Standing Waves: Formulas, Characteristics, Types, Examples

Standing waves 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 in fixed strings, either at one end or both. They are also created on tight membranes such as drums or inside pipes and structures such as bridges and buildings.

When you have a fixed string at both ends, like that of a guitar, for example, waves of identical amplitude and frequency are created, traveling in opposite directions and combining 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.

Formulas and equations

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 is related 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

Thus:

f = 1 / T

Furthermore, the waves move with speed v according to:

v = λ.f

Mathematical expression of the standing wave

Mathematically we can express a wave by the sine function or the cosine function. Suppose that we have waves of equal amplitude A, wavelength λ and frequency ω, propagating along a string and in opposite directions:

y 1 = A sin (kx – ωt)

y 2 = A sin (kx + ωt)

When adding them we find the resulting wave and R :

y R = y 1 + y 2 = 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

Through this identity, the resulting wave and R remain:

and R = [2A sin kx]. cos ωt

Location of nodes and bellies

The resulting wave has amplitude A R = 2A sin 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 …

At such points destructive interference occurs and are called nodes . They are separated by a distance equal to λ / 2, as follows 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,…

Normal modes on a string

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 at 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 the:

(λ / 2) = L, with n = 1, 2, 3, 4….

Thus:

λ = 2L / n

Harmonics

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.

Speed ​​and frequency

Even though the standing wave does not seem to move, the equation is still valid:

v = λ. F

Thus:

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 on its linear density of mass μ (mass per unit length) as:

Original text

$v=\sqrt{\frac{T}{\mu&space;}}$

Thus:

$f=\frac{n}{2L}\sqrt{\frac{T}{\mu&space;}}$

Characteristics of standing waves

-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 A R 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 that go from one side to the other are out of phase there.

-Just in the nodes, no energy is transported, 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 rope is fixed are also considered nodes.

Types

Standing waves in one dimension

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 in two and three dimensions

Standing waves can also be presented in two and three dimensions, being their mathematical description a bit more complex.

Examples of standing waves

Fixed ropes

-A fixed rope at one end that is oscillated by hand or with a piston on the other generates standing waves along its length.

Musical instruments

-When playing string instruments such as the guitar, the harp, the violin and the piano, standing waves are also created, since they have strings adjusted to different tensions and fixed at both ends.

Standing waves are also created in tubes of air, like the tubes in organs.

Buildings and bridges

Standing waves arise in structures such as bridges and buildings. A notable case was the Tacoma Narrows suspension bridge near the city of Seattle, United States. Shortly after being inaugurated in 1940, this bridge collapsed because of 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.

Seiches

In the ports a very curious phenomenon called seiche occurs , 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, as well as ocean waters. If both waters equal their frequencies, a large standing wave is produced by resonance, as happened with the Tacoma bridge.

The seiche can also occur in lakes, reservoirs, swimming pools and other water bodies limited surfaces.

Fish tanks

Standing waves can be created in a fish tank carried by a person, if the frequency with which the person walks is equal to the frequency of the swaying of the water.

Exercise resolved

A guitar string has L = 0.9 m and 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.

Solution to

Through:

$v=\sqrt{\frac{T}{\mu&space;}}$

Is obtained;

v = [72 N / (0.005 kg / m)] 1/2  = 120 m / s.

Solution b

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.

Solution c

The equation is:

and R = [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.

Thus:

and R = 5 x 10 -3 m. sin [(10π / 3) x]. cos (400πt) =

= 0.5 cm. sin [(10π / 3) x]. cos (400πt)

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 7. Waves and Quantum Physics. Edited by Douglas Figueroa (USB).
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
4. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7ma. Ed. Cengage Learning.
5. Tipler, P. (2006) Physics for Science and Technology. 5th Ed. Volume 1. Editorial Reverté.
6. Wikipedia. Seiche. Recovered from: es.wikipedia.org.