# Not HW: Why are light waves in the form of the sine wave, instead of some other wave?

#### physixer

Why are light waves/X-rays/gamma rays/etc. in the form of sine waves, rather than, say, a zig zag wave, or even a straight line?

I recently watched this youtube video explaining how to visualize ten dimensions:
And wondered if photons behave as a sine wave because they were going in circles in some other dimension (or something) and that (somehow)( I'm just in precalculus II I can't explain my guess any further) is the reason for why the wave is a sine wave. I just made the connection, but I'm 50% sure circles have nothing to do with the sine wave behavior of light.

I'm just curious. Thanks for your insight.

edit: actually it was carl sagan's explanation of the 4th dimension that got me thinking:

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#### physixer

Why are light waves in the form of the sine wave, instead of some other wave?

Why are light waves/X-rays/gamma rays/etc. in the form of sine waves, rather than, say, a zig zag wave, or even a straight line?

I recently watched this youtube video explaining how to visualize ten dimensions:
And wondered if photons behave as a sine wave because they were going in circles in some other dimension (or something) and that (somehow)( I'm just in precalculus II I can't explain my guess any further) is the reason for why the wave is a sine wave. I just made the connection, but I'm 50% sure circles have nothing to do with the sine wave behavior of light.

I'm just curious. Thanks for your insight.

edit: actually it was carl sagan's explanation of the 4th dimension that got me thinking:

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#### Born2bwire

Gold Member
Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

They are and they aren't. We simply think of them as being sinusoidal because it is an easy mathematical basis to work with. When you work out the wave equations from Maxwell's equations, assuming a time-harmonic wave (sinusoidal) greatly simplifies the mathematics. Most materials are linear in regards to light, this means that frequency in == frequency out, and so if we can describe a signal, any signal, as a superposition of frequencies, then the analysis of the system can be greatly simplified. And so it is with light since classical electrodynamics follows the principle of linear superposition.

So, can I have an electromagnetic signal that is not a pure sine wave? Sure, to an extent. We can send approximations of square waves and saw-tooth waves. I say approximation because the necessary bandwidth for these signals is infinite and thus our ability to reproduce them is restricted by the bandwidth of our own signals. But when you get down to it, even a square wave is nothing but a superposition of sine waves. Fourier series is a good means of showing that you can pretty much decompose any real world signal into a summation of sine waves. The limit to his theorem, being that you cannot correctly reproduce discontinuous signals (like the square wave) is generally reflected in the real world too.

Gold Member
Re: Why are light waves in the form of the sine wave, instead of some other wave?

Sine waves are the only fundemental type of wave there is. All waveforms other than sine such as saw, square, or any random waveshapes, always contain the sum of sine waves of different frequencies. In other words, any waveshape other than a sine wave contains several sine waves of different frequencies added together.

Light is represented as a sine wave because it's light of a single frequency so it's the simplist thing to draw on paper. Laser light is a single frequency. White light however is light made up of the sum of different wavelengths of light.

Keep in mind however that no one really knows if light is a wave, only that it seems that way because of things like diffraction and interference. Since light can also be a particle and has what is known as particle/wave duality, there is still a mystery surrounding what light or any electromagnetic radiation really is.

#### Phrak

Re: Why are light waves in the form of the sine wave, instead of some other wave?

Light is not a single frequency, as said above. Propagating electromagnetic fields can have just about any form.

Light doesn't come in sine waves. However, light may be decomposed into sums of sine waves or square waves, or triangular waves (zig-zag), or whatever. A "straight line" is also a physically meaningful solution. It means no elecromagnetic radiation.

Gold Member
Re: Why are light waves in the form of the sine wave, instead of some other wave?

Light is not a single frequency, as said above. Propagating electromagnetic fields can have just about any form.

Light doesn't come in sine waves. However, light may be decomposed into sums of sine waves or square waves, or triangular waves (zig-zag), or whatever. A "straight line" is also a physically meaningful solution. It means no elecromagnetic radiation.
A filtered laser beam is a single frequency, hence it is a sine wave. Shine a laser into a spectrometer and you will see a single spike in the spectrum.

#### Bacat

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

Light is approximated by sinusoids because the electric and magnetic fields are oscillating in time. It might be easier to conceptualize this if you ignore light and think about radio waves. It's the same energy, just a different wavelength. So, in the radio wave, you have a slowly oscillating (say 3000 times per second) electric field. This oscillation could be described in a lot of ways but the easiest is a sinusoid.

If you study about light some more, you'll get into Maxwell's Equations which basically show you that the electric field and the magnetic fields are turning into each as the light propagates. At any time, the sum of the two fields is the same- the energy is just being distributed in two different ways. This would seem to indicate that it is not necessary to have a circular motion in a light wave to account for the sinusoid. Rather, it's like a pendulum that is converting kinetic (motion) energy into potential (stored in the height of the pendulum against gravity) and back again. It's very natural to think of light this way.

If you wish to persist with your notion of sinusoids being circular motions viewed from the side, study some complex exponentials. In the complex plane, exponentials are rotations around the plane. This seems strange because they increase/decrease to infinity in the real plane, but when you understand how imaginary numbers work it is quite natural for exponentials to rotate.

Keep thinking!

#### Phrak

Re: Why are light waves in the form of the sine wave, instead of some other wave?

A filtered laser beam is a single frequency, hence it is a sine wave. Shine a laser into a spectrometer and you will see a single spike in the spectrum.
No, it isn't. A laser cavity supports many frequencies. Say you do have a single mode--I think there are ways to do this. But the laser has not been on for an infinite time, nor will it persists for an infinite time in the future. Therefore this would be a square wave modulated sine wave, that decomposes into many frequencies.

By the way, the electric and magnetic fields of propagating waves are in phase, not out. Their amplitudes peak at the same time.

An ideal single frequency, or a frequency component that is polarized, may be decomposed into left and right circularly polarized parts. Each part has a constant electric and magnetic field amplitude. The direction of the field rotates circularly.

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#### Phrak

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

Does anyone know what the 4-vector potential is doing in circularly polarized light?

#### Cleonis

Gold Member
Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

Why are light waves/X-rays/gamma rays/etc. in the form of sine waves, rather than, say, a zig zag wave, or even a straight line?
As remarked in an earlier posting, electromagnetic waves are oscillations. Certainly in the case of propagation in vacuum the oscillations are harmonic oscillation.

For any oscillation: to be sustained a force is required that tends to restore the instantaneous state to some equilibrium state.

An oscillation is called a 'harmonic oscillation' when the strength of the restoring force is in exact linear proportion to the deviation from equilibrium state. For instance, in mechanical oscillation: if the force that is exerted upon the oscillating object is proportional to the distance to the midpoint of the oscillation the oscillation will be harmonic.

Therefore it's possible to have an emitting device, such as a radio antenna, emit genuine sine waves.

In order to emit a signal that is very close to, say, a square signal, you have to try and combine sine waves of different frequencies in such a way that the superposition of all of them is a close approximation to the desired square signal.

Cleonis

#### mikeph

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

My answer- because the 2nd derivative of a sine wave is a negative sine wave, allowing the electromagnetic field to maintain itself independent of charges, therefore radiating light. No other wave (apart from derivations from sine, such as cos, or sin(x+pi/2)) can claim this feature, and so no other wave satisfies Maxwell's equations.

#### Redbelly98

Staff Emeritus
Homework Helper
Re: Why are light waves in the form of the sine wave, instead of some other wave?

A filtered laser beam is a single frequency, hence it is a sine wave. Shine a laser into a spectrometer and you will see a single spike in the spectrum.
Look at the spectrum with fine enough resolution, and that "spike" will appear as a broad peak. So, not single frequency.

Describing EM waves as sinusoids (or as the sum of many sinusoids) is just a mathematical tool for describing them. We could instead use any complete, orthogonal set of basis functions, but sinsusoids have nice mathematical properties that make it a convenient way to represent EM waves.

#### shespuzzling

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

As a follow up to this: to create radio waves you need an oscillating electric field...is it this oscillation that creates the EM wave? How are EM waves emitted from the sun then, if the electric field isn't necessarily oscillating?

#### Naty1

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

And wondered if photons behave as a sine wave because they were going in circles in some other...dimension....
There is no evidence for this so far. String theory seems to hint that gravitational waves might pass to other dimensions but not the electromagnetic force....

#### Redbelly98

Staff Emeritus
Homework Helper
Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

As a follow up to this: to create radio waves you need an oscillating electric field...is it this oscillation that creates the EM wave? How are EM waves emitted from the sun then, if the electric field isn't necessarily oscillating?
Not quite.
The radio (or any EM) wave is an oscillating electric field. What creates it are charged particles that oscillate somewhere, for example in the radio station's transmitting antenna, or in the sun.

#### kcdodd

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

Sine waves are the only fundemental type of wave there is. All waveforms other than sine such as saw, square, or any random waveshapes, always contain the sum of sine waves of different frequencies. In other words, any waveshape other than a sine wave contains several sine waves of different frequencies added together.

Light is represented as a sine wave because it's light of a single frequency so it's the simplist thing to draw on paper. Laser light is a single frequency. White light however is light made up of the sum of different wavelengths of light.

Keep in mind however that no one really knows if light is a wave, only that it seems that way because of things like diffraction and interference. Since light can also be a particle and has what is known as particle/wave duality, there is still a mystery surrounding what light or any electromagnetic radiation really is.
I don't know if there is anything fundamental about a sine wave. I'm pretty sure you can decompose a function using any complete set of orthogonal functions. As an example, suppose you were given an infinite set of square waves of continuous frequencies. You could reproduce a sine wave by a linear combination of those, just like making a square wave from combinations of sine waves. Or another example, a continuous set of delta functions.

A filtered laser beam is a single frequency, hence it is a sine wave. Shine a laser into a spectrometer and you will see a single spike in the spectrum.
But that is an example to get the desired result. As my previous example, could theoretically set up a filter for a distribution of "sine wave frequencies", thus creating a filter of a single "square wave frequency". Although energy does seem to be quantized in terms of "sine wave frequency".

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#### Bob S

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

If you keep on putting neutral density filters in a visible light beam, you will eventually get to a point where there are only a few photons per second. These individual light photons have to satisfy the two Maxwell curl equations, and if you substitute one into the other, you will get (without conduction)

curl curl E+ ε0μ0E = 0

or del2E - ε0μ0E = 0

A sine wave is one of the few (only?) function that satisfies this wave equation (without attenuation).

From PHRAK: By the way, the electric and magnetic fields of propagating waves are in phase, not out. Their amplitudes peak at the same time.
I would like to believe this, but then if we look at the curl equation

curl E = - μ0H/∂t

E and H appear to be 90 degrees out of phase (E=-μ0H in EE parlance), just like in the Faraday induction Law..

Bob S

#### kcdodd

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

If a sine wave is a solution, then any wave is a solution, due to Fourier decomposition. A photon can be any shape. There seems to be 'something' special about sine waves, and it'd be interesting if someone had a physical reason why the universe decided not to quantize in terms of square, triangle, or teddy-bear waves...

#### Bob S

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

If a sine wave is a solution, then any wave is a solution, due to Fourier decomposition. A photon can be any shape. There seems to be 'something' special about sine waves, and it'd be interesting if someone had a physical reason why the universe decided not to quantize in terms of square, triangle, or teddy-bear waves...
I thought all the atomic transition lines, like the hydrogen transition 4d->2p, was a sine wave. See (Enter H for spectrum and 4000 to 6000 A for range):

http://physics.nist.gov/PhysRefData/ASD/lines_form.html

Here is the Fourier decomposition for a square wave.

http://mathworld.wolfram.com/FourierSeriesSquareWave.html

Note that all the components of the square wave are sine waves. If any atomic transition line were a square wave, there would have to be the frequency harmonics sin(nωt) for n= 2, 3, 4 etc. But n>1 would violate the requirement that the transition energy is fixed at a single value. Are there any visible lines in light i.e., (photons) that are square waves, and have the necessary frequency harmonics?

Bob S

#### kcdodd

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

Thats what I meant by quantize in terms of sine wave frequency. An energy transition in an atom gives a single energy, and that corresponds to a single sine wave frequency. I am fairly sure all properties eventually derive from this idea, as in dispersive media, involving photons interacting with the atom. Of course, it is not perfect because photons do not immediately occupy the entire universe. Due to uncertainty principle all photons would have a distribution of energy/momentum (ie frequencies). The uncertainty probably comes from the final kinetic energy of the atom imparted when the photon kicked out (conservation of momentum).

Now, you could artificially make a 'square' photon. Imagine it travelling along and someone traps it in a box. Well, it's shape has to conform for it to be inside. However, since we always quantize energy by sine waves, for whatever reason, all the modes are sine waves. But you can still get something fairly square like given enough energy bandwidth. At least, I think so...

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#### Born2bwire

Gold Member
Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

If you keep on putting neutral density filters in a visible light beam, you will eventually get to a point where there are only a few photons per second. These individual light photons have to satisfy the two Maxwell curl equations, and if you substitute one into the other, you will get (without conduction)

curl curl E+ ε0μ0E = 0

or del2E - ε0μ0E = 0

A sine wave is one of the few (only?) function that satisfies this wave equation (without attenuation).

I would like to believe this, but then if we look at the curl equation

curl E = - μ0H/∂t

E and H appear to be 90 degrees out of phase (E=-μ0H in EE parlance), just like in the Faraday induction Law..

Bob S
The curl of the electric field would bring out a -jk factor on the left-hand side cancelling out the imaginary factor on the right-hand side.

#### Bob S

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

The curl of the electric field would bring out a -jk factor on the left-hand side cancelling out the imaginary factor on the right-hand side.
Thanks. You are absolutely correct.

Now we have a slight ambiguity.

Faraday's law is commonly written (in air)

[1] E·dl = -(d/dt)μ0H·n da

which is closely related to the EE equation

[2] V = L dI/dt

where, if E(t) = E0sin(ωt), clearly E (and V) are 90 degrees out of phase with H (and I).

Using Stokes Law for the vector E

[3] E·dl = ∫(curl·E)·n da

which, by comparing to [1], leads to the Maxwell equation

[4] curl E = -μ0H/∂t

It was established in an earlier post that in [4], E and H were in phase and curl·E and H (and hence E) were 90 degrees out of phase, while in [1] E and H are 90 degrees out of phase.

Furthermore, in [2] (Stokes Law) we have a vector E which clearly is in phase on both sides of the equation, implying that E and curl·E are also in phase. How can this be?

Bob S

Before [4], add where E(x,t) = E0·sin(ωt - kx)

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#### kcdodd

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

I think perhaps when you integrate E under stokes theorem and you draw your loop, it is over differing spatial locations, and so differing phase of E.

#### bjacoby

Re: Not HW: Why are light waves in the form of the sine wave, instead of some other w

If you keep on putting neutral density filters in a visible light beam, you will eventually get to a point where there are only a few photons per second. These individual light photons have to satisfy the two Maxwell curl equations, and if you substitute one into the other, you will get (without conduction)

curl curl E+ ε0μ0E = 0

or del2E - ε0μ0E = 0

A sine wave is one of the few (only?) function that satisfies this wave equation (without attenuation).

I would like to believe this, but then if we look at the curl equation

curl E = - μ0H/∂t

E and H appear to be 90 degrees out of phase (E=-μ0H in EE parlance), just like in the Faraday induction Law..

Bob S
1. Mathematics is not more real than reality!

2. A sine wave is a mathematical construct which does not exist in reality. All real waves start and stop at some point making them a pulsed function. Such "modulation" creates a broadened bandwith and a "pure" single frequency is hence impossible.

3. Also even in a pure single frequency laser ignoring the pulsed nature of the output, there is the additional problem of limited coherence. This essentially amounts to phase modulation which like amplitude modulation broadens the frequency spectrum of the light.

4. What you guys are doing is deriving that the uncertainty principle is at work here. It is related to Fourier transform relations that are mutually exclusive.

5 Hence, it follows that your sine wave solution to the wave equation is correct but totally abstract. It is a mythological construct with no basis in reality. Mathematics is NOT more real than reality!

Part II.

Sorry. Magnetic and Electric fields in EM radiation are in phase. And No, they DO NOT "create each other".

Nor is Faraday's law correct as usually expressed where a changing magnetic field creates an electric field. Bzzzt. Wrong. The truth is that Maxwell's equations as usually expressed are NOT causal relationships! Yes, one side of the equation EQUALS the other side, but one side does not CAUSE the other side! The cause of BOTH E and B fields is CHARGE and it's variations. Hence in Faraday induction a changing electric current (charge) creates BOTH the changing magnetic field AND the electric fields resulting in EMF. Likewise for EM radiation the charges simultaneously create BOTH Electric and Magnetic fields which are propagated out through space from the source together and as it happens, in phase.

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