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: