7 junio, 2024

Linear waves: concept, characteristics, examples

The linear waves They are those in which the superposition principle is applicable, that is, those in which the waveform and its spatiotemporal evolution can be achieved as the sum of basic solutions, for example of a 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 terms involving the dependent variable or its derivatives are raised to the first power.

For their part, non-linear waves satisfy wave equations that have quadratic terms or higher degrees in the dependent variable or in its derivatives.

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

But longitudinal waves, as well as transverse ones, 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.

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.

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Differential equation in linear waves

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

Linear waves are always associated with a differential equation of the linear type, the solution of which represents the prediction of what the disturbance will be like in later moments of an initial disturbance spatially located at the initial moment.

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

In the above equation or represents the perturbation of a certain physical quantity in the position x and in the instant youthat is to say or is a function of x and you:

u = u(x,t)

For example, if it is a sound wave in air, or can represent the variation of the pressure with respect to its undisturbed value.

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, or represents the transverse displacement with respect to the equilibrium position of the rope, as shown in the following figure:

Differential Equation Solutions

If there are 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:

u1= A⋅sin(k⋅x – ω⋅t) and u2= A⋅sin(k⋅x + ω⋅t)

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

The first solution represents a progressive wave that advances to the right, while the second to the left with rapidity. 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 = A1 cos(k1⋅x – ω1⋅t) + A2 sin(k2⋅x – ω2⋅t) is a solution.

The most relevant characteristic of linear waves is that any waveform, however complex it may be, can be obtained by adding simple harmonic waves in sine and cosine:

u(x,t) = A0 + ∑n still cos(kn)⋅x – ωn⋅t) + ∑m bm sin(km⋅x – ωm⋅t).

Dispersive and non-dispersive linear waves

In the classical linear wave equation, c represents the propagation velocity 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 Shape f(x) spreads 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 speed of propagation c can depend on the wavelength λ, that is: c = c(λ).

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

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

dispersion relations

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

Some characteristic dispersion relations of linear waves are the following:

In ocean waves where the wavelength (distance between crests) is much greater than the depth hbut that its amplitude is much smaller than the depth, the dispersion relation is:

ω = √(gH)⋅k

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

But waves in very deep water are dispersive, since their dispersion relation 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 move at different velocities, then the group (ie wave packet) velocity does not match the phase velocity.

The group velocity vg is defined as the derivative of the frequency with respect to the wavenumber in the dispersion relation: vg = ω'(k).

The following figure shows the superposition or sum of two harmonic waves u1= A⋅sin(k1⋅x – ω1⋅t) and u2= A⋅sin(k2⋅x – ω2⋅t) traveling at different speeds v1= ω1/k1 and v2= ω2/k2. Notice how the group velocity is different from the phase velocity, in this case the group velocity is ∆ω/∆k.

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

Examples of Linear Waves

Electromagnetic waves

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

Schrodinger’s equation

It is the equation that describes the dynamics of particles on an atomic scale, where the 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 is much greater than the depth. Waves in deep water follow the linear theory (known as Airy wave theory).

However, the wave that approaches the shore and forms the characteristic curling crest (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 string

The waves propagated by a taut string are linear, as long as the initial pulsation is of small amplitude, that is, the elastic limit of the string 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 characteristic harmonic and subharmonic tones of string instruments.

References

Griffiths G and Schiesser W. Linear and Nonlinear Waves. Retrieved from: sholarpedia.org.
Whitham GB (1999) “Linear and Nonlinear Waves”. wiley.
Wikipedia. nonlinear waves. Recovered from: en.wikipedia.com
Wikipedia. Nonlinear acoustic. Retrieved from: en.wikipedia.com
Wikipedia. Waves. Retrieved from: en.wikipedia.com
Wikiwaves. Nonlinear waves. Retrieved from: wikiwaves.org

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