
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The main elements of the wave to represent it in space and time are its amplitude A, its wavelength λ and its angular frequency ω.
In the mathematical representation it is preferred to use k, than the wave number or number of times the wave occurs per unit length. That is why it is defined through the wavelength λ which is the distance between two valleys or two ridges:
k = 2π / λ
While the angular frequency relates to the period or duration of a complete oscillation, such as:
ω = 2π / T
And also the frequency f is given by:
f = ω / 2π
Therefore:
f = 1 / T
Also the waves move with speed v according:
v = λ.f
Mathematically we can express a wave using the sine function or the cosine function. Suppose we have waves of equal amplitude A, wavelength λ and frequency ω, propagating along a string and in opposite directions:
Y1 = A sin (kx - ωt)
Ytwo = A sin (kx + ωt)
When adding them we find the resulting wave andR:
YR = and1 + Ytwo = A sin (kx - ωt) + A sin (kx + ωt)
There is a trigonometric identity to find the sum:
sin α + sin β = 2 sin (α + β) / 2. cos (α - β) / 2
By this identity, the resulting wave yR remains:
YR = [2A sin kx]. cos ωt
The resulting wave has amplitude AR = 2Asen kx, which depends on the position of the particle. Then, at the points for which sin kx = 0, the amplitude of the wave vanishes, that is, there is no vibration.
These points are:
kx = π, 2π, 3π…
Since k = 2 π / λ:
(2 π / λ) x = π, 2π, 3π…
x = λ / 2, λ, 3λ / 2 ...
Destructive interference occurs at such points and is called nodes. They are separated by a distance equal to λ / 2, as deduced from the previous result.
And between two consecutive nodes are the antinodes or bellies, in which the amplitude of the wave is maximum, since constructive interference occurs there. They occur when:
sin kx = ± 1
kx = ± π / 2, 3π / 2, 5π / 2…
Again k = 2 π / λ and then:
x = λ / 4, 3λ / 4, 5λ / 4,…
The boundary conditions in the string determine what the wavelengths and frequencies are like. If a string of length L is fixed at both ends, it cannot vibrate with any frequency, because the points where the string is fixed are already nodes.
Furthermore, the separation between adjacent nodes is λ / 2, and between node and belly is λ / 4, in this way only for certain wavelengths are stationary waves produced: those in which an integer n of λ / 2 fits within of:
(λ / 2) = L, with n = 1, 2, 3, 4 ... .
Therefore:
λ = 2L / n
The different values that λ takes are called harmonics. Thus we have:
-First harmonic: λ = 2L
-Second harmonic: λ = L
-Third harmonic: λ = 2 L / 3
-Fourth harmonic: λ = L / 2
And so on.
Even though the standing wave does not seem to move, the equation is still valid:
v = λ. F
Therefore:
v = (2L / n). F
f = nv / 2L
Now, it can be shown that the speed with which a wave travels in a string depends on the tension T in it and its linear density of mass μ (mass per unit length) as:
Therefore:
-When the waves are stationary, the resulting wave does not propagate the same as its components, which go from one side to the other. There are points where y = 0 because there is no vibration: the nodes, in other words, the amplitude AR it becomes zero.
-The mathematical expression of a standing wave consists of the product of a spatial part (which depends on the x coordinate or spatial coordinates) and a temporal part.
-Between the nodes the resulting black wave oscillates in one place, while the waves going from one side to the other are out of phase there..
-Energy is not transported right at the nodes, since this is proportional to the square of the amplitude, but it is trapped between the nodes.
-The distance between adjacent nodes is half the wavelength.
-The points at which the chord is fixed are also considered nodes.
The waves in a fixed string are examples of standing waves in one dimension, whose mathematical description we offered in the previous sections..
Standing waves can also be presented in two and three dimensions, their mathematical description being a bit more complex.
-A rope fixed at one end that is oscillated by hand or with a piston on the other generates standing waves along its length.
-Playing stringed instruments such as the guitar, harp, violin and piano also create standing waves, as they have strings adjusted to different tensions and fixed at both ends..
Standing waves are also created in tubes with air, such as organ tubes..
Standing waves arise in structures such as bridges and buildings. A notable case was that of the Tacoma Narrows suspension bridge near the city of Seattle, United States. Shortly after being inaugurated in 1940, this bridge collapsed due to the standing waves created inside by the wind..
The frequency of the wind was paired with the natural frequency of the bridge, creating standing waves in it, which were increasing in amplitude until the bridge collapsed. The phenomenon is known as resonance.
In ports there is a very curious phenomenon called seiche, in which the waves of the sea produce large oscillations. This is due to the fact that the waters in the port are quite enclosed, although the oceanic waters penetrate from time to time through the entrance of the port..
Port waters move with their own frequency, just like those of the ocean. If both waters equal their frequencies, a large standing wave is produced by resonance, as happened with the Tacoma bridge..
The seiches They can also occur in lakes, reservoirs, swimming pools, and other surface-limited bodies of water..
Standing waves can be created in a fish tank carried by a person, if the frequency with which the person walks equals the frequency of the swaying of the water.
A guitar string has L = 0.9 m and a linear mass density μ = 0.005 kg / m. It is subjected to 72 N of tension and its mode of vibration is the one shown in the figure, with amplitude 2A = 0.5 cm.
Find:
a) Velocity of propagation
b) Wave frequency
c) The corresponding standing wave equation.
Through:
Is obtained;
v = [72 N / (0.005 kg / m)]1/2 = 120 m / s.
The distance between two adjacent nodes is λ / 2, therefore:
(2/3) L - (1/3) L = λ / 2
(1/3) L = λ / 2
λ = 2L / 3 = 2 x 0.90 m / 3 = 0.60 m.
Since v = λ.f
f = (120 m / s) / 0.60 m = 200 s-1= 200 Hz.
The equation is:
YR = [2A sin kx]. cos ωt
We need to substitute the values:
k = 2π / λ = k = 2π / 0.60 m = 10 π / 3
f = ω / 2π
ω = 2π x 200 Hz = 400 π Hz.
The amplitude 2A is already given by the statement:
2A = 0.5 cm = 5 x 10 -3 m.
Therefore:
YR = 5 x 10 -3 m. sin [(10π / 3) x]. cos (400πt) =
= 0.5 cm. sin [(10π / 3) x]. cos (400πt)
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