Heat Equation
The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is
also written
or sometimes
where α is a positive constant and or denotes the Laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.Let a thin square-formed plate of heat conducting homogeneous material be in the - plane with sides on the -axis (isolated), on the line (held at the constant temperature ), and on the vertical lines and (both held at the constant temperature ). Determine the temperature function on the plate, when the faces of the plate are isolated.
The equation of the heat flow in this stationary case is
(1)
under the boundary conditions
We first try to separate the variables, i.e. seek the solution of (1) of the form
Then we get
and thus (1) gets the form
(2)
and the boundary conditions
We separate the variables in (2):
This equation is not possible unless both sides are equal to a same negative constant , which implies for the solution
and for the solution
The two first boundary conditions give , , and since , we must have , i.e.
Therefore
The fourth boundary condition now yields that ; thus and So (1) has infinitely many solutions
(3)
with and they all satisfy the boundary conditions except the third. Because of the linearity of (1), also the sum
of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges. The third boundary condition requires that
on the interval . But this is the Fourier sine series of the constant function on the half-interval , whence
The even 's here give 0 and the odd give
Thus we obtain the solution
It can be shown that this series converges in the whole square of the plate.
Applications
I. Particle diffusion
One can model particle diffusion by an equation involving either:
• the volumetric concentration of particles, denoted c, in the case of collective diffusion of a large number of particles, or
• the probability density function associated with the position of a single particle, denoted P.
In either case, one uses the heat equation
or
Both c and P are functions of position and time. D is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second. If the diffusion coefficient D is not constant, but depends on the concentration c (or P in the second case), then one gets the nonlinear diffusion equation.
II. Brownian motion
The random trajectory of a single particle subject to the particle diffusion equation (or heat equation) is a Brownian motion. If a particle is placed at at time t = 0, then the probability density function associated with the position vector of the particle will be the following:
which is a (multivariate) normal distribution evolving in time.
III. Schrodinger equation for a free particle
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
, where i is the unit imaginary number, and is Planck's constant divided by 2π, and ψ is the wavefunction of the particle.
This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t=0:
, with
Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.
IV. Thermal diffusivity in polymers
A direct practical application of the heat equation, in conjunction with Fourier theory, in spherical coordinates, is the measurement of the thermal diffusivity in polymers (Unsworth and Duarte). The dual theoretical-experimental method demonstrated by these authors is applicable to rubber and various other materials of practical interest.
V. Further applications
The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The famous Black–Scholes option pricing model's differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the Crank–Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995).An abstract form of heat equation on manifolds provides a major approach to the Atiyah–Singer index theorem, and has led to much further work on heat equations in Riemannian geometry.
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.
The wave equation is the prototypical example of a hyperbolic partial differential equation. In its simplest form, the wave equation refers to a scalar function u = u (x1, x2, …, xn; t) that satisfies:
where is the (spatial) Laplacian and where c is a fixed constant equal to the propagation speed of the wave. This is known as the non-dispersive wave equation. For a sound wave in air at 20°C this constant is about 343 m/s (see speed of sound). For the vibration of a string the speed can vary widely, depending upon the linear density of the string and the tension on it. For a spiral spring (a slinky) it can be as slow as a meter per second. More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. In such a case, c must be replaced by the phase velocity:
Another common correction in realistic systems is that the speed can also depend on the amplitude of the wave, leading to a nonlinear wave equation:
Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar u will contain a Mach factor (which is positive for the wave moving along the flow and negative for the reflected wave).
The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:
where:
• and are the so-called Lamé parameters describing the elastic properties of the medium,
• is the density,
• is the source function (driving force),
• and is the displacement vector.
Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.
Variations of the wave equation are also found in quantum mechanics, plasma physics and general relativity.
string has been strained between the points (0 0) and (p 0) of the x -axis. The transversal vibration of the string in the xy -plane is determined by the one-dimensional wave equation
(1)
satisfied by the ordinates u(x t) of the points of the string with the abscissa x on the time moment t( 0) . The boundary conditions are thus u(0 t)=u(p t)=0 We suppose also the initial conditions u(x 0)=f(x) u t(x 0)=g(x) which give the initial position of the string and the initial velocity of the points of the string.
For trying to separate the variables, set u(x t):=X(x)T(t) The boundary conditions are then X(0)=X(p)=0 , and the partial differential equation (1) may be written
(2)
This is not possible unless both sides are equal to a same constant −k2 where k is positive; we soon justify why the constant must be negative. Thus (2) splits into two ordinary linear differential equations of second order:
(3)
The solutions of these are, as is well known,
(4)
with integration constants Ci and Di .
But if we had set both sides of (2) equal to +k2 , we had got the solution T=D1ekt+D2e−kt which can not present a vibration. Equally impossible would be that k=0 .Now the boundary condition for X(0) shows in (4) that C1=0 , and the one for X(p) that C2sinckp=0 If one had C2=0 , then X(x) were identically 0 which is naturally impossible. So we must have sinckp=0 which implies ckp=n (n +) This means that the only suitable values of k satisfying the equations (3), the so-called eigenvalues, are k=pn c(n=1 2 3 ) So we have infinitely many solutions of (1), the eigenfunctions u=XT=C2sinpn x D1cospn ct+D2sinpn ct or u= Ancospn ct+Bnsinpn ct sinpn x (n=1 2 3 ) where An 's and Bn 's are for the time being arbitrary constants. Each of these functions satisfy the boundary conditions. Because of the linearity of (1), also their sum series
(5)
is a solution of (1), provided it converges. It fulfils the boundary conditions, too. In order to also the initial conditions would be fulfilled, one must have n=1Ansinpn x=f(x) n=1Bnpn csinpn x=g(x) on the interval [0 p] . But the left sides of these equations are the Fourier sine series of the functions f and g , and therefore we obtain the expressions for the coefficients: An=2p 0pf(x)sinpn xdx Bn=2n c 0pg(x)sinpn xdx
Applications of wave equation:-
The ideal-string wave equation applies to any perfectly elastic medium which is displaced along one dimension. For example, the air column of a clarinet or organ pipe can be modeled using the one-dimensional wave equation by substituting air-pressure deviation for string displacement, and longitudinal volume velocity for transverse string velocity. We refer to the general class of such media as one-dimensional waveguides. Extensions to two and three dimensions (and more, for the mathematically curious), are also possible .
For a physical string model, at least three coupled waveguide models should be considered. Two correspond to transverse-wave vibrations in the horizontal and vertical planes (two polarizations of planar vibration); the third corresponds to longitudinal waves. For bowed strings, torsional waves should also be considered, since they affect bow-string dynamics.
In the piano, for key ranges in which the hammer strikes three strings simultaneously, nine coupled waveguides are required per key for a complete simulation (not including torsional waves); however, in a practical, high-quality, virtual piano, one waveguide per coupled string (modeling only the vertical, transverse plane) suffices quite well. It is difficult to get by with fewer than the correct number of strings, however, because their detuning determines the entire amplitude envelope as well as beating and aftersound effects .
The wave equation with a source is
( ,t) = - 4 f ( ,t) . (1)
and the Green's function is the solution with a delta function source,
( ',t'; ,t) = - 4 ( - r') (t - t') (2)
where, as before, the Green's function is generally the solution of the adjoint equation, but the wave equation is self adjoint.
To use the harmonic oscillator result, we want to eliminate the spatial operators. We can do this exactly as we did for the Green's function for Poisson's Equation. Expanding in the eigenfunctions of . We write
- ( ) = kn2 ( ) , (3)
where the eigenfunctions are determined by the spatial boundary conditions that we enforce at the boundaries of our system. For free space, the eigenfunctions are plane waves, and the transformation to eigenfunctions is the same Fourier transform that we used for the Coulomb case. Eigenfunctions in a finite volume such as a cavity, are typically normalized to one,
d 3r ( ) ( ) = (4)
while the free space Fourier transform is typically normalized to Dirac delta functions using
d 3re - i e i ' = (2 )3 ( - ') (5)
which was derived in class by taking the limit of the periodic box normalized solutions.
The general solution of the wave equation is given by expanding the Green's function in the complete set of eigenfunctions,
G( ,t', ,t) = an(t) ( ) (6)
where an(t) are the expansion coefficients which also depend on and t' , but to keep the notation simple we will not display that dependence explicitly. Substituting the expansion into the wave equation gives
( ) = - kn2c 2an(t) ( ) - 4 c 2 ( - ) (t - t') . (7)
We multiply this by ( ) and integrate using the orthogonality to get
= - km2c 2am(t) + 4 c 2 ( ) (t - t') . (8)
This equation is identical with our harmonic oscillator Green's function equation, with chosen appropriately. As before, we can choose any boundary conditions we like. We will choose either advanced or retarded boundary conditions. Substituting our harmonic oscillator result, the advanced and retarded Green's functions are
G (R)( ,t, ,t') = 4 c 2 (t - t') sin(knc[t - t']) .
G (A)( ,t, ,t') = - 4 c 2 (t' - t) sin(knc[t - t']) . (9)
where we have used the reality of the solution to put the complex conjugate on the other eigenfunction. We have also enforced the retarded boundary and advanced conditions by using the theta function defined by
(x) = . (10)
Let's work out the result for free space. The eigenfunctions of are plane waves, (that is you can simply Fourier transform the space part of the wave equation.) The result is
G (R)( ,t, ,t') = 4 c (t - t') e i ( - ') . (11)
Integrating over the angular part gives
G (R)( ,t, ,t') = (t - t') dksin(k| - |)sin (kc[t - t'])
= (t - t') dk
= (t - t') dk
=
= (t - t' - c - 1| - '|)
where in the last line, the theta function restricts the time such that the second delta function can never contribute. The calculation for the advanced delta function goes through identically, except in that case the theta function restricts the time so the first delta function never contributes. The minus signs cancel and the result is
G (A)( t, ,t') = (t - t' + c - 1| - '|) . (12)
Notice these agree with Jackson Eq. 6.44.
Let's calculate the solution of the wave equation given that ( ,t) is zero at all times before the source f ( ,t) is turned on. We repeat the same mathematics as before, where we take the wave equation with a source and the Green's function equation, cross multiply, subtract, and use Green's theorem (i.e. the divergence theorem or integration by parts) to arrive at
( ,t) = d 3r' dt'G( ,t, ,t')f ( ',t')
+ dt' da' .
By taking infinite space, with the initial field zero everywhere, the disturbance never makes it to the ``edge'' of the volume, so we can drop the spatial surface terms. If we were calculating the response in a cavity where was specified (e.g. zero) on the cavity surface, we would want to choose our Green's function to go to zero on the surface so that we could drop the surface terms proportional to the normal derivative of , just as in the Poisson case. Similarly, as in the harmonic oscillator, if we use the retarded Green's function the upper limits of the time surfaces give zero, while our boundary condition that the field is zero at ti allows us to drop the time surface terms. Therefore for this case, we can write
( ,t = d 3r' . (13)
A related but alternative method to using the Green's function is to use the eigenfunction expansion directly for ( ,t) ,
( ,t = bn(t) ( ) (14)
where bn(t) are time dependent coefficients. Substituting into the wave equation gives
- 4 f ( ,t) (15)
multiplying by ( ) and integrating gives the equations of motion for the coefficient bm(t) ,
= - kn2c 2bm(t) + 4 d 3rf ( ,t) ( ) (16)
which shows that each mode amplitude bm(t) behaves exactly like a driven harmonic oscillator. Any method to solve these differential equations can be used to produce a solution, and by summing over the modes a complete solution is obtained.
The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is
also written
or sometimes
where α is a positive constant and or denotes the Laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.Let a thin square-formed plate of heat conducting homogeneous material be in the - plane with sides on the -axis (isolated), on the line (held at the constant temperature ), and on the vertical lines and (both held at the constant temperature ). Determine the temperature function on the plate, when the faces of the plate are isolated.
The equation of the heat flow in this stationary case is
(1)
under the boundary conditions
We first try to separate the variables, i.e. seek the solution of (1) of the form
Then we get
and thus (1) gets the form
(2)
and the boundary conditions
We separate the variables in (2):
This equation is not possible unless both sides are equal to a same negative constant , which implies for the solution
and for the solution
The two first boundary conditions give , , and since , we must have , i.e.
Therefore
The fourth boundary condition now yields that ; thus and So (1) has infinitely many solutions
(3)
with and they all satisfy the boundary conditions except the third. Because of the linearity of (1), also the sum
of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges. The third boundary condition requires that
on the interval . But this is the Fourier sine series of the constant function on the half-interval , whence
The even 's here give 0 and the odd give
Thus we obtain the solution
It can be shown that this series converges in the whole square of the plate.
Applications
I. Particle diffusion
One can model particle diffusion by an equation involving either:
• the volumetric concentration of particles, denoted c, in the case of collective diffusion of a large number of particles, or
• the probability density function associated with the position of a single particle, denoted P.
In either case, one uses the heat equation
or
Both c and P are functions of position and time. D is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second. If the diffusion coefficient D is not constant, but depends on the concentration c (or P in the second case), then one gets the nonlinear diffusion equation.
II. Brownian motion
The random trajectory of a single particle subject to the particle diffusion equation (or heat equation) is a Brownian motion. If a particle is placed at at time t = 0, then the probability density function associated with the position vector of the particle will be the following:
which is a (multivariate) normal distribution evolving in time.
III. Schrodinger equation for a free particle
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
, where i is the unit imaginary number, and is Planck's constant divided by 2π, and ψ is the wavefunction of the particle.
This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t=0:
, with
Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.
IV. Thermal diffusivity in polymers
A direct practical application of the heat equation, in conjunction with Fourier theory, in spherical coordinates, is the measurement of the thermal diffusivity in polymers (Unsworth and Duarte). The dual theoretical-experimental method demonstrated by these authors is applicable to rubber and various other materials of practical interest.
V. Further applications
The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The famous Black–Scholes option pricing model's differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the Crank–Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995).An abstract form of heat equation on manifolds provides a major approach to the Atiyah–Singer index theorem, and has led to much further work on heat equations in Riemannian geometry.
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.
The wave equation is the prototypical example of a hyperbolic partial differential equation. In its simplest form, the wave equation refers to a scalar function u = u (x1, x2, …, xn; t) that satisfies:
where is the (spatial) Laplacian and where c is a fixed constant equal to the propagation speed of the wave. This is known as the non-dispersive wave equation. For a sound wave in air at 20°C this constant is about 343 m/s (see speed of sound). For the vibration of a string the speed can vary widely, depending upon the linear density of the string and the tension on it. For a spiral spring (a slinky) it can be as slow as a meter per second. More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. In such a case, c must be replaced by the phase velocity:
Another common correction in realistic systems is that the speed can also depend on the amplitude of the wave, leading to a nonlinear wave equation:
Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar u will contain a Mach factor (which is positive for the wave moving along the flow and negative for the reflected wave).
The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:
where:
• and are the so-called Lamé parameters describing the elastic properties of the medium,
• is the density,
• is the source function (driving force),
• and is the displacement vector.
Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.
Variations of the wave equation are also found in quantum mechanics, plasma physics and general relativity.
string has been strained between the points (0 0) and (p 0) of the x -axis. The transversal vibration of the string in the xy -plane is determined by the one-dimensional wave equation
(1)
satisfied by the ordinates u(x t) of the points of the string with the abscissa x on the time moment t( 0) . The boundary conditions are thus u(0 t)=u(p t)=0 We suppose also the initial conditions u(x 0)=f(x) u t(x 0)=g(x) which give the initial position of the string and the initial velocity of the points of the string.
For trying to separate the variables, set u(x t):=X(x)T(t) The boundary conditions are then X(0)=X(p)=0 , and the partial differential equation (1) may be written
(2)
This is not possible unless both sides are equal to a same constant −k2 where k is positive; we soon justify why the constant must be negative. Thus (2) splits into two ordinary linear differential equations of second order:
(3)
The solutions of these are, as is well known,
(4)
with integration constants Ci and Di .
But if we had set both sides of (2) equal to +k2 , we had got the solution T=D1ekt+D2e−kt which can not present a vibration. Equally impossible would be that k=0 .Now the boundary condition for X(0) shows in (4) that C1=0 , and the one for X(p) that C2sinckp=0 If one had C2=0 , then X(x) were identically 0 which is naturally impossible. So we must have sinckp=0 which implies ckp=n (n +) This means that the only suitable values of k satisfying the equations (3), the so-called eigenvalues, are k=pn c(n=1 2 3 ) So we have infinitely many solutions of (1), the eigenfunctions u=XT=C2sinpn x D1cospn ct+D2sinpn ct or u= Ancospn ct+Bnsinpn ct sinpn x (n=1 2 3 ) where An 's and Bn 's are for the time being arbitrary constants. Each of these functions satisfy the boundary conditions. Because of the linearity of (1), also their sum series
(5)
is a solution of (1), provided it converges. It fulfils the boundary conditions, too. In order to also the initial conditions would be fulfilled, one must have n=1Ansinpn x=f(x) n=1Bnpn csinpn x=g(x) on the interval [0 p] . But the left sides of these equations are the Fourier sine series of the functions f and g , and therefore we obtain the expressions for the coefficients: An=2p 0pf(x)sinpn xdx Bn=2n c 0pg(x)sinpn xdx
Applications of wave equation:-
The ideal-string wave equation applies to any perfectly elastic medium which is displaced along one dimension. For example, the air column of a clarinet or organ pipe can be modeled using the one-dimensional wave equation by substituting air-pressure deviation for string displacement, and longitudinal volume velocity for transverse string velocity. We refer to the general class of such media as one-dimensional waveguides. Extensions to two and three dimensions (and more, for the mathematically curious), are also possible .
For a physical string model, at least three coupled waveguide models should be considered. Two correspond to transverse-wave vibrations in the horizontal and vertical planes (two polarizations of planar vibration); the third corresponds to longitudinal waves. For bowed strings, torsional waves should also be considered, since they affect bow-string dynamics.
In the piano, for key ranges in which the hammer strikes three strings simultaneously, nine coupled waveguides are required per key for a complete simulation (not including torsional waves); however, in a practical, high-quality, virtual piano, one waveguide per coupled string (modeling only the vertical, transverse plane) suffices quite well. It is difficult to get by with fewer than the correct number of strings, however, because their detuning determines the entire amplitude envelope as well as beating and aftersound effects .
The wave equation with a source is
( ,t) = - 4 f ( ,t) . (1)
and the Green's function is the solution with a delta function source,
( ',t'; ,t) = - 4 ( - r') (t - t') (2)
where, as before, the Green's function is generally the solution of the adjoint equation, but the wave equation is self adjoint.
To use the harmonic oscillator result, we want to eliminate the spatial operators. We can do this exactly as we did for the Green's function for Poisson's Equation. Expanding in the eigenfunctions of . We write
- ( ) = kn2 ( ) , (3)
where the eigenfunctions are determined by the spatial boundary conditions that we enforce at the boundaries of our system. For free space, the eigenfunctions are plane waves, and the transformation to eigenfunctions is the same Fourier transform that we used for the Coulomb case. Eigenfunctions in a finite volume such as a cavity, are typically normalized to one,
d 3r ( ) ( ) = (4)
while the free space Fourier transform is typically normalized to Dirac delta functions using
d 3re - i e i ' = (2 )3 ( - ') (5)
which was derived in class by taking the limit of the periodic box normalized solutions.
The general solution of the wave equation is given by expanding the Green's function in the complete set of eigenfunctions,
G( ,t', ,t) = an(t) ( ) (6)
where an(t) are the expansion coefficients which also depend on and t' , but to keep the notation simple we will not display that dependence explicitly. Substituting the expansion into the wave equation gives
( ) = - kn2c 2an(t) ( ) - 4 c 2 ( - ) (t - t') . (7)
We multiply this by ( ) and integrate using the orthogonality to get
= - km2c 2am(t) + 4 c 2 ( ) (t - t') . (8)
This equation is identical with our harmonic oscillator Green's function equation, with chosen appropriately. As before, we can choose any boundary conditions we like. We will choose either advanced or retarded boundary conditions. Substituting our harmonic oscillator result, the advanced and retarded Green's functions are
G (R)( ,t, ,t') = 4 c 2 (t - t') sin(knc[t - t']) .
G (A)( ,t, ,t') = - 4 c 2 (t' - t) sin(knc[t - t']) . (9)
where we have used the reality of the solution to put the complex conjugate on the other eigenfunction. We have also enforced the retarded boundary and advanced conditions by using the theta function defined by
(x) = . (10)
Let's work out the result for free space. The eigenfunctions of are plane waves, (that is you can simply Fourier transform the space part of the wave equation.) The result is
G (R)( ,t, ,t') = 4 c (t - t') e i ( - ') . (11)
Integrating over the angular part gives
G (R)( ,t, ,t') = (t - t') dksin(k| - |)sin (kc[t - t'])
= (t - t') dk
= (t - t') dk
=
= (t - t' - c - 1| - '|)
where in the last line, the theta function restricts the time such that the second delta function can never contribute. The calculation for the advanced delta function goes through identically, except in that case the theta function restricts the time so the first delta function never contributes. The minus signs cancel and the result is
G (A)( t, ,t') = (t - t' + c - 1| - '|) . (12)
Notice these agree with Jackson Eq. 6.44.
Let's calculate the solution of the wave equation given that ( ,t) is zero at all times before the source f ( ,t) is turned on. We repeat the same mathematics as before, where we take the wave equation with a source and the Green's function equation, cross multiply, subtract, and use Green's theorem (i.e. the divergence theorem or integration by parts) to arrive at
( ,t) = d 3r' dt'G( ,t, ,t')f ( ',t')
+ dt' da' .
By taking infinite space, with the initial field zero everywhere, the disturbance never makes it to the ``edge'' of the volume, so we can drop the spatial surface terms. If we were calculating the response in a cavity where was specified (e.g. zero) on the cavity surface, we would want to choose our Green's function to go to zero on the surface so that we could drop the surface terms proportional to the normal derivative of , just as in the Poisson case. Similarly, as in the harmonic oscillator, if we use the retarded Green's function the upper limits of the time surfaces give zero, while our boundary condition that the field is zero at ti allows us to drop the time surface terms. Therefore for this case, we can write
( ,t = d 3r' . (13)
A related but alternative method to using the Green's function is to use the eigenfunction expansion directly for ( ,t) ,
( ,t = bn(t) ( ) (14)
where bn(t) are time dependent coefficients. Substituting into the wave equation gives
- 4 f ( ,t) (15)
multiplying by ( ) and integrating gives the equations of motion for the coefficient bm(t) ,
= - kn2c 2bm(t) + 4 d 3rf ( ,t) ( ) (16)
which shows that each mode amplitude bm(t) behaves exactly like a driven harmonic oscillator. Any method to solve these differential equations can be used to produce a solution, and by summing over the modes a complete solution is obtained.
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