Question about standing waves

[PsxDcSquall]

---

Members don't see this ad.

I am working through some of the problems in the Kaplan High Yield problem solving guide and I'm having trouble wrapping my head around part 1 of the standing waves problem on page 46.

It states that the general formula for a pipe open at one end is λn=4L/n where n=1, 3, 5 thus λ5=(3/5)λ3...

It then goes onto state that for a pipe open at both ends that the general formula is λn=2L/n and that λ3=(2/3)λ2

I am having a hard time seeing this relationship between the second and third harmonic the standing waves in a pipe. As a result I am unable to derive a relationship that would relate the wave lengths of the first and second harmonic, the 3rd and forth harmonic and so on.

If any one could help explain this relationship to me I would appreciate it!

 

[baller99]

kaplan goes way to in-depth on this i think, but i hope this answers your question:

λn=2L/n , right? so the first wavelength will be the 2L. the next (second) harmonic wavelength will be L.this is because if the L wavelength repeats twice, then it will have the same nodes as the 2L wavelength (the definition of a harmonic). so the third harmonic will have 2L/3. if that repeats three times, then that will be the same length as the 2L first wavelength.

maybe there's someone else out there who can explain it a little better...

 

[Dibs]

λ1= first harmonic

then λ2=1/2 λ1,λ3=1/3 λ1 ..etc. for open or closed at both ends

also, λ2=1/3 λ1,λ3=1/5 λ1 etc. for closed at one end

 

[AZFutureDoc]

Try drawing out a few pipes for each case. That is what really shows you the big picture.

 

[tmntdonjuan]

I am working through some of the problems in the Kaplan High Yield problem solving guide and I'm having trouble wrapping my head around part 1 of the standing waves problem on page 46.

It states that the general formula for a pipe open at one end is λn=4L/n where n=1, 3, 5 thus λ5=(3/5)λ3...

It then goes onto state that for a pipe open at both ends that the general formula is λn=2L/n and that λ3=(2/3)λ2

I am having a hard time seeing this relationship between the second and third harmonic the standing waves in a pipe. As a result I am unable to derive a relationship that would relate the wave lengths of the first and second harmonic, the 3rd and forth harmonic and so on.

If any one could help explain this relationship to me I would appreciate it!

You know what a standing wave looks like right? The crest and trough of the wave are called nodes while the neutral point is called an anti-node. Realize that for pipes, waves always associate a node with the open end and an anti-node with the closed end. Therefore if you're working with the fundamental frequency of a wave on a pipe with one end open and one end closed, then on the closed end you have an anti-node and on the open end on the opposite side you have a node. Therefore the length of the pipe is equivalent to 1/4 of a full wavelength (think of a sin wave and remember a full wavelength goes from [1]anti-node to node; [2]node to anti-node; [3]anti-node to node (trough); [4]node (trough) to anti-node). Going from anti-node to node only covers 1/4 of a full wavelength thus giving the equation for λ=4L. the "n" that's included is for the frequency level. The reason why it's only in odd numbers is because of the relationship with node and anti-node. If "n" is an even number, then the equation basically becomes λ=2L which means the length of the pipe is only 1/2 of a full wavelength. Remember the shape of a wavelength though? If a pipe is 1/2 of a full wavelength, then both ends would either be nodes or anti-nodes (draw out a wavelength if this gets confusing). This can't work though since only one end of the pipe is open. The closed end must have an anti-node and the open end must have a node. Hope this helps. Make sure you're familiar with how the relationship works and memorizing the formula becomes easier.

 

[futuredoctor10]

Quick question - can someone check/verify this?

Standing Waves (OR) Open Pipe at Both Ends
λ=2L/n where n = 1, 2, 3.....

Closed Pipe at One End
λ=4L/n where n = 1, 3, 5.....

As you go from n=1 to n=2 and so on (increasing in harmonic #):
frequency increases / wavelength decreases / period decreases

Thanks 🙂

Also for pictures just be able to recognize these 3 types and/or draw them if it came up and you had to identify... right?