standing waves

[RichD]

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?


--
Rich

 

[whit3rd]

On Sunday, February 8, 2015 at 12:40:20 PM UTC-8, RichD wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?

Waves (as solutions to a wave equation) are standing waves when
they have a recurring state (i.e. when they always return to a
prior configuration). It's easy to do this with a box made of
mirrors, and the mirrors are nodes in a sense, but it can
also be done with other propogation-of-waves conditions.
Standing-wave solutions are the resonances of bell, for instance.

A bell resonance is a three-D standing wave. The physical bell
has some thermalization (the sound dies away), so it's not a
perfect standing wave solution.

 

[Phil Allison]

RichD wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?

** Standing waves on a drum skin are interesting:

http://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane

And so is the math....


..... Phil

 

[Jasen Betts]

On 2015-02-08, RichD <r_delaney2001@yahoo.com> wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?

a wave that fills a space with no energy being transferred by it

I would include ring lasers and surface waves on a droplet

--
umop apisdn

 

[Phil Hobbs]

On 2/8/2015 3:40 PM, RichD wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?

One whose shape is time invariant, i.e. it just gets bigger and smaller
by a uniform time-varying factor, generally cos(omega t + phi). (Folks
might argue that that's too restrictive, and that linear combinations of
such waves should be included, e.g. the wave you get by plucking a
clothesline.)

Does to have to include nodes, where the amplitude is
always zero?

Depends on the boundary conditions. For instance, you could concoct a
boundary condition that acted like a fixed point someplace outside the
domain.

And fixed end points?
That's the special case of Dirichlet boundary conditions. Neumann
conditions make the derivative at the ends constant, and there are also
mixed boundary conditions.

Can it be 2-D, or 3-D?

A higher order transverse mode of a closed-end organ pipe is a 3-D
standing wave. So are many of the modes in a water balloon, or a
"breathing mode" in acoustics.

Cheers

Phil Hobbs


--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

 

[John Larkin]

On Mon, 09 Feb 2015 15:48:53 -0500, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:

On 2/8/2015 3:40 PM, RichD wrote:
We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?

One whose shape is time invariant, i.e. it just gets bigger and smaller
by a uniform time-varying factor, generally cos(omega t + phi). (Folks
might argue that that's too restrictive, and that linear combinations of
such waves should be included, e.g. the wave you get by plucking a
clothesline.)

Does to have to include nodes, where the amplitude is
always zero?

Depends on the boundary conditions. For instance, you could concoct a
boundary condition that acted like a fixed point someplace outside the
domain.

And fixed end points?
That's the special case of Dirichlet boundary conditions. Neumann
conditions make the derivative at the ends constant, and there are also
mixed boundary conditions.

Can it be 2-D, or 3-D?

A higher order transverse mode of a closed-end organ pipe is a 3-D
standing wave. So are many of the modes in a water balloon,

or a liquid tin droplet.


--

John Larkin Highland Technology, Inc
picosecond timing laser drivers and controllers

jlarkin att highlandtechnology dott com
http://www.highlandtechnology.com

 

[szczepan bialek]

"RichD" <r_delaney2001@yahoo.com> napisal w wiadomosci
news:56b7a642-7be9-4040-ac05-68f07db1dfbe@googlegroups.com...

We know what standing waves look like, on a string, or
EM waves in a cavity.

Like this?
http://resource.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-standing-nature.htm

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?

Amplitude of what?
The particles of medium always are moving (but the directions are
different). So the amplitudes never are zero.
S*

 

[RichD]

On February 8, whit3rd wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.
But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?

Waves (as solutions to a wave equation) are standing waves when
they have a recurring state (i.e. when they always return to a
prior configuration).

But that would include any oscillator, which doesn't
sound right.

Standing-wave solutions are the resonances of bell, for instance.
A bell resonance is a three-D standing wave.

A bell is a 2-D surface, curved into a 3rd dimension,
which leaves it still 2-D. Though of course it can
support standing waves.

But still, it's an example, a picture. Given a
ringing bell, what's the mathematical expression
which defines 'standing wave'?

--
Rich

 

[RichD]

On February 9, Phil Hobbs wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.
But what's the mathematical definition of a standing wave?

One whose shape is time invariant, i.e. it just gets bigger and smaller
by a uniform time-varying factor, generally cos(omega t + phi).

hmmmm...

You mean, any function transferred through
a time varying linear amp, is a standing wave?

Does to have to include nodes, where the amplitude is
always zero?

Depends on the boundary conditions. For instance, you could concoct a
boundary condition that acted like a fixed point someplace outside the
domain.

But then the domain could be extended to the
boundary conditions, and you're back to square one -

And fixed end points?
That's the special case of Dirichlet boundary conditions. Neumann
conditions make the derivative at the ends constant, and there are also
mixed boundary conditions.

ok
So, if you specify the boundary conditions by its
derivatives, does that change anything essential
regarding the nature of a standing wave?

Can it be 2-D, or 3-D?

A higher order transverse mode of a closed-end organ pipe is a 3-D
standing wave.

--
Rich

 

[Phil Hobbs]

On 2/9/2015 11:11 PM, John Larkin wrote:

On Mon, 09 Feb 2015 15:48:53 -0500, Phil Hobbs
pcdhSpamMeSenseless@electrooptical.net> wrote:

On 2/8/2015 3:40 PM, RichD wrote:
We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?

One whose shape is time invariant, i.e. it just gets bigger and smaller
by a uniform time-varying factor, generally cos(omega t + phi). (Folks
might argue that that's too restrictive, and that linear combinations of
such waves should be included, e.g. the wave you get by plucking a
clothesline.)

Does to have to include nodes, where the amplitude is
always zero?

Depends on the boundary conditions. For instance, you could concoct a
boundary condition that acted like a fixed point someplace outside the
domain.

And fixed end points?
That's the special case of Dirichlet boundary conditions. Neumann
conditions make the derivative at the ends constant, and there are also
mixed boundary conditions.

Can it be 2-D, or 3-D?

A higher order transverse mode of a closed-end organ pipe is a 3-D
standing wave. So are many of the modes in a water balloon,

or a liquid tin droplet.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

 

[Phil Hobbs]

On 2/11/2015 4:43 PM, RichD wrote:

On February 9, Phil Hobbs wrote:
We know what standing waves look like, on a string, or
EM waves in a cavity.
But what's the mathematical definition of a standing wave?

One whose shape is time invariant, i.e. it just gets bigger and smaller
by a uniform time-varying factor, generally cos(omega t + phi).

hmmmm...

You mean, any function transferred through
a time varying linear amp, is a standing wave?

No, that isn't what I said. The output of an amp doesn't have a wave
shape, because it's zero-dimensional. If you put it into a long
transmission line, it will be a travelling wave--the peaks won't have
the same position at all times, which is required for the wave to meet
the definition above.

Does to have to include nodes, where the amplitude is
always zero?

Depends on the boundary conditions. For instance, you could concoct a
boundary condition that acted like a fixed point someplace outside the
domain.

But then the domain could be extended to the
boundary conditions, and you're back to square one -

So what? That was just an example. The physically oscillating thing
doesn't have to have stationary points.

And fixed end points?
That's the special case of Dirichlet boundary conditions. Neumann
conditions make the derivative at the ends constant, and there are also
mixed boundary conditions.

ok
So, if you specify the boundary conditions by its
derivatives, does that change anything essential
regarding the nature of a standing wave?

Well, if you think the shape of the wave is important, yes it does.

Cheers

Phil Hobbs


--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

 

[George Herold]

On Sunday, February 8, 2015 at 3:40:20 PM UTC-5, RichD wrote:

We know what standing waves look like, on a string, or
EM waves in a cavity.

But what's the mathematical definition of a standing wave?
Does to have to include nodes, where the amplitude is
always zero? And fixed end points? Can it be 2-D, or 3-D?


--
Rich

Maybe this is too simplistic, but in one dimension a standing wave
is really two waves of the same frequency going in opposite directions.
It 2 dimensions it gets more complicated, but a similar idea.

George H.