# What is Plane waves: Definition and 58 Discussions

In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.For any position

x

{\displaystyle {\vec {x}}}
in space and any time

t

{\displaystyle t}
, the value of such a field can be written as

F
(

x

,
t
)
=
G
(

x

n

,
t
)
,

{\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}},t),}
where

n

{\displaystyle {\vec {n}}}
is a unit-length vector, and

G
(
d
,
t
)

{\displaystyle G(d,t)}
is a function that gives the field's value as from only two real parameters: the time

t

{\displaystyle t}
, and the displacement

d
=

x

n

{\displaystyle d={\vec {x}}\cdot {\vec {n}}}
of the point

x

{\displaystyle {\vec {x}}}
along the direction

n

{\displaystyle {\vec {n}}}
. The latter is constant over each plane perpendicular to

n

{\displaystyle {\vec {n}}}
.
The values of the field

F

{\displaystyle F}
may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave.
When the values of

F

{\displaystyle F}
are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector

n

{\displaystyle {\vec {n}}}
, and a transverse wave if they are always orthogonal (perpendicular) to it.

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1. ### I Plane wave decomposition of the fields of a dipole antenna

A dipole antenna will have near fields and far fields. Can both the near and far fields can be decomposed into an infinite sum of plane waves? If so, are the plane waves for far fields and near fields of different type or class? Near fields must die off at infinity but far fields do not. Thanks.
2. ### I Why no plane waves of macroscopic bodies? The micro-macro threshold...

One of the strange features of Quantum Mechanics is that for his formulation one needs the classical physics that actually should emerge as its macroscopic limit. All experiences with quantum objects have to be analyzed through classical "glasses". Naturally, then the question arises: where...
3. ### I EM Power transmitted from one region to another (normal incidence)

High! I have a EM plane wave hitting normally a surface dividing universe in media 1 and 2, both without losses. So we have incident, reflected and transmitted waves. It's a simple exercise in which you are given the basic data about two media and wave incident amplitude H in medium 1. I get...
4. ### Interference pattern of a fan of plane waves

So I've kind of made the assumption that there will be an odd number of plane waves and the same amount above and below the z-axis. Then, using the diagram below, I determined the angle the nth plane wave makes with respect to the z-axis to be the angle it makes with respect to the n =1 plane...
5. ### Show that the given electric field is a plane wave

A wavefront is defined as a surface in space where the argument of the cosine has a constant value. So I set the argument of the cosine to an arbitrary constant s. ## k(\hat{u} \cdot r - c t) + \phi = s ## The positional information is is in r, so I rearrange the equation to be ## \hat{u}...
6. ### Electromagnetic plane waves from a current sheet

I have an infinite sheet (in lossless, homogeneous medium) of time-harmonic current in ##yz##-plane at ##x=−d##. The current density on this sheet is given by $$\mathbf{J}=\hat{z}J_0\delta(x+d)$$ ##δ(x+d)## is delta function. Moreover, there is a perfect electric conductor (PEC) half space at...
7. ### "Total" destructive interference of plane waves

Hello. Let's suppose that we have a Michelson interferometer to study interference patterns of light. This time we use plane waves. If we set the whole thing up so that the two separated beams have a phase difference of π when they superpose, destructive interference ensues. Since we're...
8. ### I Why is e^ikx considered a plane wave?

In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes11.pdf, in the middle of page 5, it is mentioned: We will be interested in bound states namely, energy eigenstates that are normalizable. For this the energy E of the states...
9. ### I Are plane waves actually spherical waves at large distance?

I am doubting that any plane wave is generated from a spherical wave. At large distance the radii of curvature becomes so large that we can think it as plane. Like we se Earth surface as plane though it is spherical. Is it true? I have a mathematical proof for my argument!
10. ### Maximizing Localization in Adding Multiple Plane Waves

To whom it may concern, I am having issues with a given assignment in my quantum mechanics class. The instructions listed below are all I have to go on since the prof. is not available for discussion and I have searched through at least 15 articles regarding plane waves and a dozen textbooks...
11. ### I What is the difference between transverse and plane waves?

Hi. Are all transverse waves plane waves ? Are all plane waves transverse ? I'm confused about the difference. I know the difference between transverse and longitudinal waves but I'm not sure on how plane waves fit into the picture ? Thanks
12. ### I Generating Aperiodic Tilings with Plane Waves

Hello, I saw an applet awhile ago during a late-night mathematical web-surfing marathon that used the addition of plane waves at different angles to generate aperiodic tilings (like the penrose tiling.) I haven't been able to find it again. I'm trying to make my own version of it, using the same...
13. ### Dielectric slab and angle of incidence

Hello! Let's consider a plane wave represented by a ray, propagating in a 2D dielectric slab. It has a medium with refractive index n_1 as its core and a medium with refractive index n_2, n_2 < n_1, as its cladding. In order for this ray to represent a mode, it must satisfy two conditions: -...
14. ### Equation for the boundary of rays turning to plane waves

Homework Statement I am given the following figure: These are converging rays that appear to be going to a point F convert to a plane wave upon hitting the boundary between n2 and n1, and I am asked to find the equation for the boundary between n1 and n2 that perfectly accomplishes this...
15. ### Inertial frame where plane waves have the same frequency

Homework Statement Plane harmonic waves of 1/p, 1/q, 1/r and 1/s are travelling, respectively, in the directions of the (non-unit) vectors (1,1,1), (1,-1,-1), (-1,1,-1) and (-1,-1,1). Show that there exists an inertial coordinate system in which they have the same frequency if and only if...
16. ### Can small condensation drops produce an enlarged image?

Homework Statement This is not really a homework or coursework question, but might be of interest to readers of this forum. It concerns what I have viewed on two separate occasions which baffles me. The setup was different it the two instances, but there are similar elements in both. In both...
17. ### I Wave-particle duality and localization

I have read recently that the motion of an electron of momentum p must be described by the means of a plane waves :\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r -wt)}=Ae^{i(\vec p\cdot \vec r -Et)/\hbar} de Broglie hypothesis states that every particle of momentum p has a wavelength lamda. I will...
18. ### Curvature polynomials vanish for plane waves?

Geroch 1968 touches on the Kundt type I and II curvature invariants. If I'm understanding correctly, then type I means curvature polynomials. Type II appears to be something else that I confess I don't understand very well. (I happen to own a copy of the book in which the Kundt paper appeared. I...
19. ### Plane Wave Equation Propagation and Oscillation Directions

Homework Statement Write down the equation for a plane wave traveling in perpendicular to the plane x+y+z=constant traveling in the direction of increasing x, y, and z. Homework Equations From the given information how do I determine the unit vector that goes next to E(0)? How do I determine...
20. ### Are plane waves real/physical?

When finding solutions to Maxwells equations we always cosider the case of a plane wave. But are plane waves real/physical solutions we can consider in real life? My guess is not because it is required to propagate infinitely. So why do we use plane waves to solve Maxwell's equations?
21. ### About wavefronts/wavelets and the formation of plane waves

The photography book I have talks about waveforms, but it doesn't do a good job of explaining them. So, here's my understanding from the book and google searches (it could be unbelievably wrong.) In an isotropic homogenous medium, light will spread out in all directions, essentially forming a...
22. ### E and B Fields of Monochromatic Plane Waves

Homework Statement 560 nm light is collimated and passes as a parallel beam in a direction perpendicular to the ##x+y+z=0## plane. It is polarized parallel to the ##(y-z)## plane. Treating it as a plane wave, what are the real E and B fields? Intensity of the beam is ##1 \ mW/cm^2##. Make...
23. ### Exploring Interference of Plane Waves Using the Residue Theorem

Homework Statement I have to show that the interference of plane waves: f^{(\pm)}(\vec r,t)=\int \frac {d^3k}{(2\pi)^{3/2}}\int \frac {d\omega}{(2\pi)^{1/2}}e^{i(\vec k \cdot \vec r - \omega t)}\tilde f^{(\pm)}(\vec k, \omega) where the amplitudes are given as: \tilde f^{(\pm)}(\vec k...
24. ### Poynting theorem for harmonic em plane waves

explain this derivation: Source: microwave engineering, pozar Why did pozar use those conjugates? What does the source current mean?
25. ### Exploring Elliptical Polarization of Plane Waves

Homework Statement Hi! The entire problem is this: (a) Two plane-polarized harmonic plane waves having the same propagation constant are polarized, respectively, along two perpendicular directions. Show that if the phases of the two waves are different, their superposition yields generally an...
26. ### How to use Wave equations for uniform plane waves

Homework Statement Home work 3 Q1 Study the E field in free space and a source-free region, E= (a + b)exp(-jkx), where a and b are nonzero real constants, and in the x,y plane respectively. Does it satisfy Maxwell’s equations? If so, find the k and H fields . If not, explain why not...
27. ### Interference of two plane waves in any point of space

Homework Statement For two plane EM waves with identical amplitude, frequency, polarization and phase traveling in the directions of ##\hat{k_1}## and ##\hat{k_2}##, show that their average intensity is given by ##2I_0[1+\cos((\vec{k_2}-\vec{k_1}) \vec{r})]##The Attempt at a Solution Ugh...
28. ### How Can Plane Wave Solutions Apply in Charged Particle Interactions?

Hi all, While I am reading interaction of atoms with radiation the following doubt came to me...While solving Maxwell equations in a charge and current free region, we get solutions for Vector potential,Electric field and so on which are plane wave solutions...However, when we study the...
29. ### How Do B and Gamma Values Affect Polarization in Electromagnetic Waves?

Homework Statement I'm currently trying to understand linear and circular polarization of electromagnetic plane waves. Let's say I have an electric field given by \vec{E}=Acos(kx-\omega t)\hat{x}+Bcos(kx-\omega t - \gamma)\hat{y} A is given and nonzero. I want to find what values of...
30. ### Show the loop integral of Poynting vector is zero for plane waves

Homework Statement Show that for plane waves, the following result holds: \oint \textbf{S}\cdot d \ell = 0. Homework Equations -- The Attempt at a Solution \oint \textbf{S}\cdot d \ell = \frac{1}{\mu_{0}}\oint (\textbf{E} \times \textbf{B})\cdot d \ell Now do I just use...
31. ### Maxwell's equations and plane waves

Homework Statement Show that the general relationship from Maxwell's equations for the conservation of energy \nabla \cdot \textbf{S} + \frac{\partial u}{\partial t} = 0, where u = \frac{1}{2} \epsilon _{0} \left| \textbf{E} \right| ^{2} + \frac{1}{2 \mu _{0}} \left| \textbf{B}...
32. ### Polarizations of plane waves propagating in anisotropic media

Hey guys! (I am not sure if I should post this thread in Physics or Mathematics) I have had some issues with developing expressions for the polarizations (material displacement) of waves propagating in anisotropic media. To bring you guys up to speed I have to start a few steps before the...
33. ### TEM plane waves, decay and attenuation

I've been trying to get reantiquated with electormagnetics to understand RF communications better. I have a question about TEM plane waves. The funtions which describe the plane waves in the z dirrection are: e-α cos(ωt-βz) ; in the time domain where is the rate of decay. In free...
34. ### Plane waves and sites distance in a lattice

Hi, i would ask you an opinion about a (maybe stupid) doubt. Let us think of a 1D lattice whose sites distant from each other "a"; a plane wave in the lattice is given by e^{ikja} where k is the momentum and j an integer label for each site. Now, we modify the lattice in this way: between...
35. ### Plane waves: sign of Re(ε), Re(μ) in passive media, attenuation angle

Hello, I'm having some issues with plane waves propagating through a medium which is: - linear - spatially and temporally homogeneous - spatially non-dispersive - isotropic - temporally dispersive - passive I know that permittivity, permeability and the k-vector are complex in...
36. ### Calculating Relative Phase of Reflected Plane Waves in Lossless Materials

not relating to any specific homework question: how can i go about calculating the relative phase of reflected / transmitted fields for normally incident plane waves? for example, i know how to calculate the relative amplitude of the reflected field from the reflection coefficient...
37. ### An optics problem - elliptical lens and plane waves

Homework Statement I must show that a point source placed at one of the focii will produce a plane wave (or vise versa) when the light is refracted through the interface. This problem is really kicking my ***, I've spent at least three hours trying to figure out different ways to do it...
38. ### Plane Waves and the Poynting Vector

Why is it that that poynting vector is independent of distance from the source? Is it because EM waves are plane waves? Furthermore I do not fully understand why EM waves have to be plane waves. I understand that changing magnetic fields give rise to electric fields and vice versa, but does...
39. ### How Does Orientation Affect EMF Induction in a Loop by a Plane Wave?

[b]1. The problem statement A plane electromagnetic wave traveling in a vacuum is given by E=(0, E_0*exp[i(kz-ωt)], 0) where E_0 is real. A circular loop of raduis a, N turns, and resistance R is located with its center at the origin. The loop is oriented so that a diameter lies along the z...
40. ### How to Find the Resultant Wave Equation of Two Plane Waves?

Homework Statement I have two plane waves, one of the form: 4 sin(20t + (pi/3)x + pi), and the other one: 2 sin(20t +(pi/4)y + pi) .. with the same frequency and vibrations in the z direction .. I am asked to find the resultant wave equation at x = 5 and y =2 Homework Equations...
41. ### Is it acceptable to use plane waves?

In classical mechanics, scattering depends on initial momentum and position. In quantum mechanics, the initial condition that is specified is just the momentum. But if this were strictly true, then definite momentum implies the particle is somewhere out of the lab and therefore doesn't...
42. ### Applying Maxwell's equations to plane waves

Homework Statement For a harmonic uniform plane wave propagating in a simple medium, both \vec{E} and \vec{H} vary in accordance with the factor exp(-i \vec{k}.\vec{R}) Show that the four Maxwell’s equations for a uniform plane wave in a source-free region reduce to the following: \vec{k}...
43. ### Plane waves, phase difference question

Homework Statement Consider a lightwave having a phase velocity of 3 x 10^8 m/s and a frequency of 6 x 10^14 hz. What is the shortest distance along the wave between any two points that have a phase difference of 30 degrees ? What phase shift occurs at a given point in 1 microsecond and...
44. ### Orthogonality of Matsubara Plane Waves

Hi there! In thermal field theory, the Matsubara frequencies are defined by \nu_n = \frac{2n\pi}{\beta} for bosons and \omega_n = \frac{(2n+1)\pi}{\beta} for fermions. Assuming discrete imaginary time with time indices k=0,\hdots,N, it is easy to obtain the following orthogonality relation...
45. ### EM Field strength and plane waves

Hi ! I've a question. Where is the connection between the (kinetic) Lagrangian - \dfrac{1}{4} F_{\mu \nu} F^{\mu \nu} and a plane wave of the form \vec{\varepsilon} exp(i \vec{k} \cdot \vec{x}) / \sqrt{V} (the epsilon is a polarization vector) confined in a box with a finite volume V ? I...
46. ### Electromagnetic plane waves in vacuo

Consider a plane wave f = cos(kz - wt). Applying Maxwell's equation (divE=0 in vacuo) gives kcos(kz - wt) = 0 which means that k = 0. This surely doesn't make sense?
47. ### Superposition of two plane waves

Hi, I'm not after much help here. I already have an answer, but I want to make sure that I haven't made any stupid mistakes and that I understand the question. I also have a query about the second part of the question. Homework Statement What is the probability density of the...
48. ### Non-Uniform Plane Waves, Multiple values of E and H?

I have a question in Electromagnetics, precisely about non-uniform plane waves. In the lecture, the professor made a strange assumption, he first used two of Maxwell's equations to get an expression for E and H (which he called E1 and H1). Then he used the remaining two to derive what he...
49. W

### Plane Waves vs. Waves on a String

I already took 3 quarters of EM and I'm ashamed to say I didn't learn anything the final quarter, where we covered the most interesting topics. Bah. One thing that I'm still confused about are plane waves. I understand the description of a regular sine wave on string. If you wiggle it once...
50. ### Expansion of polarized plane waves into spherical harmonics,

expansion of polarized plane waves into spherical harmonics, please help! Hi all, I would like to get some guidance in how to expand a polarized (i.e. linear polarization) plane wave into a series of spherical harmonics. I am aware of the formula applying to scalar plane waves (please see...