Why are the harmonics of a stretched string and that of an open pipe the same?

Question

In page-376 of the NCERT physics class-11 chapter on waves (see here), the frequency relations for the harmonic of a stretched string are derived as(eq.15.42):

$$ \nu_n = \frac{nv}{L}$$

Now, in page-382, example 15.5, it is written that harmonics for an open pipe is given as:

$$ v_n = \frac{nv}{L}$$

Which is the exact same form as the first equation, so my question is why the equation modelling the harmonics of open pipe and string the the same?

I don't understand it because for string we need that both ends must be nodes (it is clamped at both ends) while for the pipe, it must be anti node at both end.

Answer 1

Where the nodes are in the case of standing sound waves in a pipe depend on what whether you are thinking of displacement nodes or pressure nodes. The diagrams in the text to which you refer show the displacement nodes and antinodes which closely parallels the examples of waves in strings that it presented earlier. In this case at the closed end of a pipe the air particles can not move so we draw a node there.

The other way to represent standing sound waves in a pipe is to represent pressure nodes and antinodes. This is less common in texts at the secondary school level but I have seen it occasionally. At an open end the air pressure is constrained to be equal to the atmospheric pressure an so in this sort of representation we would draw nodes at the open ends. Conversely at a closed end the air pressure can vary resulting in an antinode.

If you draw the standing waves by representing pressure rather than displacement then the diagrams for a pipe open at both ends ends up looking just like the diagrams for a rope fixed at both ends.

Even if you draw the, more common, displacement wave diagram it leads to the same relationship between the wavelength and the length of the tube. Think of, for example, the second harmonic: in a rope you would have a node at either end and a node in the centre giving you two loops which means two half wavelengths so $L = \lambda$, in a pipe open at both ends you would have an antinode at each end and one in the middle resulting in one complete loop in the middle of the pipe and half a loop at each end still giving you two loops in total so once again $L = \lambda$.

For me the first time I really appreciated the issue of pressure wave diagrams vs. displacement wave diagrams was when I visited the Ontario Science Center. Outside the main entrance is a huge plastic tube through which a person can crawl. In the tube a speaker maintains a standing sound wave. On the side of the tube the standing wave is marked. When I first saw it I thought to myself that they had incorrectly shown nodes at the ends but when I crawled through it I saw (heard really) that that was were the sound was actually the quietest. It turns out our ears detect fluctuations in air pressure rather than in position of air molecules.

Answer 2

In both cases the harmonics are the standing waves, which can be thought of as waves reflected multiple times from the ends of the string/pipe and interfering constructively with themselves. The only such a waves are those with the integer number of half-wavelengths fitting between the two ends of the string/pipe.

What might seem confusing here is that the pipe is open. However, the pressure wave is still partially reflected from its ends - it is this reflected wave that we use - everything else is just losses, which is why the pipe should not be too wide.