Untitled

I apologize for the EPICLY long post, but it is rather necessary so please bear with me.

 

>So, are sin(t+4-x), sin(t+5-x), and 1.76*sin(t+9/2-x) all simple waves? They seem to be.

 

Okay.  Let's start from the beginning here.

 

A [sine wave](http://en.wikipedia.org/wiki/Sine_wave) (i.e. simple wave) by definition has the following equation:

 

A(t) = A_0 * sin(ωt + φ)

 

Let's define what each part is:

 

A_0 is the amplitude; a constant.

ω is the angular frequency; also a constant.

t is the time; a variable.

φ is the phase; a constant.

 

Note that angular frequency has a unit of radians per second (rad/s).  What we really want to talk about when referring to frequency is known as the "ordinary frequency," or "temporal frequency," which is measured in inverse seconds (1/s), also known as Hertz.  The frequency f is defined in terms of the angular frequency as:

 

f = ω / 2π

 

Now, let's move forward.

 

We see in the definition of a sine wave above that there is only one variable on the right-hand side of the equation, t.  All of your equations have two variables, x and t.  Therefore, none of your waves are (1-spatial-dimensional) sine waves.

 

So what could they be?  A suitable definition may be a [plane wave](http://en.wikipedia.org/wiki/Plane_wave), a higher-dimensional analogue of the sine wave, which has the following equation:

 

A(x,t) = A_0 * cos(kx - ωt + φ)

 

Where:

 

A_0 is the amplitude; a constant.

k is the angular wavenumber; also a constant.

x is the position in space; a variable.

ω is the angular frequency; also a constant.

t is the time; a variable.

φ is the phase; a constant.

 

But, this definition has a cosine function instead of a sine function.  Okay, so that's not a terrible problem, as we can rewrite your functions as a cosine function, and match it exactly.  I will only do this for the first function you provided, for right now:

 

y(x,t) = 1 * cos(-1x - (-1t) + 4)

 

I have explicitly written out the ones and negatives for completeness' sake.

 

At this point, we need to introduce a new type of frequency.  Now we need to define the [spatial frequency](http://en.wikipedia.org/wiki/Spatial_frequency), ν (Greek letter nu), which is measured in units of inverse metres (1/m):

 

ν = k / 2π

 

So, now we have two different notions of frequency; temporal, and spatial.  This of course makes sense since our plane wave function is of two variables, x and t.  But now we must speak about which notion of frequency we mean.  Generally, when we say frequency we are talking about temporal frequency, and in the future, when I say "frequency" I mean "temporal frequency."

 

Anyhow, to break off from this discussion, I want to point out that an electromagnetic plane wave is a function of four variables, and has an equation as follows:

 

A(x, y, z, t) = sin(k_x*x + k_y*y + k_z*z - ωt + φ)

 

So an electromagnetic wave, which is what I originally referenced, is quite a bit more complex, but I am pointing this out so you can see the pattern of generalization.

 

We can of course, hold two of those spatial variables constant at the value 0, and recover the two-variable plane wave equation described previously.  There's much more to it than that, of course -- part of that "more" is that an electromagnetic plane wave actually has two equations, one corresponding to the electric field amplitude, and one corresponding to the magnetic field amplitude, and that the two amplitudes are inexplicably linked.

 

In any case, if we generalize the definition of a "simple wave" to include higher-dimensional analogues of sine waves, such as plane waves, then yes, your equations are "simple waves," even though they are hardly simple to describe. :P

 

>How about (sin(t+4-x) + sin(t+5-x)), is it, the interference of two simple waves, a complex wave?

>How about if you simplify it mathematically and write it as 1.76*sin(t+9/2-x)?

 

In short, no, because it only has a single frequency, which makes it a simple wave, as you demonstrated by rewriting it with a single sine function.

 

>Either your statement wasn't categorically true, or there's no distinction between complex and simple waves.

 

I think you missed the point entirely here.  You can decompose any single-frequency wave into any number of waves *with the same frequency*, arbitrarily.  It's still a simple wave, because it still has only a single frequency.  Any combination of single-frequency waves that have the same frequency, still gives you a single-frequency wave.

 

Remember that I said, "a simple wave is otherwise known as a sine wave, which contains only a single frequency."  Granted my definition there was overly simplistic in that it did not include the generalization to higher-dimensional analogues of a sine wave, it still made explicit mention of the requirement that it contains a single frequency.

 

I never said that adding any two single-frequency waves *always* gives a complex wave -- only that a complex wave is by definition made up of more than one single-frequency wave.

 

That does, unfortunately, make this the **third time in a row** now that you have misread what I said ...

 

...

 

>So then, what about (sin(t+4+x) + sin(t+4-x))? It contains only a single frequency, so it's a simple wave?

 

Can it be rewritten in the form A(x,t) = A_0 * cos(kx - ωt + φ)?  If so, then it is a simple plane wave.  If not, then it is complex.

 

I'm going to cheat a little bit here and plug it into [Wolfram Alpha](http://www.wolframalpha.com/input/?i=y%28t%2C+x%29+%3D+%28sin%28t%2B4%2Bx%29+%2B+sin%28t%2B4-x%29%29), which shows an alternate form of the equation derived from the [sum-to-product identity](http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities):

 

A(t,x) = 2 * sin(t + 4) * cos(x)

 

Now, I see that in your original sum-of-two-simple-plane-waves, the variable x had the same sign, and you were able to reduce the cosine argument to (1/2) which is constant and can therefore be multiplied by the leading 2 to get approximately 1.76, by using the [sum-to-product identity](http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities) to derive as follows:

 

sin(t+4-x) + sin(t+5-x) =

 

2 * sin( ( (t+4-x) + (t+5-x) ) / 2) * cos( ( (t+5-x) - (t+5-x) ) / 2 ) =

 

2 * sin( ( t + 4 - x + t + 5 - x ) / 2 ) * cos( ( t + 5 - x - t - 5 + x ) / 2 ) =

 

2 * sin( ( 2t - 2x + 9 ) / 2 ) * cos( ( 1 / 2 ) ) =

 

2 * sin( t - x + (9/2) ) * cos(1/2) =

 

2 * cos(1/2) * sin(t - x + 9/2) =

 

~1.76 * sin(t + 9/2 - x)

 

I am also seeing that in deriving a standing wave reduction the same way, the cosine argument is x, which is not constant, and thus cannot be factored out in this way:

 

sin(t+4+x) + sin(t+4-x) =

 

2 * sin( ( (t+4+x) + (t+4-x) ) / 2) * cos( ( (t+4+x) - (t+4-x) ) / 2 ) =

 

2 * sin( ( t + 4 + x + t + 4 - x ) / 2 ) * cos( ( t + 4 + x - t - 4 + x ) / 2 ) =

 

2 * sin( ( 2t + 8 ) / 2 ) * cos( 2x / 2 ) =

 

2 * sin(t + 4) * cos(x)

 

It seems quite clear to me that your former plane wave sum has component waves with both the same temporal frequency and the same spatial frequency, which explains why they can be added into a single plane wave, but the latter plane wave sum has component waves with the same temporal frequency, *but different spatial frequencies*, and thus cannot possibly be described as a single wave.

 

Now, I've been sitting here for two hours now, thinking that perhaps my math-fu is just weak/rusty, and trying to use every trigonometric identity I can think of to simplify the equation you gave into a single sine function, under the assumption that you are right.  Without success.

 

To address your earlier statement:

 

>Based on this, since the coefficient on t is 1 on all those waves, then they're all simple waves.

 

In other words, what I am saying is, in higher-dimensional analogues of the sine wave, having different spatial frequencies but having the same temporal frequency is not sufficient to combine two waves into a single simple wave.  *ALL* of the frequencies must be the same.  That means both the coefficient of t AND the coefficient of x must be the same.  (And that includes y, z, and so-on into the even-higher-dimensional analogues.)

 

Now, if you believe you are able to reduce it to a single sine wave, then I would like to see your derivation.  If you can provide one, and I can verify that it is correct, then I will eat my hat and go back to correct what false statements I have made thus far.  However, I am convinced that there is no such derivation, which would mean that your standing wave cannot be described as a simple plane wave.

 

That said, if you believe you can prove that a standing wave is in fact a simple plane wave, then I leave the burden of proof to you, good sir.

 

>>>>[From your original post:] Total destructive interference refers to the cancellation of one quantity, but to achieve this, you must have constructive interference of the other quantity.

 

>>>[me] standing wave in a guitar string, the nodes have neither potential nor kinetic energy

 

>>a standing wave is made up of much more than just nodes. Antinodes have all of the potential/kinetic energy that nodes do not

 

>Your latest statement is right, but the original was wrong (or at least misleading). At points where destructive interference occurs (the nodes), both quantities are zero; they both physically appear in a different location, where constructive interference occurs (the antinodes).

 

The very equations that describe electromagnetic wave propagation demand that you are wrong.  If you were correct, then electromagnetic waves would not be able to propagate in free space.

 

I actually spent my last semester in university learning this in my Elementary & Classical Physics course.

 

Some sources for you:

 

My old physics textbook, [Fundamentals of Physics, 9th Edition](http://www.amazon.com/Fundamentals-Physics-Edition-David-Halliday/dp/0470556536) by Halliday, Resnick, and Walker

[Wikipedia: Electromagnetic wave](http://en.wikipedia.org/wiki/Electromagnetic_wave)

 

From the latter source:

 

"According to Maxwell's equations, a spatially varying electric field causes the magnetic field to change over time. Likewise, a spatially varying magnetic field causes changes over time in the electric field. In an electromagnetic wave, the changes induced by the electric field shift the wave in the magnetic field in one direction; the action of the magnetic field shifts the electric field in the same direction."

 

[Graphic](http://en.wikipedia.org/wiki/File:Onde_electromagnetique.svg)

 

Look at the above diagram.  You will see two waves -- one electric and one magnetic.

 

Now, superfically looking at the diagram, it would seem you are correct -- the electric field and magnetic field are both zero at the same point.

 

But there is a critical mistake you've made, and I will admit it is clear that you have made it because I actually did say something wrong in the very beginning, out of a need to simplify the concept.  Originally, I said:

 

>You can either define the electric field amplitude as kinetic and the magnetic field amplitude as potential, or the other way around.

 

However, this really isn't the full story.  Both the electric field amplitude and the magnetic field amplitude should be thought of as potential, while the *change* in electric and magnetic *flux* (not shown in the graphic) should be thought of as kinetic.

 

[Faraday's law](http://en.wikipedia.org/wiki/Faraday%27s_law_of_induction) describes how a changing magnetic flux generates an electric field.

 

The [Maxwell-Ampère euqation], on the other hand, describes how a changing electric flux generates a magnetic field.

 

These equations, together with the equations for [electric](http://en.wikipedia.org/wiki/Gauss%27s_law) and [magnetic flux](http://en.wikipedia.org/wiki/Gauss%27s_law_for_magnetism), form a set of equations known as [Maxwell's equations](http://en.wikipedia.org/wiki/Maxwell%27s_equations).

 

One point that my instructor really *forcefully* hammered home time and time again, was how the change in these four quantities cause them to cycle:

 

A changing electric field generates an electric flux;

a changing electric flux in turn generates a magnetic field;

a changing magnetic field in turn generates a magnetic flux;

a changing magnetic flux in turn generates an electric field.

 

At any point where the electric and magnetic fields are zero, the *change* in electric flux and *change* in magnetic flux are at their maximum.  (It is very important to note that it is the *change* in flux, not the flux itself, that is nonzero and drives the fields to regenerate.  My instructor was absolutely *relentless* in stressing this; I will never forget it as long as I live, haha.)

 

So, you caught me.  I was wrong when I said electric and magnetic fields should be thought of as kinetic and potential, and I have done my instructor a great disservice in saying so.  Clearly the time I've spent away from the books has taken its toll weathering away my understanding of the subject.

 

But, my point does still ultimately stand, that a propagating wave oscillates between a kinetic and a potential energy.

 

Another point I feel I should be clear about:

 

The amplitude of a sine wave (or higher-dimensional analogue thereof) is not related to the total energy, only either one of potential or kinetic energy.  To describe total energy, two *dual* sine waves are needed, and there are two *dual* forces at work.

 

Consider [Hooke's law](http://en.wikipedia.org/wiki/Hooke%27s_law) for a spring system, which describes the exerted force (manifestation of potential energy) in relation to the amplitude of the sine wave describing position:

 

F = -kx

 

The potential energy stored in the spring is described by the equation:

 

U = (1/2)kx^(2)

 

Where the force is zero, the potential energy is zero, and where the force is maximal, the potential energy is maximal.

 

And we have the wave equation that then describes how the position varies with time:

 

x(t) = A * cos(ωt) ; or equivalently,

x(t) = A * sin(ωt - π/2)

 

However, where the force and potential energy are zero, the velocity (and thus kinetic energy) is maximal, as is the kinetic energy:

 

K = (1/2)mv^(2)

 

p = mv   ; p is momentum

 

In the above equation, the momentum described is that which the mass on the end of the spring acquires (manifestation of kinetic energy), just like how for potential energy the force described is that which would be exerted on the mass.

 

And of course, there is a wave equation that then describes how the velocity varies with time:

 

v(t) = A * sin (ωt) = dx(t)/dt  ; or equivalently,

v(t) = A * cos (ωt + π/2)

 

As you can see, they are the same sine wave, but phase shifted by 1/4 of the period.

 

And if you graph both the position and the velocity together, you get a graph that looks like [this](http://electron9.phys.utk.edu/phys135d/modules/m9/images/oscill2.gif).

 

The TOTAL energy of the system, is held constant by the *covariance* of the position and the velocity.  Equivalently stated, it is held constant by the *covariance* of the kinetic and potential energy.

 

I think I've gone on enough about this for now, so I'll stop here.

 

>Your original post is incorrect in that the phenomenon of destructive interference at a point does not mean that the energy at that location is transferred into a different form at that same point.

 

You're correct, and I misspoke.  Destructive interference at a point means that either the *kinetic* or the *potential* energy (depending on which is related to the sine wave we're talking about) at that location is converted into the opposite type.

 

>My clearest counterexample was that of the guitar string above

 

And as common sense will tell you, when a guitar string is at its maximal displacement, it is at its minimal velocity, and vice versa.  If your sine wave describes the displacement of the guitar string, there is another phase-shifted sine wave which describes the velocity of the guitar string.

 

>I'll also add that your Poynting vector analysis is wrong (but also inconsequentially wrong because of another error). Even if your statement about energy at an interference minimum being all in E or in B were true, then S=ExB=0 since one of E or B would be 0 at that point. If both were to become nonzero at some later point, S would also change. However, this is inconsequential to your argument, since the Poynting vector only specifies energy flow and not any stored energy, and there's nothing that says it has to stay constant.

 

You're correct that my analysis is wrong -- at least in part because I made the error of associating the electric and magnetic fields with opposite energy types, when I should have associated the fluxes with opposite energy types.

 

Given the proper association, you are correct, and the Poynting vector describes not the total energy, but the total energy *flux* through the third dimension of space.  You've got me there. :)

 

Now then.  I feel I must again apologize for the length of this reply, but I believe it is comprehensive enough to end the discussion.  If there is anything yet outstanding, feel free to point it out in a reply and I will address it.

 

Good job in challenging me.  While you were wrong on a number of points, I was wrong on a number of points as well.

 

(Although I do wish to point out, except for the Poynting vector and associating the wrong quantities with kinetic/potential energy, the rest of my original post was correct, including the answer to the OP's question, "where does the energy go?")