Wave shape

[Quartile]

What is the shape of physical waves? I would assume they are usually sin waves, but is it possible to have a shaped wave like square or triangle occur naturally? How would the shape of a wave be determined?

[swansont]

You can get square and triangle waves by adding together certain sine waves (Fourier components) and that's why you can synthesize different tones. As to whether these occur naturally, it depends on what is considered "natural." You get different wave shapes by striking vs plucking a taut string — are these natural?

[Quartile]

I am not referring to sound as I do understand how different wave shapes are produced in music. I am referring specifically to other physical phenomena.

 

If everything can be expressed as a wave function then it certainly has a wave form? Maybe I'm missing something.

[5614]

I don't understand your question, there are many different wave shapes... Generally waves observed in physics will follow some sort of a sinusoidal shape. When you talk about a photon it is comprised of two oscillating waves (one electric, one magnetic) that are both sinusoidally shaped. What exactly are you asking? (try to be specific!)

[Quartile]

More specifically then, is it possible for physical waves (other than sound) of the same type (electric, magnetic, transverse, or otherwise) to combine together as Fourier components to form different wave shapes like square, triangle, saw, etc?

[swansont]

More specifically then, is it possible for physical waves (other than sound) of the same type (electric, magnetic, transverse, or otherwise) to combine together as Fourier components to form different wave shapes like square, triangle, saw, etc?

 

Yes. Signal generators for electrical signals are standard lab equipment, and making square, triangle and sawtooth waves are typical functions. Programmable generators are available, too, to make arbitrary waveforms.

[Quartile]

So it is possible to formulate different shapes of waves artificially using equipment and whatnot, but is it possible for these shapes to be formed in nature by "accident"?

[swansont]

Any signal that's not comprised of an infinitely long sine wave will have harmonics in it.

 

heart arrhythmia

electroencephalogram

[foodchain]

Any signal that's not comprised of an infinitely long sine wave will have harmonics in it.

 

heart arrhythmia

electroencephalogram

 

I remember that stuff. Some machine recorded my heartbeat on a running machine. I think it was between 172-174 beats per minute. It was a two mile run on a machine though, but I did do it in under ten minutes so the pace must have been good.

[Quartile]

Any signal that's not comprised of an infinitely long sine wave will have harmonics in it.[/Quote]

 

What is an example of a signal that is comprised of an infinitely long sine wave?

 

Could two infinitely long sine waves combine in the same way that finite waves combine? (I assume that the definition of infinite here applies in much the same way as it does to the universe being infinite if it is continually expanding outward; that is, since the sine wave continually oscillates through time it is said to be infinitely long.)

 

Thanks for having patience with me through these questions!

 

congrats on 1000 posts foodchain

[thedarkshade]

Could two infinitely long sine waves combine in the same way that finite waves combine?

They could go on combining infinitely.

[Mike Dubbeld]

Hi, I only posted once before a few months ago. I am playing around with DE and Fourier Series and FFT also. You can create shapes with any sort of waves you like. Sine waves are popular because no matter how many times you differentiate them they still come back as sinewaves with a phase shift. But you could use tuba waves or piano waves if you like. I believe spherical waves/harmonics are particularly important in understanding electron orbits. Fourier transforms are a powerful means of solving differential equations and nowhere will you find more analysis using Fourier than in Electrical Engineering. Fourier analysis can take any form and deconstruct it into a set of harmonic waveforms and its converse - Fourier synthesis can take a set of waves and use wave superposition which is just wave addition to add them. You just add their amplitudes together. Waves are linear in this way and are therefore a popular means of analysis. Before discussing waves in any sort of detail an understanding of the level of mathematics someone is comfortable with is necessary. Fourier Series is easiest involving the average value (a0) which is 1/T int from 0 to T of y(t)dt + an = 2/T int from 0 to T of f(t) cos(nwt)dt (the cosine waves) + bn = 2/T int from 0 to T of f(t) sin(nwt)dt where f(t) is your particular function and I am using int to mean the calculus integral of.

 

If you have a square wave for instance you have y(t) = 2 on the interval [0,1) and y(t) = 0 on the interval (1,2] there is obviously a discontinuous jump at x = 1 from 2 to 0. Without going into the details of the integrations of this wave, the square wave function can be represented by the Fourier Series 1 + 4/pi(cos(pit) - 1/3 cos(3pit) + 1/5 cos(pit)..... the more terms you add to the series the more this combination of cosine waves looks like a square wave when added together.

 

All the above may sound like a lot of work but thats life in the big city when you need to represent a discontinuous function by waves - for instance when you have an input of square waves from a signal generator and you have an electrical network function and the product of the input wave x network function = the output function things like Fourier analysis become necessary. And there are tables of waveforms you can select from and get a Fourier Series representation from/the Fourier Series equation at the same time and there are standard computer algorithms for calculating any such Fourier Series out as far as you like along with the resulting waveform plot. Associated with any Fourier Series comes the truncation error but for a series that converges rapidly the error is small with only a few terms. The minimum requirements for representing a complicated waveform including one that is discontinuous like a square wave or sawtooth wave is the Direchlet Conditions where the waveform has to have a finite number of discontinuities, have a finite number of maxima and minima and be absolutely convergent. (calculus subjects)

 

One of the most fastinating things about Fourier Series is that they are analogous to vector analysis to some extent. When you want to construct a Fourier Series for a function you need to determine the coefficients or amplitudes of the harmonics. You could consider these amplitudes the vector components in each direction. But the really nice thing about using waves is that the frequencies of the waves are equivalent to the dimensions in a vector space. While you cannot visualize more than 3 dimensions you can visualize any number of frequencies....... But see, the thing is, you could construct a 'mathematical universe' where each dimension/frequency is some aspect of that 'universe' like say gravity or the electric force (or quantum-weird entity) as represented by vector component x with dot product coefficient A = a Fourier harmonic frequency x and amplitude A and adding up the vector components gives a final 'entity' consisting of the components and in the same way the harmonic Fourier frequency 'components' represent the directions while the Fourier coefficients represent the magnitude of the vector components in each direction!! The book 'Who Is Fourier?' points this out comparing orthogonality in vectors with orthogonality in Fourier series in terms of Euler's Formula and they are equivalent. (just construct the Maclaurin Series for each). There they use the analogy of vegetable juice to illustrate the same thing where the amount of carrot is the vector magnitude and carrot would be in the x direction for example and celery say in the y direction with B magnitude and there being a resultant vegtable juice when they are added. But in Fourier terms it would be carrot is one frequency with its coefficient and celery another frequency with its coefficient representing vegtable juice in terms of waves instead of vectors.

 

Another very interesting thing about Fourier is you can extend it to include waves of infinite periods using the Fast Fourier Transform. You can take a snapshot of a wave of some length and infer/predict what the rest of the wave will look like based on that snapshot where the longer the snapshot wave - the more closely it will resemble the original wave. And this is where the Heisenberg Uncertainty Principle comes in - from the uncertainty found in waves. Same principle.

[Quartile]

Thanks for your post Mike!

 

Although I would never trust myself to produce any of the math underlying some of the things you have addressed I do feel that I have a working understanding of many of the concepts.

 

I find it interesting that a truly square wave cannot be produced using a finite number of Fourier transforms. It would seem that in this way there is an "asymptote" governing absolute vertical motion in any wave. I use the word asymptote in quotes to point out that I understand this is an incorrect usage of the term, but it does serve to illustrate the point. There is also asymptotic behavior governing absolute horizontal motion in any wave; "flat waves" do not exist.

 

I found this after typing the above. Can anyone tell me if its true?

 

The reason a square-wave can't actually be generated is fairly simple. The amount of energy required to create a transition in the wave is related to the derivative of that change. So when a square wave goes from low to high or vice versa it has an infinite derivative and, thus, requires infinite energy.

 

If this is the reason for no vertical waves, then a horizontal wave represents one that carries zero energy, which is, by definition, no longer a wave.

[Mike Dubbeld]

The above is true except consider that you can add enough waves to make it as square as you like in appearance - any number of decimal places. Something called Gibbs phenomena happens at the junction of the vertical and horizontal intersections where there is a spiked over shoot that cannot be avoided but still you can make it as small as you like. Since there is no such thing as a perfect anything in the universe to begin with and the most accurate thing measured in science is about 10^-13 meters who really cares? There is no such thing as a straight line or perfect circle or any other geometric shape (in the universe) because you can always go smaller and smaller to find deviations. Everything there is - is what it is to some number of decimal places for things in the universe. I used to know where on the web you could plug in numbers and create waveforms and there is one for spherical waves also.

[Quartile]

Could zero-point energy provide what is necessary to produce a vertical wave at the Planck scale?