Question

The sixth harmonic is set up in a pipe. (a) How many open ends does the pipe have (it has at least one)? (b) Is there a node, antinode, or some intermediate state at the midpoint?

Short answer

1. Both the ends of the pipe are open.
2. There is antinode at the midpoint of the pipe.

Step-by-step solution

Step 1: Given

The sixth harmonic is set up in a pipe.

Determining the concept

The standing sound wave is set up in a pipe if the sound for proper wavelength is sent through the pipe. The pattern of the standing wave will change depending on how many ends of the pipe are open. In a pipe open at both ends, all harmonics are present, and in a pipe with one end open, only odd harmonics are present.

Formulae are as follow:

For a pipe open at both ends

$\lambda =\frac{2L}{n}$

Where,

L is length,lis wavelength.

(a) Determining how many open ends does the pipe have

Since the harmonic that is set up is an even integer, the pipe must be open at both ends.

Hence, both the ends of the pipe are open.

(b) Determining is there a node, antinode or some intermediate state at the midpoint

$\lambda =\frac{2L}{n}$

$\therefore \lambda =\frac{2L}{6}$

$\therefore \lambda =\frac{L}{3}$

$\therefore L=3\lambda$

$\therefore \frac{L}{2}=\frac{3\lambda }{2}$

Thus, the midpoint of the pipe (L/2) corresponds to half the wavelength of the wave. For a pipe open at both ends, antinodes are present at the ends of the pipe. At half wavelength, the position of the particle is also going to be an antinode.

Hence, there is an antinode at the midpoint.

Therefore, the sound wave can produce a standing wave in the pipe. The pipe open at both ends can produce the harmonics of all integer multiples of the given frequency and the pipe with one open end can produce only odd harmonics. Thus, the number of harmonic present can give us the information about the nature of the pipe.