In a resonance-tube experiment a long vertical glass tube is partially filled with water so that the water surface acts as a movable closed end. The top of the tube remains open to the air. A 426\,Hz tuning fork is struck and held above the open end.
The first two resonances that are clearly audible occur when the distance from the open end to the water surface is 19.5\,cm and 64.5\,cm, respectively.
(a) Explain why the two observed resonances correspond to successive odd-numbered standing-wave modes in the tube.
(b) Neglecting any end correction, use the measured distances to obtain an experimental value for the speed of sound in the laboratory. Show all steps in your reasoning.
(c) A more accurate analysis includes an "end correction" e, which accounts for the displacement antinode lying slightly above the physical end of the tube. Derive an expression for e in terms of the two measured resonance lengths and calculate its numerical value.
(d) Using the value of e found in part (c), determine an improved value for the speed of sound. Compare this value with the accepted speed 343\,m\,s−1 and comment on at least two possible sources of discrepancy between the two values.