Clarification on resonant wavelength formulas

[dcnetsfan77]

Hey guys,

I just had a quick question about the formulas for 2 fixed ends of a pipe and 1 fixed end of a pipe and how it relates to this AAMC problem below....

1st off I just want to make sure my formulas are correct:

2 fixed ends= wavelength= 2L/n
1 fixed end= wavelength =wavelength/n

what about = wavelength=4L/n

Im definitely mixing things up....

Here is the question: What is the resonant wavelength of a 1m long pipe OPEN at both ends is .1 diameter.

answer: pipes and tubes have their resonant wavelengths when standing waves develop. An open pipe has its fundamental resonant wavelength at twice the length of the rope. Both end have displaced antinodes with a node in the middle of the pipe. Thus the pipe is half a wavelength long. the resonant wavelength is independent of diameter.

answer: 2m

Is this a separate formula? or the same as for 2 fixed ends (which it seems like to me), although both ends are open in the problem which confuses me..maybe I dont understand what exactly a "resonant wavelength is"?

Thanks!

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[Dipraman]

Pipes open at both ends or closed at both ends have the same formula i.e. wavelength= 2L/n
Resonant wavelength just basically means fundamental frequency/1st harmonic (n=1)

So for your problem wavelength will be 2(1)/1= 2m.

Hope this helps!

 

[Vybe]

Also, if you want to understand the formulas a little bit more intuitively...

If both ends are either open or closed, it basically means that the wave has completed half its wavelength. That is why you calculate wavelength by multiplying L by 2. It takes 2 lengths of the pipe to measure the length of the wavelength, if that makes any sense. If you look at a diagram of a 2-open-ended wave in a pipe, you will see that the end and start of each amplitude has been visited on each end of the pipe (I dont know if I just made it more complicated). You will be able to see that it is half the wavelength

When one end is open or closed, it is only 1/4 of the wavelength in the length of the pipe. That is why, for 1 close and 1 open end in a pipe, the wavelength is calculated...

wavelength = 4L/n.

Hope this helps also.

If anyone sees a mistake in what I have said, please address it!

 

[dcnetsfan77]

thanks! I think i get it a little better....

so...
we have the formula for both ends open or closed
we have the formula for one end open or closed

but what is the formula wavelength=wavelength/n and the equivalent for frequency (f=fn) used for exactly?

 

[Vybe]

but what is the formula wavelength=wavelength/n and the equivalent for frequency (f=fn) used for exactly?

I'm not exactly sure where you're getting that from.

 

[dcnetsfan77]

Ok let me exactly type what the TPR physics book says about it....sheesh, if i were to do this again I would use EK

"The number n is called the harmonic number. The first harmonic is usually called the fundamental because once we know the fundamental λ1, we automatically know all the other harmonic wavelengths because we can write λn in terms of λ1, like this: λn= λ1/n"

next page....

"In the same way that the fundamental wavelength can be used to figure out all the other harmonic wavelengths, the fundamental frequency can be used to figure out all other harmonic frequencies : f=nf1.Regardless of the type of standing wave, the formula to find the appropriate harmonic from the fundamental frequency still holds (fn=nf1)"


...yea soooo......thought I knew what this was saying for the past month and now I dont? lol thanks again for the help

 

[Vybe]

"In the same way that the fundamental wavelength can be used to figure out all the other harmonic wavelengths, the fundamental frequency can be used to figure out all other harmonic frequencies : f=nf1.Regardless of the type of standing wave, the formula to find the appropriate harmonic from the fundamental frequency still holds (fn=nf1)"


...yea soooo......thought I knew what this was saying for the past month and now I dont? lol thanks again for the help

Alright.. Yeah I can't really comprehend that understanding. Just know that to find frequency... you need to have those variables plus the velocity.

Frequency of both open/close ended = nv / 2L
wavelength of both open/close ended = 2L/n
harmonic number [n]--> 1,2,3,4

Frequency of one open/close ended = nv / 4L
wavelength of one open/close ended = 4L/n
harmonic number [n] --> 1,3,5,7,9

Just have those in your arsenal. Have a deep understanding why each of those variables matter, and I think you should be good to go!