I Vibrations of waves with pinned vs free boundary conditions

[member 428835]

Hi PF!

Can someone explain to me why in math/physics the frequencies associated with waves (or say drum heads) tend to be larger when the boundaries are pinned as opposed to free? If possible, do you know any published literature on this?

Thanks!

 

[Dale]

Consider a resonant column of air (like a flute). If you have an unpinned boundary it turns into an antinode and a pinned boundary turns into a node when you are thinking of the fundamental frequency. So with two pinned boundaries you have a wavelength equal to twice your tube length and with only one pinned boundary you have a wavelength equal to four times your tube length. The shorter wavelength corresponds to a higher frequency.

 

[sophiecentaur]

I don't think this is all consistent. An unpinned boundary at each end will have a half wave resonance - same as with a pinned boundary at each end. There are practical difficulties achieving this, though. Open ends make it difficult to mount things like vibrating strings and diaphragms.

 

[Dale]

I was thinking pinned at one end and then the other end either pinned or unpinned. With most musical instruments it is hard to make both ends unpinned.

 

[member 428835]

Thank you both! Dale, is this in the literature anywhere formally?

 

[etotheipi]

You can read about the boundary conditions here, look at section 5.6: https://www2.physics.ox.ac.uk/sites/default/files/2012-09-04/waves2_pdf_16294.pdf. It talks about waves on a string rather than sound waves, but it is similar in concept.

 

[sophiecentaur]

I was thinking pinned at one end and then the other end either pinned or unpinned. With most musical instruments it is hard to make both ends unpinned.

Blowing on (across) the end of an open tube would qualify as that which (I suddenly remembered) applies to a flute and others. So actually all combinations are possible with common musical instruments.

I do remember that the lowest rank of organ pipes are closed end, which gives them an octave of range below the rest, for a given length. Quantity rather than sound quality perhaps.

That link again: But nothing is simple because (and again I heard this years ago and had forgotten) an oboe has a double reed and a nominally closed end, yet it has the same range as a flute, which is definitely open ended. The link seems to fudge that issue a bit but I guess it has to do with the difference between the balanced and unbalanced excitation of double and single reeds. Actually it cannot be simple (as if!) because a clarinet has a much more bass sounding note than an oboe, which is not what the basic statement about musical range would suggest. Oboe is very 'harmonicky', compared with the deep reedyness of a clarinet - but that's a whole new layer of complexity.

 

[Dale]

@sophiecentaur well, your symphonic knowledge certainly exceeds mine!

 

[hutchphd]

Actually it cannot be simple (as if!) because a clarinet has a much more bass sounding note than an oboe, which is not what the basic statement about musical range would suggest. Oboe is very 'harmonicky', compared with the deep reedyness of a clarinet

I believe the bore of a clarinet is cylindrical whereas the oboe is conical; this is also true for the trumpet and the flugelhorn. In each case the length is the same.

 

[sophiecentaur]

@sophiecentaur well, your symphonic knowledge certainly exceeds mine!

I got to Grade 2 on Violin. The rest of my life has been played by ear!

 

[hutchphd]

This is sort of fun. Apparently each horn is more "conical" trumpet<cornet< flugelhorn< mellophone. Live and learn: