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The Wave Equation

By Christina Marshall,2014-05-07 19:24
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The Wave Equation

The Wave Equation:

The displacement of a wave propagating in space is a function both of position and of

    time: . Let the wave have propagation speed c. Then we can make a Galilean ??(x,t)

    ?transformation , to observe the disturbance of the medium as the wave xxct

    progresses through space. This removes the dependence of ? on time:

    ?, say. ?(x,t)?(x)f(xct)

    Here we assume that the wave is travelling in the positive x sense (w.r.t. some inertial frame), so we have that . c0

    . (1) ?(x,t)f(xct)

    A wave propagating with constant velocity c has a profile that is uniform in space and constant in time. Thus, . ;??x)x,t)t?(x,t)

We can derive a wave equation from these few considerations which relates the change of

    with the change of x and of t. ?

?xxct

    ????(x,t)(x)

    ???????x????x?x?x?x

    ???????x?;c ???t?x?t?x

    2222????????????????2;;;;ccccc??22??????t?x?t?x?x?t?x?t?t?x??

    22??1??222??xc?t

By Galilean invariance, this is also the wave equation in the first inertial frame:

22??1?? (2) 222?xc?t

A solution to this equation is of the form

;?;??? (3) ???x,tAexpikxct?

where A, k, and ? are constants.

    We want a real solution, so take

    ;????(x,t)Asinkxct)?.

Note that initial conditions determine ?: . ;??x0,t0Asin?

    Let the properties of the wave be given: let be the wavelength and ?;,;(?f. An

    ?elementary argument gives us c = f. Thus, kc = kf = . k2?

Consider equation (3): We must have that . Thus, ;?;??x,t?x,t

    ??expikxkkctexpikxkct;?;?????

    ? expik1exp2i;?;???

    ?2?k

    Definition: The quantity k is called the propagation number of the given wave.

Thus we have completely characterized the one-dimensional wave from some simple

    considerations and obtained the formula

    ??;???(x,t)Asinkxct)?

    ?cf? (4) ???2f??2?k?

Definition: We define the quantity , called the phase:

     (5) ?kxkct?

We have,

    ???k???x??t

    ???????t??x

    ????t?x??????1?????????tx??????xt

    ??t??1??k?x??

    ?x??? (6) ??c???tk??

    ?x??Definition: is the speed of propagation of the condition of constant phase. ???t??

Plane Waves:

Recall the equation of a plane in Cartesian coordinates:

    ;? :rr?k00

In Cartesian coordinates,

xkykzkaxyz axkykzk0x0y0z

Thus, is the equation of the plane . k?ra

    We can now construct a function defined on a set of planes each with normal vector k, which varies sinusoidally in space. The function will be scalar-valued, but will have a

    vector as an argument. This is the function , or, ?(r)Asin(k?r)

;? (7) ?rAexp(ik?r)

    We insist that this function be spatially repetitive:

    k??ˆ (8) ;?;?rrk?r?????k??

From (7), this requirement is the same as

    Aexp(ik?r)Aexpik?rk;???

    ? ?exp(ik)1exp(2i)

    ?2?k

    Definition: The vector k, whose magnitude k is the propagation number, is called the propagation vector.

    We introduce a time-dependence into equation (7) and get

     (9) ;?;????r,tAexpik?r;?t

As before, the phase is that quantity ,;,;. A wave front is a surface joining all k?r;?t

    points of equal phase.

    The phase is constant in time and uniform in space. Thus,

    ?))?rrttrt,,;?;?kkk

    )??)??AirkrkttAirkt;?;?exp??;;exp??;kkk

    )?);?irktexp??;1;k

    )?)rk;t0k

    )?r??ctk)

    ;

    r is the component of the position vector r in the k direction and so rk = rk. The result kk

    drkis that the magnitude of the wave velocity, , is equal simply to c. dt

To derive a three-dimensional analogue of equation (2), we recall the direction cosines:

Thus, for

    ;?;???;?, we have ???r,tAexpikxkykz;?tAexpik?x?y?z;?txyz

    ?2???22k2?x

    2??22??k2?y 2??22??k2?z

    2??2??2?t

    2222??????1?? (10) ?22222?x?y?zc?t

    Note the symmetry between the variables x, y and z in equation (10).

Light in Matter:

From Maxwell’s equations, we have

    12 (11) c?00

If we consider a homogeneous, isotropic dielectric in a region of space, equation (11) is

    modified and we get that

    12 (12) v?,

Note the convention: c denotes the speed of propagation of electromagnetic (em)

    radiation in vacuo; v denotes the speed of propagation of em radiation in any other

    medium.

?;and are related linearly to ?and to ;respectively by dimensionless constants:

??K?E0 (13) ?KM0?

    c1vThus, equation (12) becomes .

    KK?KKEM00EM

    cWe define the quantity (14) n?KKEMv

    n is called the absolute index of refraction of the dielectric.

For materials that are transparent to visible em radiation (i.e., light), K is almost unity, M

    since these materials in particular, glass are not magnetic. Thus,

     (15) n?KE

This equation is known as Maxwell’s relation.

Now K is constant, so equation (15) suggests that n is constant, once the material E

    properties of the dielectric are fixed. However, it is an experimental fact that n depends on the frequency of the incident em radiation. This dependency is called dispersion. Maxwell’s equations ignore dispersion. Clearly, it must be considered.

    Firstly, we consider the different ways in which em radiation, or, equivalently, photons, interacts with a given dielectric. This provides the key to the physical basis for the frequency-dependence of n. We consider the interaction of an incident em wave with the array of atoms which constitutes the dielectric. An atom reacts to the incoming radiation in two ways. Depending on the frequency, the incident photon may simply be scattered

    redirected without being altered. If, on the other hand, the energy of the incoming photon matches that of one of the excited states, the atom will absorb the photon, and is raised to a higher energy level. In gases under pressure and in dense materials, it is probable that this energy will be dissipated by random atomic motion, before a photon can be emitted. (This dissipation is analogous to a damping force in an oscillating system.) This absorption is thus known as dissipative absorption.

In contrast to this atomic excitation, a process called non-resonant scattering occurs

    when incoming em radiation of frequencies lower than the frequencies necessary for absorption interacts with the atoms. The energy of the photon is too small to cause a transition of the atom to any excited state. However, the em field of the incident light can drive the electron cloud of the atom into oscillation (more on this later). There is no atomic transition: the electron remains in its ground state but the electron cloud vibrates slightly at the frequency of the incident light (analogous to the driving frequency in an

    oscillator). When the electron cloud starts to vibrate w.r.t. the positive nucleus, the system constitutes an oscillating electric dipole and as such will immediately begin to

    radiate at that same frequency. The resulting emitted photon has the same frequency

    and thus energy as the incident one. Therefore, this process of scattering is completely

    elastic.

We also have to consider polarization. When a dielectric is subject to an external E field,

    the internal charge distribution is disturbed. This corresponds to the generation of many electric dipole moments, which in turn contribute to the electric field. This is

    polarization. Polarization is characterized by the dipole moment per unit volume due to the E field, called the electric polarization, denoted by the vector P. It is found that

;? (16) ??EP0

There are in fact several kinds of polarization:

; Orientational Polarization: A molecule that itself has a dipole moment is subject to

    orientational polarization. These molecules are called polar molecules. Usually, a

    collection of polar molecules will be such that the orientation of the polarization is

    random the randomness being due to thermal effects. On application of an E field,

    these dipoles align. An example is the water molecule.

    ; Electronic Polarization: In non-polar molecules and atoms, no such “internal”

    dipoles exist. But on application of an E field, the electron cloud of each atom /

    molecule shifts relative to the nucleus thereby producing a dipole moment. This is

    the most significant sort of polarization here.

    ; Ionic Polarization: If a collection of ionic molecules (e.g., NaCl) is subjected to an

    external E field, the positive and negative ions undergo a shift relative to each other,

    thereby producing dipole moments, which will be aligned in the external field.

    If the dielectric is subjected to an incident harmonic em wave, its internal structure will experience time-dependent (external) forces and / or torques. These forces are due to the E field of the incident wave only, since , and so is negligible for Fqv?Bv;;cM

Consider now the electron cloud of a nucleus in the external E field. Since the electrons

    have small inertia, they will be sensitive to the applied E field. We can assume that the

    electron cloud has some stable equilibrium position w.r.t. the nucleus (due to a minimum of the potential function, where the potential function arises because of the Coulomb interaction between the positive nucleus and the negative electron cloud). Any small disturbance from equilibrium will result in a restoring force proportional to the displacement from equilibrium. So we assume that a restoring force of the F = -kx acts

    on the system. Once disturbed, any electron in the cloud will oscillate with natural

    kfrequency . Further, if the electron oscillates at this frequency, we assume ?0me

    that it will emit a photon at precisely this frequency.

    We therefore think of the electron cloud as an oscillating system; as though it were a collection of particles attached to the nucleus by springs of spring constant k i.e., a

    collection of coupled oscillators. Applying the external E field will result in the system’s

    being driven at frequency ?, where this arises . EEcos?t0

Thus, the force on each electron due to E is . Therefore, the equation of FqEcos?tEe0

    motion for each particle in the system is

    2dx2 (17) m?qEcos?te0e02dt

We assume the (steady-state) solution . Notice that after transience has xxcos?t0

    ended, the system oscillates at the frequency of the external field.

    q/meeWe find x to be (18) xE00022??0

    q/meex(t)Ecos?t0Thus, (18’) 22??0

Recall some facts about dipoles:

     ?d?qE1 qEdsin?1

     D = 2d ???2qEdsin?qeDsintot12

    p?Dq ??p?E

If there are N molecules per unit volume, then the dipole moment per unit volume, or

    electric polarization, P, is

     (19) PqDN

    In our case, we have that P = qxN. e

Thus,

    q/meePqxNqEcos?tN, from (18’) 0ee22??0

Now P = (?;,;?E.

    2qm/ee??Thus, ;?EE022??0

    2qN??e?0?22;?m0e??K0E

    2qNe K1E22???;?m00e

    2K?nE

    2qNe?;?1?n22?m;???00e

We shall write

    2qN2e? (20) n;?122;??m??e00

Remarks:

; Definition: We say that ?;,;?is a resonant frequency. 222; For , we have that , so . This implies a complex value for ????0n000

    n, on which more later.

    222; For , we have that , so . ????0n000

    ; We can rearrange equation (20) to express n as a function of .

    2??????mmm411122222222eee000??, ;?;????ffc??:?:4000222222??nqNqNqN1eee??01?22?:?:02??n1 (21) 2?4??me20?c?2?qNe?

Various graphs are plotted for the values of wavelength and of n listed below:

     ;!;Nanometres n

    728.135 1.53460

    706.519 1.53520

    667.815 1.53629

    587.562 1.53954

    504.774 1.54417

    501.567 1.54473

    492.193 1.54528

    471.314 1.54624

    447.148 1.54943

    438.793 1.55026

    414.376 1.55374

    412.086 1.55402

    402.619 1.55530

    388.865 1.55767

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