Four successive resonance frequencies of a vertical air column of unknown boundary conditions were measured as 285 Hz, 475 Hz, 665 Hz and 855 Hz.
(i) Write the general expression for the frequency of the nth harmonic for each of the following physical systems:
(a) a string fixed at both ends,
(b) an organ pipe open at both ends,
(c) an organ pipe closed at one end.
(ii) Using only the pattern of the measured frequencies, explain which one of the three systems listed in part (i) the air column must represent. Give a clear reason for your choice.
(iii) For the identified system, state the harmonic number that corresponds to each of the four measured frequencies and hence determine the fundamental frequency of the column.
a. Write the expressions for the nth harmonic for three listed systems.
b. Identify the physical system represented by the measured frequencies, giving reasons.
c. Assign harmonic numbers to the measured frequencies and find the fundamental frequency.