Linear System Properties?

Randy Yates wrote:


> I gave you the wrong reason before about why sine waves were special
> in this sense. A linear system can't change the shape of a sine wave
> (other than its amplitude or phase, that is) because a sine wave only
> consists of one frequency. In any other waveform, multiple frequencies
> are involved, and the shape of the waveform is determined by not only the
> frequencies but the phases and amplitudes as well. Thus a linear system
> may change the shape of such a waveform since it may change the phases
> and amplitudes of each frequency component.

Well, it is a single frequency because sine and cosine are the
basis functions of the Fourier transform.

As derivative and integration are also linear operators, and
when applied to sin() and cos() the results are still sin()
and cos() of the same frequency, they are appropriate basis
functions for the system.

-- glen

Bob Cain wrote:
> Jerry Avins wrote:
> 
>> Till wants to know what unique property of sine waves makes them come
>> through a linear circuit unaltered (presumably in shape) while square
>> waves do not. However sharp your truths, I don't think your answer to
>> him was transparent at his level of understanding.

> I think it because they are Eigenvalues of a linear, time invariant 
> system even though I'm not sure what that means.  :-)

The modes of a vibrating string of uniform density, air column
of uniform cross section, or transmission line of uniform impedance,
for example.  There are other systems with other differential
equations and corresponding eigenfunctions.

As far as I know it, the origin of the Fourier series came
from the solutions to the differential equation for the above
systems.   There were two different ways to solve it, which
came up with different answers.   Consider a string, tube,
transmission line of length L.   One method would allow
any periodic function with period 2L.  The other method gave
an infinite sum of sines with periods integer fractions of 2L.

The differential equation in those cases is Y''+A**2 Y=0,
A is the eigenvalue,  sin(Ax) and cos(Ax) are the eigenfunctions.
Add the boundary conditions that the string is fixed at x=0
and X=L, only sin() terms are allowed, and A=n pi/L.
That is how to make a violin, guitar, piano, flute, or
many other musical instruments.

Now, consider the vibrational modes of an oboe, with a
cone shaped air column.  (The vibrating reed is at the tip.)

If you use spherical coordinates and consider the radial
modes only, you get the n=0 spherical Bessel's equation,
x**2 R'' + 2x R' + x**2 R = 0

http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html
has a good explanation of the derivation, and
http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html
the solutions.  Consider an oboe of length L.

The first solution, j0(x)=sin(x)/x, and the modes again
have frequencies that are integer multiples of the lowest mode.

One more, to show that not all equations have harmonic solutions,
consider the radial (circularly symmetric, n=0) modes of a drum,
which should result if you hit it in the center.

x**2 Y'' + x Y' + x**2 Y = 0

The solutions are called (cylindrical) Bessel functions,
J0 for the radial modes.   The second and third modes
are approximately 2.295 and 3.398 times the fundamental
http://www.gmi.edu/~drussell/Demos/MembraneCircle/Circle.html

-- glen

"glen herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
news:YAaac.133061$_w.1589940@attbi_s53...

>
> The modes of a vibrating string of uniform density, air column
> of uniform cross section, or transmission line of uniform impedance,
> for example.  There are other systems with other differential
> equations and corresponding eigenfunctions.
>
> As far as I know it, the origin of the Fourier series came
> from the solutions to the differential equation for the above
> systems.   There were two different ways to solve it, which
> came up with different answers.   Consider a string, tube,
> transmission line of length L.   One method would allow
> any periodic function with period 2L.  The other method gave
> an infinite sum of sines with periods integer fractions of 2L.

Hello Glen and others,

The following link gives some details about how Fourier came up with his
series.

http://www.math.carleton.ca/~amingare/calculus/fourier-series.pdf

Clay

Clay S. Turner wrote:

(snip)

> The following link gives some details about how Fourier 
 > came up with his series.

> http://www.math.carleton.ca/~amingare/calculus/fourier-series.pdf

Somehow I always thought it was the wave equation instead of
the diffusion equation.  The X part is the same in both cases.

Most likely not much later it would have been used for the
wave equation.  Vibrating strings were also in interesting
problem around that time.

For the wave equation it is easy to show the general solution
of the form Af(x-vt)+Bg(x+vt) for arbitrary functions f and g.

I still remember a physics quiz where we had to show that was
the solution to the wave equation.  I had never done partial
derivatives before but had one hour to solve this problem.
I had somehow heard that partial derivatives are like ordinary
ones with everything else constant, and went on trying to do
the problem.  After a little while, the right answer came out!

So now I never forget that problem.

-- glen