Three ideal one-dimensional systems of the same physical length L are set up in a laboratory:
• System S: a stretched string fixed at both ends.
• System O: an air column in a pipe open at both ends.
• System C: an air column in a pipe that is open at one end and closed at the other.
Assume the waves on the string travel with speed v_s and sound in air travels with speed v_a. Ignore end corrections and damping.
(a) For each system, describe the standing-wave pattern (nodes and antinodes) associated with its third harmonic. Sketches are acceptable but must be accompanied by verbal descriptions.
(b) For systems O and C only, determine the ratio of the fundamental wavelength of system C to that of system O, explicitly stating all reasoning.
(c) Explain why the harmonic series generated by system C contains only odd-numbered harmonics, whereas systems S and O include all integer harmonics.
(d) A student claims that the sixth harmonic of system S has the same frequency as the fourth harmonic of system O provided the wave speeds are equal (v_s = v_a). Evaluate this claim and justify your conclusion.