Helmholtz equation

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Helmholtz_equation

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation:

$${\displaystyle {\mathrm{\nabla }}^{2}f=-k^{2}f,}$$

where ∇² is the Laplace operator, k² is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

In optics, the Helmholtz equation is the wave equation for the electric field.

The equation is named after Hermann von Helmholtz, who studied it in 1860.

Motivation and uses

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

For example, consider the wave equation

$${\displaystyle \left({\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)u(\mathbf{r},t)=0.}$$

Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:

$${\displaystyle u(\mathbf{r},t)=A(\mathbf{r})T(t).}$$

Substituting this form into the wave equation and then simplifying, we obtain the following equation:

$${\displaystyle \frac{{\mathrm{\nabla }}^{2}A}{A}=\frac{1}{c^{2}T}\frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}.}$$

Notice that the expression on the left side depends only on r, whereas the right expression depends only on t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for A(r), the other for T(t):

$${\displaystyle \frac{{\mathrm{\nabla }}^{2}A}{A}=-k^2}$$

$${\displaystyle \frac{1}{c^{2}T}\frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}=-k^2,}$$

where we have chosen, without loss of generality, the expression −k² for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k² is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

$${\displaystyle {\mathrm{\nabla }}^{2}A+k^{2}A=({\mathrm{\nabla }}^2+k^2)A=0.}$$

Likewise, after making the substitution ω = kc, where k is the wave number, and ω is the angular frequency (assuming a monochromatic field), the second equation becomes

$${\displaystyle \frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}+{\omega }^{2}T=\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}+{\omega }^2\right)T=0.}$$

We now have Helmholtz's equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation:

$${\displaystyle {\mathrm{\nabla }}^{2}A=-k^{2}A}$$

can be obtained for simple geometries using separation of variables.

Vibrating membrane

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu's differential equation.

If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).

If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form

$${\displaystyle A_{rr}+\frac{1}{r}{A}_r+\frac{1}{r^2}{A}_{\theta \theta }+k^{2}A=0.}$$

We may impose the boundary condition that A vanishes if r = a; thus

$${\displaystyle A(a,\theta )=0.}$$

the method of separation of variables leads to trial solutions of the form

$${\displaystyle A(r,\theta )=R(r)\mathrm{\Theta }(\theta ),}$$

where Θ must be periodic of period 2π. This leads to

$${\displaystyle {\mathrm{\Theta }}^{″}+n^2\mathrm{\Theta }=0,}$$

$${\displaystyle r^2{R}^{″}+r{R}^{\prime }+r^2{k}^{2}R-n^{2}R=0.}$$

It follows from the periodicity condition that

$${\displaystyle \mathrm{\Theta }=\alpha \cos n\theta +\beta \sin n\theta ,}$$

and that n must be an integer. The radial component R has the form

$${\displaystyle R(r)=\gamma J_n(\rho ),}$$

where the Bessel function J_n(ρ) satisfies Bessel's equation

$${\displaystyle {\rho }^2{J}_n^{″}+\rho J_n^{\prime }+({\rho }^2-n^2)J_n=0,}$$

and ρ = kr. The radial function J_n has infinitely many roots for each value of n, denoted by ρ_m,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by

$${\displaystyle k_{m,n}=\frac{1}{a}{\rho }_{m,n}.}$$

The general solution A then takes the form of a generalized Fourier series of terms involving products of J_n(k_m,nr) and the sine (or cosine) of nθ. These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutions

In spherical coordinates, the solution is:

$${\displaystyle A(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\left(a_{\ell m}{j}_{\ell }(kr)+b_{\ell m}{y}_{\ell }(kr)\right)Y_{\ell }^m(\theta ,\phi ).}$$

This solution arises from the spatial solution of the wave equation and diffusion equation. Here j_ℓ(kr) and y_ℓ(kr) are the spherical Bessel functions, and Y^m
_ℓ(θ, φ) are the spherical harmonics (Abramowitz and Stegun, 1964). Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. For infinite exterior domains, a radiation condition may also be required (Sommerfeld, 1949).

Writing r₀ = (x, y, z) function A(r₀) has asymptotics

$${\displaystyle A(r_0)=\frac{e^{ik{r}_0}}{r_0}f\left(\frac{{\mathbf{r}}_0}{r_0},k,u_0\right)+o\left(\frac{1}{r_0}\right)\text{ as }{r}_0\to \mathrm{\infty }}$$

where function f is called scattering amplitude and u₀(r₀) is the value of A at each boundary point r₀.

Three-dimensional solutions given the function on a 2-dimensional plane

Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:^[3]

$${\displaystyle A(x,y,z)=-\frac{1}{2\pi }{\iint }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{A}^{\prime }(x^{\prime },y^{\prime })\frac{e^{ikr}}{r}\frac{z}{r}\left(ik-\frac{1}{r}\right)d{x}^{\prime }d{y}^{\prime },}$$

where

• ${\displaystyle A^{\prime }(x^{\prime },y^{\prime })}$ is the solution at the 2-dimensional plane,
• ${\displaystyle r=\sqrt{(x-x^{\prime }{)}^2+(y-y^{\prime }{)}^2+z^2},}$

As z approaches zero, all contributions from the integral vanish except for r=0. Thus ${\displaystyle A(x,y,0)=A^{\prime }(x,y)}$ up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates ${\displaystyle (\rho ,\theta )}$.

This solution is important in diffraction theory, e.g. in deriving Fresnel diffraction.

Paraxial approximation

Further information: Slowly varying envelope approximation

In the paraxial approximation of the Helmholtz equation, the complex amplitude A is expressed as

$${\displaystyle A(\mathbf{r})=u(\mathbf{r})e^{ikz}}$$

where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves

$${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}u+2ik\frac{\mathrm{\partial }u}{\mathrm{\partial }z}=0,}$$

where ${\mathrm{\nabla }}_{\perp }^2\stackrel{\text{ def }}{=}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}$ is the transverse part of the Laplacian.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form.

The assumption under which the paraxial approximation is valid is that the z derivative of the amplitude function u is a slowly varying function of z:

$${\displaystyle \left|\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right|\ll \left|k\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\right|.}$$

This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: θ ≪ 1.

The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:

$${\displaystyle {\mathrm{\nabla }}^2(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.}$$

Expansion and cancellation yields the following:

$${\displaystyle \left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)u(x,y,z)e^{ikz}+\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}u(x,y,z)\right)e^{ikz}+2\left(\frac{\mathrm{\partial }}{\mathrm{\partial }z}u(x,y,z)\right)ik{e}^{ikz}=0.}$$

Because of the paraxial inequality stated above, the ∂²u/∂z² term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting u(r) = A(r) e^−ikz then gives the paraxial equation for the original complex amplitude A:

$${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}A+2ik\frac{\mathrm{\partial }A}{\mathrm{\partial }z}+2k^{2}A=0.}$$

The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.

Inhomogeneous Helmholtz equation

Two sources of radiation in the plane, given mathematically by a function f, which is zero in the blue region

 

The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation (∇² + k²) A = −f.

The inhomogeneous Helmholtz equation is the equation

$${\displaystyle {\mathrm{\nabla }}^{2}A(\mathbf{x})+k^{2}A(\mathbf{x})=-f(\mathbf{x})\text{ in }{\mathbb{R}}^n,}$$

where ƒ : Rⁿ → C is a function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) were switched to a minus sign.

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

$${\displaystyle \underset{r\to \mathrm{\infty }}{lim}{r}^{\frac{n-1}{2}}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }r}-ik\right)A(\mathbf{x})=0}$$

in ${\displaystyle n}$ spatial dimensions, for all angles (i.e. any value of ${\displaystyle \theta ,\varphi }$). Here ${\displaystyle r=\sqrt{\sum _{i=1}^n{x}_i^2}}$ where ${\displaystyle x_i}$ are the coordinates of the vector ${\displaystyle \mathbf{x}}$.

With this condition, the solution to the inhomogeneous Helmholtz equation is

$${\displaystyle A(\mathbf{x})={\int }_{{\mathbb{R}}^n}G(\mathbf{x},{\mathbf{x}}^{\prime })f({\mathbf{x}}^{\prime })\mathrm{d}{\mathbf{x}}^{\prime }}$$

(notice this integral is actually over a finite region, since f has compact support). Here, G is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with f equaling the Dirac delta function, so G satisfies

$${\displaystyle {\mathrm{\nabla }}^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })+k^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })=-\delta (\mathbf{x},{\mathbf{x}}^{\prime })\in {\mathbb{R}}^n.}$$

The expression for the Green's function depends on the dimension n of the space. One has

$${\displaystyle G(x,x^{\prime })=\frac{i{e}^{ik|x-x^{\prime }|}}{2k}}$$

for n = 1,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{i}{4}{H}_0^{(1)}(k|\mathbf{x}-{\mathbf{x}}^{\prime }|)}$$

for n = 2, where H⁽¹⁾
₀ is a Hankel function, and

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{e^{ik|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{4\pi |\mathbf{x}-{\mathbf{x}}^{\prime }|}}$$

for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for |x| → ∞.

Finally, for general n,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=c_d{k}^p\frac{H_p^{(1)}(k|\mathbf{x}-{\mathbf{x}}^{\prime }|)}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^p}}$$

where ${\displaystyle p=\frac{n-2}{2}}$ and ${\displaystyle c_d=\frac{1}{2i(2\pi {)}^p}}$.

Quadratic irrational number

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Quadratic_irrational_number

In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

${\displaystyle \frac{a+b\sqrt{c}}{d},}$

for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.

Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c). For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers:

${\displaystyle \frac{d}{a+b\sqrt{c}}=\frac{ad-bd\sqrt{c}}{a^2-b^{2}c}.}$

Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms. For example

${\displaystyle \sqrt{3}=1.732\dots =[1;1,2,1,2,1,2,\dots ]}$

The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers. The correspondence is explicitly provided by Minkowski's question mark function, and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map ${\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }$ for continued fractions.

Real quadratic irrational numbers and indefinite binary quadratic forms

We may rewrite a quadratic irrationality as follows:

${\displaystyle \frac{a+b\sqrt{c}}{d}=\frac{a+\sqrt{b^{2}c}}{d}.}$

It follows that every quadratic irrational number can be written in the form

${\displaystyle \frac{a+\sqrt{c}}{d}.}$

This expression is not unique.

Fix a non-square, positive integer ${\displaystyle c}$ congruent to ${\displaystyle 0}$ or ${\displaystyle 1}$ modulo ${\displaystyle 4}$, and define a set ${\displaystyle S_c}$ as

${\displaystyle S_c=\left\{\frac{a+\sqrt{c}}{d}:a,d\text{ integers, }d\text{ even},a^2\equiv c(\mathrm{mod}2d)\right\}.}$

Every quadratic irrationality is in some set ${\displaystyle S_c}$, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.

A matrix

${\displaystyle \left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)}$

with integer entries and ${\displaystyle \alpha \delta -\beta \gamma =1}$ can be used to transform a number ${\displaystyle y}$ in ${\displaystyle S_c}$. The transformed number is

${\displaystyle z=\frac{\alpha y+\beta }{\gamma y+\delta }}$

If ${\displaystyle y}$ is in ${\displaystyle S_c}$, then ${\displaystyle z}$ is too.

The relation between ${\displaystyle y}$ and ${\displaystyle z}$ above is an equivalence relation. (This follows, for instance, because the above transformation gives a group action of the group of integer matrices with determinant 1 on the set ${\displaystyle S_c}$.) Thus, ${\displaystyle S_c}$ partitions into equivalence classes. Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix. Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.

There are finitely many equivalence classes of quadratic irrationalities in ${\displaystyle S_c}$. The standard proof of this involves considering the map ${\displaystyle \varphi }$ from binary quadratic forms of discriminant ${\displaystyle c}$ to ${\displaystyle S_c}$ given by

${\displaystyle \varphi (t{x}^2+uxy+v{y}^2)=\frac{-u+\sqrt{c}}{2t}}$

A computation shows that ${\displaystyle \varphi }$ is a bijection that respects the matrix action on each set. The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.

Through the bijection ${\displaystyle \varphi }$, expanding a number in ${\displaystyle S_c}$ in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.

Square root of non-square is irrational

The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ax² + bx + c = 0 are

${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}.}$

Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since b and 2a are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational.

The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.

Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational.

Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in Euclid's Elements Book X Proposition 9.

The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by Richard Dedekind, among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by Theodor Estermann in 1975.

If D is a non-square natural number, then there is a natural number n such that:

n² < D < (n + 1)²,

so in particular

0 < √D − n < 1.

If the square root of D is rational, then it can be written as the irreducible fraction p/q, so that q is the smallest possible denominator, and hence the smallest number for which q√D is also an integer. Then:

(√D − n)q√D = qD − nq√D

which is thus also an integer. But 0 < (√D − n) < 1 so (√D − n)q < q. Hence (√D − n)q is an integer smaller than q which multiplied by √D makes an integer. This is a contradiction, because q was defined to be the smallest such number. Therefore, √D cannot be rational.