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WAVE EQUATIONS AND ITS APPLICATIONS
STEPHY PHILIP
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WAVE EQUATIONS AND ITS
APPLICATIONS
Stephy philip
MSc Mathematics
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Preface
Wave equations play an important role in increasing the applied sciences as
well as in mathematics itself. This book is intended to familiarize the reader
with the basic concepts, principles, different types of wave equations and its
applications .The book that contains details about the different types of wave
equations and its applications in the field of sciences. Applications are given at
the last page of the text. Chapter 1 contain the basic results, chapter 2-chapter
4 contains different types of wave equations. Notations are also explained.
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Acknowledgement
I would like to express my gratitude to many people. Their comments and
suggestions are influenced the preparation of the text. I want to thank teachers
and friends for their effective cooperation and great care for preparing this
book.
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INTRODUCTION
The wave equation is the simplest example of a hyperbolic equation of
second order.If x∊ Rⁿ represents the space variable and t the time variable it can
model waves in pipes or vibrating strings when n=1, waves on the surface of
water when n=2, and waves in optics or acoustics when n=3.The initial value
problem for the wave equation is written as:
∂2
u/∂t2
(x,t)−Δu(x,t)=0
U(x, 0) =f(x)
∂u/∂t(x,0)=g(x)
Where Δ is the Laplace operator in the space variables alone and f and g
are initial displacement and initial velocity respectively. In the case when x ∊Ω
subset of Rⁿ we may also prescribe boundary conditions. The wave equation is
used to discuss the physical problems is of first order in time and second order
in space co-ordinates. This contradicts the space time symmetry requirements of
relativity
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CHAPTER 1
BASIC RESULTS
Relativistic wave equations
Relativistic wave equations predict the behavior of particles at high
energies and velocities comparable to the speed of light
Non relativistic
The quantum mechanics of particles without accounting for the effect of special
relativity, for example particles propagating speeds much less than light is
known as non-relativistic wave equations
Homogeneous
If a body interact with no other bodies it is various in position in space and
different orientations would be mechanically equivalent
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Isotropic
All the properties like electrical conductivity, thermal conductivity, refractive
index …….are same in all directions
The Laplace operator
The Laplace operator is a scalar operator defined as the dot product (inner
product) of two gradient vector operators:
Euler Formula
The Euler formula, sometimes also called the Euler identity, states
Where i is the imaginary unit. Note that Euler's polyhedral formula
issometimes also called the Euler formula, as is the Euler curvature formula
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CHAPTER 2
WAVE EQUATIONS
The wave equation is an important second-order linear partial differential
equation for the description of waves – as they occur in physics – such as
soundwaves, light waves and water waves. It arises in fields like acoustics,
electromagnetics, and fluiddynamics.
The wave equation is a hyperbolic partial differential equation. It
typically concerns a time variable t, one or more spatial variables x1, x2,…..xn,
and a scalar function u = u (x1, x2, …, xn; t), whose values could model the
displacement of a wave. The wave equation for u is
Where ∇2
is the (spatial) Laplacian and where c is a fixed constant.
Solutions of this equation that are initially zero outside some restricted
region propagate out from the region at a fixed speed in all spatial directions, as
do physical waves from a localized disturbance; the constant c is identified with
the propagation speed of the wave. This equation is linear, as the sum of any
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two solutions is again a solution: in physics this property is called the
superpositionprinciple.
The equation alone does not specify a solution; a unique solution is
usually obtained by setting a problem with further conditions, such as initial
conditions, which prescribe the value and velocity of the wave. Another
important class of problems specifies boundary conditions, for which the
solutions represent standing waves, or harmonics, analogous to the harmonics of
musical instruments.
Scalar wave equation in one space dimension
The wave equation in one space dimension can be written like this:
This equation is typically described as having only one space dimension
"x", because the only other independent variable is the time "t". Nevertheless,
the dependent variable "y" may represent a second space dimension, as in the
case of a string that is located in the x-y plane.
Derivation of the wave equation
The wave equation in one space dimensioncan be derived in a variety of
different physical settings. Most famously, it can be derived for the case of a
string that is vibrating in a two-dimensional plane, with each of its elements
being pulled in opposite directions by the force of tension.
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The wave equation in the one-dimensional case can be derived from
Hooke's Law in the following way: Imagine an array of little weights of mass m
interconnected with massless springs of length h. The springs have a
spring constant of k:
Here the dependent variable u(x) measures the distance from the
equilibrium of the mass situated at x, so that u(x) essentially measures the
magnitude of a disturbance (i.e. stress) that is traveling in an elastic material.
The forces exerted on the mass m at the location x+h are:
The equation of motion for the weight at the location x+h is given by
equating these two forces:
Where the time-dependence of u(x) has been made explicit.If the array of
weights consists of N weights spaced evenly over the length L = Nh of total
mass M = Nm, and the total spring constant of the array K = k/N we can write
the above equation as:
Taking the limit N → ∞, h → 0 and assuming smoothness one gets:
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(KL2
)/M is the square of the propagation speed in this particular case.
General solution
Algebraic approach
The one-dimensional wave equation is unusual for a partial differential
equation in that a relatively simple general solution may be found. Defining new
variables:
Changes the wave equation into
This leads to the general solution
Or equivalently:
In other words, solutions of the 1D wave equation are sums of a right
traveling function F and a left traveling function G. "Traveling" means that the
shape of these individual arbitrary functions with respect to x stays constant,
however the functions are translated left and right with time at the speed c. This
was derived by Jean le Rond d' Alembert.]
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Another way to arrive at this result is to note that the wave equation may be
"factored":
and therefore:
These last two equations are advection equations, one left traveling and one
right, both with constant speed c.
For an initial value problem, the arbitrary functions F and G can be determined
to satisfy initial conditions:
The result is d' Alembert's formula:
Scalar wave equation in three space dimensions
The solution of the initial-value problem for the wave equation in three
space dimensions can be then be used to obtain the solution in two space
dimensions.
Scalar wave equation in two space dimensions
In two space dimensions, the wave equation is
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We can use the three-dimensional theory to solve this problem if we
regard u as a function in three dimensions that is independent of the third
dimension. If
then the three-dimensional solution formula becomes
whereα and β are the first two coordinates on the unit sphere, and dω is
the area element on the sphere. This integral may be rewritten as an integral
over the disc D with center (x,y) and radius ct:
It is apparent that the solution at (t,x,y) depends not only on the data on the light
cone where
but also on data that are interior to that cone.
Scalar wave equation in general dimension and Kirchhoff's formulae
Odd dimensions
Assume n ≥ 3 is an odd integer and g∈Cm+1
(Rn
), h∈Cm
(Rn
) for m = (n+1)/2. Let
and let
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then
u∈C2
(Rn
× [0, ∞))
utt−Δu = 0 in Rn
× (0, ∞)
Even dimensions
Assume n ≥ 2 is an even integer and g∈Cm+1
(Rn
), h∈Cm
(Rn
), for m = (n+2)/2.
Let and let
then
u∈C2
(Rn
× [0, ∞))
utt−Δu = 0 in Rn
× (0, ∞)
Inhomogeneous wave equation in one dimension
The inhomogeneous wave equation in one dimension is the following:
With initial conditions given by
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The function s(x, t) is often called the source function because in practice
it describes the effects of the sources of waves on the medium carrying them.
Physical examples of source functions include the force driving a wave on a
string, or the charge or current density in the Lorenzgauge of electromagnetism.
To model dispersive wave phenomena, those in which the speed of wave
propagation varies with the frequency of the wave, the constant c is replaced by
the phase velocity:
The elastic wave equation in three dimensions describes the propagation
of waves in an isotropichomogeneouselastic 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:
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Where:
λ and μ are the so-called Lamé parameters describing the elastic
properties of the medium,
ρ is the density,
f is the source function (driving force),
andu 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
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CHAPTER 3
ELECTROMAGNETIC WAVE EQUATIONS
The electromagnetic wave equation is a second-order partial differential
equation that describes the propagation of electromagnetic waves through a
medium or in a vacuum. It is a three-dimensional form of the wave equation.
The homogeneous form of the equation, written in terms of either the electric
fieldE or the magnetic fieldB, takes the form:
where
is the speed of light in a medium with permeability ( ), and permittivity
( ), and ∇2
is the Laplace operator. In a vacuum, c = c0 = 299,792,458 meters
per second, which is the speed of light in free space.[1]
The electromagnetic
wave equation derives from Maxwell's equations. It should also be noted that in
most older literature, B is called the magnetic flux density or magnetic
induction.
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The origin of the electromagnetic wave equation
In his 1864 paper titled A Dynamical Theory of the Electromagnetic
Field, Maxwell utilized the correction to Ampère's circuital law that he had
made in part III of his 1861 paper On Physical Lines of Force. In Part VI of his
1864 paper titled Electromagnetic Theory of Light,[2]
Maxwell combined
displacement current with some of the other equations of electromagnetism and
he obtained a wave equation with a speed equal to the speed of light. He
commented:
The agreement of the results seems to show that light and magnetism are
affections of the same substance, and that light is an electromagnetic
disturbance propagated through the field according to electromagnetic
laws.
Maxwell's derivation of the electromagnetic wave equation has been replaced in
modern physics education by a much less cumbersome method involving
combining the corrected version of Ampère's circuital law with Faraday's law of
induction.
To obtain the electromagnetic wave equation in a vacuum using the
modern method, we begin with the modern 'Heaviside' form of Maxwell's
equations. In a vacuum- and charge-free space, these equations are:
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Where ρ = 0 because there's no charge density in free space.
Taking the curl of the curl equations gives:
We can use the vector identity
WhereV is any vector function of space. And
Where∇V is a dyadic which when operated on by the divergence operator
yields a vector. Since
then the first term on the right in the identity vanishes and we obtain the wave
equations:
is the speed of light in free space.
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Covariant form of the homogeneous wave equation
These relativistic equations can be written in contravariant form as
Where the electromagnetic four-potential is
With the Lorenz gauge condition:
Where
is the d' Alembertian operator. (The square box is not a typographical error; it is
the correct symbol for this operator.)
Homogeneous wave equation in curved spacetime
The electromagnetic wave equation is modified in two ways, the
derivative is replaced with the covariant derivative and a new term that depends
on the curvature appears.
where is the Ricci curvature tensor and the semicolon indicates covariant
differentiation.
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The generalization of the Lorenz gauge condition in curved spacetime is
assumed:
Inhomogeneous electromagnetic wave equation
Localized time-varying charge and current densities can act as sources of
electromagnetic waves in a vacuum. Maxwell's equations can be written in the
form of a wave equation with sources. The addition of sources to the wave
equations makes the partial differential equations inhomogeneous.
Solutions to the homogeneous electromagnetic wave equation
The general solution to the electromagnetic wave equation is a linear
superposition of waves of the form
for virtually any well-behaved function g of dimensionless argument φ,
where ω is the angular frequency (in radians per second), and k = (kx, ky, kz) is
the wave vector (in radians per meter).
Although the function g can be and often is a monochromatic sine wave,
it does not have to be sinusoidal, or even periodic. In practice, g cannot have
infinite periodicity because any real electromagnetic wave must always have a
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finite extent in time and space. As a result, and based on the theory of
Fourier decomposition, a real wave must consist of the superposition of an
infinite set of sinusoidal frequencies.
In addition, for a valid solution, the wave vector and the angular
frequency are not independent; they must adhere to the dispersion relation:
Where k is the wave number and λ is the wavelength. The variable c can
only be used in this equation when the electromagnetic wave is in a vacuum.
Acoustic wave equation
The acoustic wave equation governs the propagation of acoustic waves
through a material medium. The form of the equation is a second order partial
differential equation. The equation describes the evolution of acoustic pressure
or particle velocity u as a function of position r and time . A simplified form of
the equation describes acoustic waves in only one spatial dimension, while a
more general form describes waves in three dimensions.
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In one dimension Equation
Feynman derives the wave equation that describes the behavior of sound
in matter in one dimension (position ) as:
where is the acoustic pressure (the local deviation from the ambient pressure),
and where is the speed of sound.
Solution
Provided that the speed is a constant, not dependent on frequency (the
dispersion less case), then the most general solution is
where and are any two twice-differentiable functions. This may be
pictured as the superposition of two waveforms of arbitrary profile, one ( )
travelling up the x-axis and the other ( ) down the x-axis at the speed . The
particular case of a sinusoidal wave travelling in one direction is obtained by
choosing either or to be a sinusoid, and the other to be zero, giving
.
Where is the angular frequency of the wave and is its number
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In three dimensions Equation
Feynman derives the wave equation that describes the behavior of sound
in matter in three dimensions as:
Where is the Laplace operator, is the acoustic pressure (the local
deviation from the ambient pressure), and where is the speed of sound.
Solution
The following solutions are obtained by separation of variables in
different coordinate systems. They are phase or solutions, that is they have an
implicit time-dependence factor of where is the angular frequency.
The explicit time dependence is given by
Here is the wave number.
Cartesian coordinates
.
Cylindrical coordinates
.
where the asymptotic approximations to the Hankel functions, when ,
are
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.
Spherical coordinates
.
The Time-Dependent Schrödinger Equation
We are now ready to consider the time-dependent Schrödinger equation.
Although we were able to derive the single-particle time-independent
Schrödinger equation starting from the classical wave equation and the de
Broglie relation, the time-dependent Schrödinger equation cannot be derived
using elementary methods and is generally given as a postulate of quantum
mechanics.
The single-particle three-dimensional time-dependent Schrödinger equation is
(1)
Where is assumed to be a real function and represents the potential
energy of the system (a complex function will act as a source or sink for
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probability. Wave Mechanics is the branch of quantum mechanics with
equation (1) as its dynamical law. .
Of course the time-dependent equation can be used to derive the time-
independent equation. If we write the wave function as a product of spatial and
temporal terms, , then equation (1) becomes
Since the left-hand side is a function of only and the right hand side is a
function of only, the two sides must equal a constant. If we tentatively
designate this constant (since the right-hand side clearly must have the
dimensions of energy), then we extract two ordinary differential equations,
namely
(2)
and
(3)
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The latter equation is once again the time-independent Schrödinger equation.
The former equation is easily solved to yield
(4)
The Hamiltonian in equation (3) is a Hermitian operator, and the eigenvalues of
a Hermitian operator must be real, so is real. This means that the solutions
are purely oscillatory, since never changes in magnitude (recall Euler's
formula )
. Thus if
(5)
then the total wave function differs from only by a phase factor of
constant magnitude. There are some interesting consequences of this. First of
all, the quantity is time independent, as we can easily show:
(6)
Secondly, the expectation value for any time-independent operator is also time-
independent, if satisfies equation (7). By the same reasoning applied
above,
(7)
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For these reasons, wave functions of the form (7) are called stationary states.
The state is ``stationary,'' but the particle it describes is not!
Of course equation (7) represents a particular solution to equation (1). The
general solution to equation (1) will be a linear combination of these particular
solutions, i.e.
Secondly, the expectation value for any time-independent operator is also time-
independent, if satisfies equation (7). By the same reasoning applied
above,
(8)
For these reasons, wave functions of the form (7) are called stationary states.
The state is ``stationary,'' but the particle it describes is not!
Of course equation (7) represents a particular solution to equation (1).
The general solution to equation (1) will be a linear combination of these
particular solutions, i.e.
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D’Alembert solution of the wave equation
We have solved the wave equation by using Fourier series. But it is often
more convenient to use the so-called d’Alembert solution to the waveequation3
.
This solution can be derived using Fourier series as well, but it is really an
awkward use of those concepts. It is much easier to derive this solution by
making a correct change of variables to get an equation that can be solved by
simple integration.
Suppose we have the wave equation
And we wish to solve the equation given the conditions
Change of variables
We will transform the equation into a simpler form where it can be solved
by simple integration. We change variables to , and we use the
chain rule:
(1)
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We compute
In the above computations, we have used the fact from calculus that .
Then we plug into the wave equation,
Therefore, the wave equation (1) transforms into . It is easy to find
the general solution to this equation by integrating twice. Let us integrate with
respect to first4
and notice that the constant of integration depends on . We get
. Next, we integrate with respect to and notice that the constant of
integration must depend on . Thus, . The solution must,
therefore, be of the following form for some functions and :
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CHAPTER 4
Wave Equations Applications
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-wavevibrations 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. In the piano, for key ranges in
which the hammer strikes three strings simultaneously, nine coupled
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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 after sound effects .
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CONCLUSION
Wave equation is differential equation solved with the conditions for
waves in Cartesian, spherical and polar coordinates. There are different types of
wave equations and there is no general method of solutions.
Here we discuss some of the wave equations in which some have solution
and some do not have any solution.
Various assumptions have to be made as regards the structure of the wave
equation, the boundary and continuity conditions on its solutions and the
physical meaning of these solutions.
Variation of these wave equation are also found in sound, quantum
mechanics, plasma physics, and general relativity.
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REFERENCES
1. Jackson, JohnD, classical Electrodynamics,Willy,1998.
2. David.H.Stalin, Ann.W.Morgenthalar, and Jin
Au Kong, Electro, Electromagnetic Waves, Prentice.
Hall,(1994)
3. William.C.Lane, The Wave Equation and Its Solutions.
