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Linear waves: concept, characteristics, examples

We explain the Linear waves with concept, characteristics and examples. The  linear waves  are those in which the superposition principle applies, namely those in which the waveform and space-time evolution can be achieved as the sum of basic solutions, for example of harmonic type. Not all waves comply with the superposition principle, those that do not comply are called non-linear waves.

The name “linear” comes from the fact that linear waves always satisfy a differential equation in partial derivatives, in which all the terms that involve the dependent variable or its derivatives are raised to the first power.

The waves seen in the distance are linear waves, however the rough waves in the foreground are non-linear.
The waves seen in the distance are linear waves, however the rough waves in the foreground are non-linear.

On the other hand, non-linear waves satisfy wave equations that have quadratic terms or higher degrees in the dependent variable or its derivatives.

Linear waves are sometimes confused with longitudinal waves, which are those in which the vibration  occurs in the same direction of propagation, as sound waves.

But longitudinal waves, as well as transverse waves, can in turn be linear or non-linear depending on, among other factors, the amplitude of the initial disturbance and the medium in which they propagate.

The waves seen in the distance are linear waves, however the rough waves in the foreground are non-linear.
The waves seen in the distance are linear waves, however the rough waves in the foreground are non-linear.

It generally happens that when the initial disturbance is of small amplitude, the equation that describes the propagation of the wave is linear or can be linearized by means of certain approximations, although this is not always the case.

Differential equation in linear waves

In a linear medium, a waveform limited in space and time can be represented by the sum of wave functions of the sine or cosine type of different frequencies and wavelengths using Fourier series.

Linear waves always have associated a differential equation of the linear type, whose solution represents the prediction of what the disturbance will be in later instants of an initial disturbance located spatially at the initial instant.

The classical linear wave equation, in a single spatial dimension, whose solutions are linear waves is:

In the above equation u represents the disturbance of a certain physical quantity at position x and at time t , that is, u is a function of x and t :

u = u (x, t)

For example, if it is a sound wave in air,  u can represent the variation of the pressure with respect to its value without disturbing.

In the case of an electromagnetic wave, u represents the electric field or the magnetic field oscillating perpendicular to the direction of propagation.

In the case of a taut rope,  u represents the transverse displacement with respect to the equilibrium position of the rope, as shown in the following figure:

Waveform at a given instant, in the case of linear waves this form is the superposition of sinusoidal waves of different frequencies and wavelengths.
Waveform at a given instant, in the case of linear waves this form is the superposition of sinusoidal waves of different frequencies and wavelengths.

Solutions of the differential equation

If you have two or more solutions of the linear differential equation, then each solution multiplied by a constant will be a solution and so will be the sum of them.

Unlike non-linear equations, linear wave equations admit harmonic solutions of the type:

1 = A⋅sen (k⋅x – ω⋅t) and 2 = A⋅sen (k⋅x + ω⋅t)

This can be verified by simple substitution in the linear wave equation.

The first solution represents a traveling wave moving to the right, while the second solution to the left with speed c = ω / k .

Harmonic solutions are characteristic of linear wave equations.

On the other hand, the linear combination of two harmonic solutions is also a solution of the linear wave equation, for example:

u = A 1 cos (k 1 ⋅x – ω 1 ⋅t) + A 2 sin (k 2 ⋅x – ω 2 ⋅t) is a solution.

The most relevant characteristic of linear waves is that any waveform, no matter how complex, can be obtained through a summation of simple harmonic waves in sine and cosine:

u (x, t) = A 0 + ∑ n  A n  cos (k n ⋅x – ω n ⋅t) + ∑ m  B m  sin (k m ⋅x – ω m ⋅t) .

Dispersive and non-dispersive linear waves

In the classical linear wave equation,  c represents the speed of propagation of the pulse.

Non-dispersive waves

In cases where c is a constant value, for example electromagnetic waves in a vacuum, then a pulse at the initial instant t = 0 of the form f (x) propagates according to:

u (x, t) = f (x – c⋅t)

Without suffering any distortion. When this occurs, the medium is said to be non-dispersive.

Dispersive waves

However, in dispersive media the propagation speed c can depend on the wavelength λ, that is: c = c (λ).

Electromagnetic waves are dispersive when traveling through a material medium. Also the surface waves of the water travel at different speeds depending on the depth of the water.

The speed with which a harmonic wave of the type A⋅sen (k⋅x – ω⋅t) propagates is ω / k = c and is called the phase velocity. If the medium is dispersive, then c is a function of wave number k : c = c (k) , where k is related to wavelength by k = 2π / λ .

Dispersion ratios

The relationship between frequency and wavelength is called the dispersion relationship , which expressed in terms of the angular frequency ω and the wave number k is: ω = c (k) ⋅k .

Some characteristic dispersion relationships of linear waves are as follows:

In sea waves in which the wavelength (distance between crests) is much greater than the depth H , but whose amplitude is much less than the depth, the dispersion ratio is:

ω = √ (gH) ⋅k

From there it is concluded that they propagate at a constant speed  √ (gH) (non-dispersive medium).

But the waves in very deep waters are dispersive, since their dispersion ratio is:

ω = √ (g / k) ⋅k

This means that the phase velocity ω / k  is variable and depends on the wave number and therefore on the wavelength of the wave.

Group speed

If two harmonic linear waves overlap but advance at different speeds, then the group velocity (that is, of the wave packet) does not match the phase velocity.

The group velocity  g is defined as the derivative of the frequency with respect to the wave number in the dispersion relation: g  = ω ‘(k) .

The following figure shows the superposition or sum of two harmonic waves 1 = A⋅sen (k 1 ⋅x – ω 1 ⋅t) and 2 = A⋅sen (k 2 ⋅x – ω 2 ⋅t) that travel at different speeds 1 = ω 1 / k 1  and 2 = ω 2 / k 2 . Note how the group velocity is different from the phase velocity, in this case the group velocity is ∆ω / ∆k .

Linear wave (blue) in a dispersive medium. The red curve has been added to highlight that the group velocity is different from the propagation velocity
Linear wave (blue) in a dispersive medium. The red curve has been added to highlight that the group velocity is different from the propagation velocity

Depending on the dispersion ratio, it may even happen that the phase velocity and the group velocity, in linear waves, have opposite directions.

Examples of linear waves

Electromagnetic waves

electromagnetic waves that make up electromagnetic radiation
electromagnetic waves that make up electromagnetic radiation

Electromagnetic waves are linear waves. Its wave equation is derived from the equations of electromagnetism (Maxwell’s equations) which are also linear.

The Schrödinger equation

It is the equation that describes the dynamics of particles on an atomic scale, where wave characteristics are relevant, for example the case of electrons in the atom.

So the “electron wave” or wave function as it is also called, is a linear wave.

Waves in deep water

Linear waves are also those in which the amplitude is much less than the wavelength and the wavelength much greater than the depth. Waves in deep water follow the linear theory (known as Airy’s wave theory).

However, the wave that approaches the shore and forms the characteristic curling crest (and which surfers love) is a non-linear wave.

Sound

Since sound is a small disturbance of atmospheric pressure, it is considered a linear wave. However, the shock wave from an explosion or the wave front from a supersonic aircraft are typical examples of a non-linear wave.

Waves on a taut rope Linear waves

The waves that propagate through a taut rope are linear, as long as the initial pulsation is of small amplitude, that is, the elastic limit of the rope is not exceeded.

Linear waves in the strings are reflected at their ends and overlap, giving rise to standing waves or vibrational modes that give the harmonic and subharmonic tones characteristic of string instruments.

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