End correction of the pipe mouth(standing wave)

[EHT]

Consider a pipe closed at one end and open at the other.a simple model tells that resonance will occurs when the sound wavelength of the resonator is 1/4,3/4,5/4.. of pipe length,by assumption that antinode occurs at the open end.But it comes out that the air at open end isn't completely "free"(the pipe wall makes it can't expand freely) so it's not a perfect antinode .We have to apply an end correction, the pipe appears to be acoustically somewhat longer than its physical length.I'm searching for a model that can be use to find this end correction and I've trouble doing the experiment because of the non uniform ity of speed of sound due to viscosity.Do anyone knows the model to compute this correction?

sorry,my english is bad

 

[Dadface]

If my memory serves me correctly the end correction is equal to 0.6 times the pipes radius.Try googling waves in pipes.Good luck.

 

[Stonebridge]

Yes Dadface was correct. The end correction, c, is 0.6r where r is the pipe radius.

Note:
In your post you say
"when the sound wavelength of the resonator is 1/4,3/4,5/4.. of pipe length,"

It is the pipe length (plus end correction) that is 1/4, 3/4 etc of the sound wavelength.

 

[EHT]

@stonebridge:yeah there I'm explaining standing wave without correction in basic physics textbook so I put the wrong one.I've read about that 0.6 r rayleigh's correction,but what i want to know is where does it come from.how do we calculate that 0.6r?

thanks

 

[Stonebridge]

I've never seen an analytical derivation of this and have always assumed it to be an experimentally determined value. I checked in some of my books and it seems the most accurate value is 0.58r (It is independent of the wavelength but does depend on the shape of the tube. Value quoted is for circular cross section)
Normally in these experiments, you eliminate (and can calculate) the end correction by finding two consecutive resonating lengths L1 and L2 at the same frequency, for example the two cases where the L1+c is 1/4 wavelength and L2+c is 3/4 wavelength.
Subtracting gives L2 - L1 = half wavelength.

These experiments, of course, calculate the speed of sound in the tube. It must be remembered that, this is not the same as the speed in free air.