Mathematical insight about waves

In summary: Sine and cosine functions can be thought of as the amplitude and phase of the waveform, respectively.
  • #1
Mappe
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I want to have a simple and intuitive explanation of why the sin and cos waves have such a simple repetitive values for their derivatives at a specific point. Their derivative values are also periodic in respect to the derivative order. For example, e^-x is also periodic, but its derivatives are never zero. Is there a good explanation of this without involving complex numbers? Or if not, is there an easy answer to why complex numbers adds their angles when multiplied? Because that kind of answers the same question. The simpler the proof the better!
 
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  • #2
sine and cosine have repeated values and repeated derivatives because you are going around the unit circle as the arguments of each increase.

A circle is symmetric and repeats.
 
  • #3
Mappe said:
For example, e^-x is also periodic, but its derivatives are never zero.

##e^{-x}## is not periodic (unless x is imaginary, in which case you have a real periodic function and an imaginary periodic function).

As Dr. Courtney said, check out angles of inscribed triangles within the unit circle. You keep going around and around. ##1## revolution for every ##2\pi## radians.
 
  • #4
Dr Courtney, I understand but your statement implies that we know that e-i2π = 1, and from the definition of eix by its taylor expansion, how can we see on its derivatives that its going to be periodic and perhaps also how do we see from this definition that it describes a circle? The same question phrased differently can be "is there a simple proof that doesn't use knowledge about sin and cos, that shows us that complex numbers add their angles when multiplied, and multiplies their norm"?
 
  • #5
Mappe said:
Dr Courtney, I understand but your statement implies that we know that e-i2π = 1, and from the definition of eix by its taylor expansion, how can we see on its derivatives that its going to be periodic and perhaps also how do we see from this definition that it describes a circle? The same question phrased differently can be "is there a simple proof that doesn't use knowledge about sin and cos, that shows us that complex numbers add their angles when multiplied, and multiplies their norm"?

If you accept the result ##e^{i\theta}= cos(\theta)+(isin\theta) ## , then this is automatic. And if you take the polar representations ( in a region where valid) of two complex numbers ##z_1=e^{i\theta_1}, z_2 = e^{i \theta_2 } ## , then ##z_1 z_2=e^{i(\theta_1+ \theta_2)} ## , which takes ##z_1=e^{i \theta_1}=cos\theta_1+i sin \theta_2## to ## z_1z_2= cos(\theta_1+ \theta_2)+ isin(\theta_1+ \theta_2) ## , which is a rotation by an angle of ##\theta_2 ##

But it ultimately depends on your starting point/assumptions.
 
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Likes Dr. Courtney
  • #6
You should read Visual Complex Analysis. This stuff might be in the free preview here:

http://usf.usfca.edu/vca//

He just illustrates it by example. Multiplying by i is rotating by 90 degrees, so you can draw a picture of it with that it mind for some example like (4+3i)(1+i), thinking of the complex numbers as vectors. Here, you would get 4(1+i), which which is 1+i scaled by a factor of 4 and then you get 3i(1+i), which is the 1+i rotated by 90 degrees and scaled by a factor of 3, and then you add those vectors together. So you you end up with a triangle similar to the triangle formed by 4+3i, with its real imaginary components being the other sides of the triangle that gets put on top of the one representing 1+i. When you stack the two triangles, you can see that the angles are added, and by similar triangles, the modulus gets multiplied.

It's a little cumbersome to describe verbally without a picture, so you really have to draw it and work it out for yourself.

Also, for the sine and cosine derivatives, you ought to think of taking the velocity vector of a particle that is moving around a unit circle. Some ideas along these lines are presented in Visual Complex Analysis, as well.
 
  • #7
Oh, and incidentally, if you want to understand exactly why the sine and cosine functions look the way that they do, just get take a look at some Cavatappi noodles.

http://www.fabios.co.za/15-short-cut-extruded-pasta/trafilata-861-cavatappi/

A Cavatappi noodle can be interpreted as a 3-d plot (x,y,t) of particle moving around in a circle in the x-y plane, with time being the other coordinate. Viewed from the top, they'd look circular, but viewed from the side, they are sine waves (ignoring their thickness).
 

FAQ: Mathematical insight about waves

1. What is the difference between a transverse wave and a longitudinal wave?

A transverse wave is a type of wave where the particles of the medium move perpendicular to the direction of the wave. Examples include light and electromagnetic waves. On the other hand, a longitudinal wave is a type of wave where the particles of the medium move parallel to the direction of the wave. Sound waves are an example of longitudinal waves.

2. How is the wavelength of a wave calculated?

The wavelength of a wave is calculated by dividing the wave's speed by its frequency. In mathematical terms, it can be represented as: λ = v/f, where λ is the wavelength, v is the wave's speed, and f is the frequency.

3. What is the relationship between wave frequency and energy?

The frequency of a wave is directly proportional to its energy. This means that as the frequency increases, the energy of the wave also increases. This is because waves with higher frequencies have shorter wavelengths, which means more energy is packed into a smaller distance.

4. Can waves interfere with each other?

Yes, waves can interfere with each other. When two or more waves meet, their amplitudes can add together or cancel each other out, depending on their relative positions. This is known as wave interference and can result in constructive interference (when the amplitudes add together) or destructive interference (when the amplitudes cancel each other out).

5. How is the speed of a wave affected by the medium it travels through?

The speed of a wave is affected by the properties of the medium it travels through. For example, sound waves travel faster in denser mediums, while light waves travel slower in denser mediums. The speed of a wave is also affected by the temperature and pressure of the medium it travels through.

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