I do not think so; not in a 3D box under usual circumstances at least. 

Hi

You can calculate box modes here.

http://www.mh-audio.nl/standingwaveinbox/calculator.asp

The equation applies for a rigid walled cavity.

Here is a link to a page with some equations on the subject.

http://www.silcom.com/~aludwig/Physics/Main/Cavity_modes.html

To answer your question, you can have 1-d standing waves as is the case in tubes. The shortest dimension of a standing wave with sinusoidal shape that you can fit is a quarter wavelength which you have when one side is open.

That said, you can have modes at still lower frequencies when you have small openings to the cavity. These are mass-spring type modes where the cavity acts as the spring and the mass in the opening as the mass. This is the tone you hear when blowing across the opening to a bottle. On top of this, you have the possibility to couple mechanical mass to your cavity, e.g. by having a wall that moves.

Sincerely

Claes

@ Claes,

You are giving the analogy of Helmholtz resonator and resonant modes. I am not sure if I misunderstood the question, but has the box an opening for Helmholtz resonator to come into the picture? Also, resonant modes die out very soon, while standing waves don't.

Hi Vikash

A standing wave is the wave pattern which forms when forced at resonance. This same pattern stays once excitation is turned off and then is called a natural mode or eigenmode as it is the 'natural' shape the system likes when left on it self. That said, a standing wave pattern usually needs to bounce around at least 3-4 roundtrips for the pattern to form.

There was no mention about boundary conditions or whether the cavity is closed, coupled or not, hence, my answer.

/Claes

Thank you 

Assume a point source of acostic wave is in the cavity which has a wavelength equal to D and P in other word wavelength=D=P>2×L, and two dimension can resonant. In this circumstances, does cavity have standing waves?

Hi Abbas

Hard walled conditions & perfectly closed? If so, you have your answer for the standing waves here

http://www.silcom.com/~aludwig/Physics/Main/Cavity_modes.html

/Claes

Hi Abbas

Let's see if the wavelength Lambda = D = P and D > 2*L and P > 2*L and you have hard walled conditions in a closed cavity - the first acoustic mode you can squeeze in then is Lambda = 2*D or Lambda = 2*P, depending on which is the larger. 

The pressure will be uniform in the L direction, i.e. in this direction you will have a plane wave. Differently put, your wave pattern will be 2-D.

I find that there is an unfortunate mix of concepts in the discussion that migh cause headaches.

Please note that resonance is forced, imply energy accumulation (response buildup) and that forcing can be made at any frequency. Standing waves and modes are nearly the same, the former implies exitation while the latter is what is found when excitation is turned off. (Yes, we do use modal summation and this can never capture a standing wave is it always have the last transient wave riding on top of the accumulated wave pattern. This contribution becomes small after a while and hence, the approximation good)

The crux of the matter is that assume that you force excitation way below of first response, f1, say at f = f1/10. You will still input energy, albeit with low response. However, the second you turn this excitation off, the energy that is stored in your systems will shift frequency to that of your natural modes.

You can easliy give this a try using a plastic ruler which you move slowly up down with your finger or simply press it down to store some energy. The second you release it, it will start to oscillate at its natural frequencies.

/Claes

Hi! Claes,

Thanks for the feedback. I have some reservations though and will appreciate if you can clarify it.

1) You say " D > 2*L and P > 2*L and you have hard walled conditions in a closed cavity - the first acoustic mode you can squeeze in then is Lambda = 2*D or Lambda = 2*P"

What do you mean by squeeze in? Without changing the dimensions of the wave, what happens to the extra wavelength of the wave not accommodated inside the box?

Won't the extra leg of the wave when rebounced by the walls of the box create a new propagating wave which would interfere with the former wave? How can standing wave form in this case?

2) You say "forcing can be made at any frequency". Let there be a 1D bar of length 3m. And wavelength of wave be 1.25m. Can you please draw the standing wave pattern for this case?

Thanks in advance.

Hi Vikash

Sloppy phrase, the first standing wave pattern that can interact and form between two hard walls is half a wavelength.

I suggest that you look, e.g. here

http://www.silcom.com/~aludwig/Physics/Main/Cavity_modes.html#Reflecting_plane_wave_model

http://www.acs.psu.edu/drussell/Demos/waveguide/waveguides.html

There is a phenomenon called cut-on that applies for wave propagation in tubes

http://www.win.tue.nl/analysis/reports/rana04-35.pdf

I believe that it affects the above in the formation of the standing wave pattern.

Also, there is a lowest frequency for which wave motion can exist.

For forcing at any frequency - first tell me your conclusions with the plastic ruler experiment. The physics is the same as for the acoustic case.

Hope this helps

Claes

Hi! Claes,

Sadly, I disagree with you. I could not find anything relevant in those links posted by you. As I understand, for standing waves to form, certain geometric constraints must be fulfilled. If they don't, I simple cannot imagine the formation of a standing wave.

Example of rules: The applied force must be Fourier decomposed to see the various frequency components. And it is pretty obvious some of those frequencies will fulfill the geometric constraints imposed by the length of the ruler and so, standing waves would form for a short while, and then destroyed by waves of other frequencies.

@ All,

It seems to be a mix of standing wave in one direction and propagating wave in another direction, for cases when L = n lambda / 2 condition is not met. Confirmed from matlab simulation (see attached file). Thanks to my friend Anders for making this code and internet ofcourse.

Thank you dear Vikash for the MATLAB code.

yes, it is possible. We developed polymeric-based chips with a channel smaller than half a wavelength to perform particle collection at different planar pressures nodes which can be established inside the channel at different positions and parallel to the sidewalls.

these are two of our paper published:

- A polymeric chip for micromanipulation and particle sorting by ultrasounds

based on a multilayer configuration, I. Gonzalez et al., Sensors and Actuators B 144 (2010) 310–317

- Optimizing Polymer Lab-on-Chip Platforms for Ultrasonic Manipulation: Influence of the Substrate, I. Gonzalez et al., Micromachines 2015, 6, 574-591; doi:10.3390/mi6050574

Thank you Mrs.Gonzalez 

You are right.Also the standing wave in a tube of air can be created while the tube diameter can be smaller than half of wavelength.

You can create a standing wave in a cavity if any dimension, including corner-to-corner, is a half wavelength or greater. The modes can be x, y, z, and/or oblique.