compare the relation of fundamental frequencies for a pipe closed at one end and for a pipe open at both ends

To compare the relation of fundamental frequencies for a pipe closed at one end and for a pipe open at both ends, we need to understand the concept of harmonics and standing waves in pipes.

In a closed pipe, such as a flute or a pipe closed at one end, the air cannot escape from one end of the pipe. This causes a node, or a point of zero displacement, to be formed at the closed end. The fundamental frequency, or the first harmonic, occurs when the pipe vibrates in one complete wavelength, with an antinode at the open end and a node at the closed end.

In an open pipe, such as a pipe open at both ends or a clarinet, the air can freely escape from both ends. This means that both ends of the pipe have antinodes, where the amplitude of vibration is maximum. The fundamental frequency occurs when the pipe vibrates in half of a wavelength, with an antinode at each end.

Now, let's compare the fundamental frequencies for these two types of pipes.

For a pipe closed at one end, the fundamental frequency, f, can be calculated using the equation:

f = v / (4L)

Where:
- f is the frequency,
- L is the length of the pipe, and
- v is the speed of sound in the medium.

For a pipe open at both ends, the fundamental frequency is given by:

f = v / (2L)

Where all variables have the same definitions as in the previous equation.

By comparing these equations, we can see that the relation between the fundamental frequencies of a closed pipe and an open pipe is as follows:

f (closed pipe) = 2f (open pipe)

This means that the fundamental frequency of a closed pipe is half the fundamental frequency of an open pipe. In other words, the fundamental frequency of a closed pipe is lower than that of an open pipe by a factor of 2.

This relation is due to the difference in the way standing waves are formed in these two types of pipes, based on their open or closed ends.