Show that the frequency produced in a closed organ pipe are in the ratio1:3:5:7...?

That may or may not be correct depending on details.

Now learn impedance. If you strike a ball near the end of the bat, speed is high but force is low, so the ball is not accelerated as much as it could be. The unused energy is reflected down the bat and might break it. That is a high impedance, high ratio of speed to force.

If you strike a ball near the handle of the bat, force is high but speed is low, so the ball is not accelerated as much as it could be. The unused energy is reflected down the bat and might break it. That is a low impedance, low ratio of speed to force.

If you strike the ball near the center of the bat, the energy transfer is perfect and the ball is accelerated to the maximum speed. There is no reflection of unused energy.

Now look at an organ pipe. A pressure wave travels through the pipe. An open end is a sudden change in impedance, so the wave is reflected. But an open end allows high amplitude with low force. A closed end allows high force with zero amplitude. So a pipe that is open at both ends will only host a wave that has maximum amplitudes at both ends. If you draw a picture of that, you will see that the length of the pipe can be any multiple of a half wavelength.

If one end of the pipe is closed, then the wave can only exist with high amplitude at the open end and a node (zero amplitude) at the closed end, and the pipe must be an odd multiple of a quarter wave length.

A guitar string is closed at both ends, and that also forces half wave lengths. But you can touch your finger at the fifth, seventh, or twelfth fret to force a node there and that produces higher notes called "chimes".
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