As we know, the phase function for a plane wave characterizes the propagation of energy and momentum of light. Plane wave in a moving medium and resolution of the. Covariant formulation of electrodynamics uwo physics. Thus we see that the waves travel at speed cgiven by c. Tsymbal 3 first, we see that for a fixed value of k the set of equations 11 couples only those coefficients, whose wave vectors differ from k by a reciprocal lattice vector. Illustration of factors entering the addition of two plane waves with the same frequency. Wavesplane waves wikibooks, open books for an open world. In special relativity, a fourvector also known as a 4vector is an object with four components, which transform in a specific way under lorentz transformation. The large arrow is a vector called the wave vector, which defines 1 the.
We might have anticipated this from the previous discussion of lorentz invariant phase space i. Similarly, for waves whose phases are relativistic invariant we have another four vector k 1. Several other physical variables are also proved to satisfy the wave equation, which gives a strong argument to. We deal with both normal and nonnormal angles of incidence. Plane waves a uniform plane wave is a particular solution of maxwells equations with e assuming the same direction, same magnitude, and same phase in infinite planes perpendicular to the direction of propagation similarly for h. Furthermore, for any position vector in the plane the dot product with. For this reason, electromagnetic waves are usually characterized by their wavevector which specifies the direction of propagation and the wavelength and the plane of polarization i. Plane waves in a moving medium and the lorentz covariance. This lesson presents a 2d plane wave simulations and 3d plane.
We now turn to the 3dimensional version of the wave equation, which can be used to describe a variety of wavelike phenomena, e. The interpretation of the various quantities occurring in the above formulae is entirely analogous to that of the corresponding quantities for a plane. Waves, the wave equation, and phase velocity what is a wave. We can therefore assume that the k vector belongs to the first brillouin zone. Lecture 18 phys 3750 d m riffe 4 22220 that is perpendicular to k and passes through the point in space defined by the vector r.
The vacuum impedance for a plane wave is unity dimensionless. For this reason, electromagnetic waves are usually characterized by their wave vector which specifies the direction of propagation and the wave length and the plane of polarization i. As before, the plane is normal to, and its distance of. In this appendix we briefly address the concept of the wave vector and its relationship to. We have the four solutions with for a free particle with momentum. Suppose a wave source in frame s emits a plane wave of angular frequency. For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction. The very same expression can be obtained directly from the phasors by the following rule. For a given wave vector, the electric field can have any direction in the plane normal to. Chapter maxwells equations and electromagnetic waves. Thomson michaelmas 2009 48 proceeding as before to calculate the probability and current densities. Furthermore, we have a plane wave, by which we mean that a surface of constant phase is a plane. In fact, it can be shown that this is the most general form of solution possible, a result known as blochs theoremthe magnetic field must take the form of a plane wave multiplied by a function with the periodicity of the lattice. The e and bfield do not uniquely specify the potentials you have gauge freedom.
Quantum physics i, lecture note 4 mit opencourseware. The two boundary conditions can works with anisotropic pml together to realize the plane wave simulation. In the last class we had some discussion about the polarization of a plane wave, but now we need to go into it in more detail. This requires some thought because rr eexy1 1 2, 6 3 7 6 5 4 2. Wave fronts and wave vectors of two plane waves with the same wavelength but oriented in different directions. E and b are perpendicular to each other in a manner such that e.
Kg2 kg4 which, again, by comparison with the continuity equation allows us to identify for a plane wave and particle densities are proportional to e. The wave has an amplitude a 0 which is directed along the polarization unit vector. The 3d wave equation and plane waves before we introduce the 3d wave equation, lets think a bit about the 1d wave equation, 2 2 2 2 2 x q c t. Electromagnetic plane wave of frequency and wave vector suppose an electromagnetic plane wave with direction of propagation to be constructed, where is a unit vector. The vertical regions in between the bars have destructive interference, and hence define the lateral boundaries of the beams produced by the superposition. Then the variable in the exponent must be replaced by, the projection of in the direction. Similarly, the annihilation of the positron produces a distinctive backtoback twophoton signature two dashed. The fact that the wave vector of a monochromatic plane wave propagating in a conductor is necessarily complex, and that this is associated with a complex refractive index, implies characteristic phase changes for a plane electromagnetic wave reflected from the surface of the conductor, where, for. Sakurai picks a normalization of the spinor so that transforms like the fourth component of a vector. By setting the timespace fourvector and the wave fourvector x x,ct k n d, c, eq. Similarly, a wave proportional to a spherical harmonic has a unique orbital angular momentum magnitude and z component. The spatial period of the wave the distance between two successive points of identical phase is.
Wave fourvector in a moving medium and the lorentz. The multiplying constant aprovides the normalization. Calculate the poynting vector and the energy intensity of electromagnetic waves. Uniform plane waves page 3 if we use the permittivity and permeability of free space, we obtain the speed of light c 1 p 0 0 2. Electromagnetic waves university of texas at austin. For a plane wave traveling in the direction of the positive xaxis with the phase of the wave chosen so that the wave maximum is. This describes the wave oscillating in time at an angular frequency. Wavesplane superposition wikibooks, open books for an open.
Maxwells equations are reduced to a simple fourvector equation. Plane waves in a moving medium and the lorentz covariance of. The wave function solution for large rwill have the form. From the consequent transformation equations of the components of the fourvector, derive the doppler e.
Similarly, for waves whose phases are relativistic invariant we have another fourvector k 1. Since there is nothing special about the direction, it follows that if is reinterpreted as a unit vector pointing in an arbitrary direction then can be reinterpreted as the general equation of a plane. We will consider the complex poynting vector for time harmonic plane electromagnetic waves in phasor notation. Additionally, all four plane waves must have phase. Iv, by use of the planewave field solution, em momentum of light and. In physics, a plane wave is a special case of wave or field.
Electromagnetic waves travel through empty space with the speed of light c 1 0 0 b b cos kx t e e cos kx t z o y o the plane wave as represented by above is said to be linearly polarized because the electric vector is always along yaxis and, similarly, the magnetic vector is always along zaxis. The timeaverage power per unit area is onehalf of the real part. For more details of these boundary conditions, please see the technical background. Since there is nothing special about the direction, it follows that if is reinterpreted as a unit vector pointing in an arbitrary direction then 32 can be reinterpreted as the general equation of a plane. Plane wave phasors and complex poynting vector for a plane wave we know the efield and hfield phasors to be. For a given wavevector, the electric field can have any direction in the plane normal to. Generally, for z propagation, the direction of e and h in the plane perpendicular to z may vary in both time and position. Mathematically this is represented by the fact that application of the momentum operator returns a unique momentum value. This assertion can be written mathematically in terms of the vector fields involved in maxwells theory in the following way.
Dirac plane wave solution university of california, san diego. Uniform plane wave in free space we have previously established the following properties of plane waves in. Classical planewave fields we have just seen that the electric and magnetic elds in a sourcefree region of free space can be speci ed in terms of a coulombgauge vector potential according to 3, and maxwells equations will be satis ed if the vector potential satis es 10. It is useful to write the plane wave states as a spinor times an exponential.
Electromagnetic plane waves the real electric and magnetic fields in a monochromatic plane wave with propagation vector k. Introduction classical electromagnetic waves in free space. A plane wave propagates in a direction given by the wave vector k. Polarization 1 polarization vectors in the last lecture, we showed that maxwells equations admit plane wave solutions e e 0e i kxt, b b 0e i kxt 1 here, e 0 and b0 are called the polarization vectors for the electric and magnetic. Such a wave is said to be linearly polarized with polarization. The polarization describes the shape and locus of the e vector in the plane. In practice, plane wave does not exist because practical wave sources are always finite in extent. The vertical bands show regions of constructive interference where wave fronts coincide. The idea of a linear combination does more for us than just give another way to interpret a system of equations. Aeikz that is, this is a plane wave with wave number k p the frequency. B b cos kx t e e cos kx t z o y o the plane wave as represented by above is said to be linearly polarized because the electric vector is always along yaxis and, similarly, the magnetic vector is always along zaxis. In this discussion, vectors are denoted by boldfaced underscored lowercase letters, e. With electromagnetic waves, doubling the e fields and b fields quadruples the energy density u and the energy flux uc.
The sign the product depends on the direction of power. Any linear combination of plane waves with the same normal vector is also a plane wave. The 3d wave equation, plane waves, fields, and several 3d differential operators. There are actually an infinite number of possible answers. Specifically, a fourvector is an element of a fourdimensional vector space considered as a representation space of the standard representation of the lorentz group, the. A plane wave in two or three dimensions is like a sine wave in one dimension except that crests and troughs arent points, but form lines 2d or planes 3d perpendicular to the direction of wave propagation. Kshetrimayum 3192014 denotes either an electric or magnetic field f is a notation for field not for the force. Wavesplane superposition wikibooks, open books for an. Quantum mechanics in 3 dimensions department of physics. Pions four light colored dashed lines point to the annihilation point. Thus an electromagnetic plane wave with direction of.
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