# Harmonics for standing wave pattern in pipes and strings

So I was reading about standing wave patterns in strings and pipes however I could not get an intuitive sense about how a harmonic was defined in each case. In the case of a string, the harmonics made some sense to me since for $n = 1$ there is only half a wavelength between the two fixed ends and so on but when it comes to pipes it does not seem to be the case.

So how exactly is it defined?

• Possible duplicate of Why are the closed and open ends of an organ pipe nodes and anti-nodes? – sammy gerbil Apr 25 '17 at 15:42
• It would help if you draw the pipe diagrams more accurately to scale, or find some properly drawn ones on the web. Your drawings are so far out that they may be confusing you. – alephzero Apr 25 '17 at 19:04
• @sammygerbil I have seen that question before it was helpful but my question is about how the harmonics are defined for example for $n=1, \lambda = 4L$ for one end closed pipe – rahul rj Apr 26 '17 at 1:17
• What do you mean "how they harmonics are defined"? In your diagrams they are correctly labelled. What is it that you do not understand? – sammy gerbil Apr 26 '17 at 10:13

The harmonics are usually numbered consecutively from 1 upwards. This is the case on strings, pipes (open or closed) or any other instrument.

The number of the harmonic is not necessarily the same as $n$ in your diagrams, which is the number of times the fundamental (lowest frequency mode $n=1$) fits into the pipe or string. For half-open pipes only odd multiples of $n$ are possible. Usually $n=3, 5, 7$ etc are called the 2nd, 3rd, 4th, etc harmonics, because they are the 2nd, 3rd, 4th etc highest frequencies which are possible.

Beware : the fundamental can be labelled $f_0$ or $f_1$.

Overtones are the same as harmonics but they are usually numbered with 1st overtone being the next mode higher in frequency than the fundamental, which is now labelled $f_0$.

It can be very confusing because the same meanings are not used everywhere. You have to read the question carefully and look for clues about whether the number given is the multiple $n$ of the fundamental $f_0$ or $f_1$, or the sequential order of frequencies, and whether the fundamental frequency is being labelled $0$ or $1$.

• What I meant to ask is why is it a case of only odd harmonics in the open end pipe and why is the $\lambda = 4L$ and $\lambda = 2L$ in case of pipe closed at both ends. I don't seem to get the reasoning behind assigning wavelength like that. – rahul rj Apr 27 '17 at 11:29
• Your diagrams show the patterns (modes) of the standing waves in pipes which are open at both ends and closed at one end. The relation between $L$ and $\lambda$ is found simply by counting the fraction/multiple of a wavelength which occupy the pipe, bearing in mind that closed ends must be nodes (N) and open ends must be anti-nodes (A). In the one end open case for the lowest mode you need 4 pipes to make one whole wavelength (NA-AN-NA-AN), so $\lambda=4L$. In the closed at both ends case for the lowest mode you need 2 pipes (NAN-NAN) so $\lambda=2L$. – sammy gerbil Apr 27 '17 at 11:50
• Only odd values of $n$ are possible in the half-open pipe because the pipe is asymmetrical. You cannot fit an even number of quarter-wavelengths into it because this would leave N at both ends or A at both ends, and you can only have N at closed ends and A at open ends. – sammy gerbil Apr 27 '17 at 11:54
• Using the same reasoning then, for both end open pipe for $n=1$ ANA makes it a complete wavelength right? – rahul rj Apr 27 '17 at 12:13
• No, that is half a wavelength, same as closed both ends (NAN). You look at only the bold line or only the dotted line. They represent the state of the wave at different times. A complete wavelength of a sine wave is NANAN. It starts and ends at zero and has a zero at one point in between, and 2 peaks (+1, -1). – sammy gerbil Apr 27 '17 at 14:56