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This is really a question about terminology, The wavelength of a standing wave in a e.g. pipe closed at one end and open at the other is said to be $\frac{4L}{n}$, where $L$ is its length and $n$ is an odd natural number. Is it correct to say that the second mode of this pipe has wavelength $\frac{4L}{3}$ and the fourth has wavelength $\frac{4L}{7}$ etc, or is it correct to say that $\frac{4L}{3}$ and $\frac{4L}{7}$ are the wavelengths for the third and seventh mode (i.e. the nth mode) and the second and fourth modes etc don’t exist for this pipe?

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I would rather say that third harmonic of this pipe has wavelength 4L/3 and the seventh harmonic has wavelength 4L/7.
The general formula for wavelength is 4L/(2n-1). This can be derived through induction by just taking further cases and observing the pattern. So for example the first harmonic would have the wavelength of 4L.

Here mode refers to the harmonic,(I suppose), and if you would observe carefully, you would conclude that only odd harmonics occur in case of pipe closed at an end.

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  • $\begingroup$ Technically the nth mode is the (n-1)th harmonic, no? $\endgroup$
    – RC_23
    Sep 19, 2023 at 18:36
  • $\begingroup$ @RC_23 Harmonics are counted differently in different contexts. Physics generally uses $n$ while music uses $n-1$. $\endgroup$
    – John Doty
    Sep 19, 2023 at 18:49

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