I'm not sure what level of physics or how much you have to dedicate to resources, but here's a couple ideas.
1) Simple idea: Drop a massive object from a certain height. Measure the time it takes for the object to reach the ground (or whatever material it lands on). Have someone drop the object for you and stand a certain distance away from the experiment with a stopwatch. Measure the time difference between seeing the object land and hearing it land. Repeat both procedures several time and apply some statistics. You could take it in many different directions from here (example: Compare your self-derived gravitation and sound formulas versus established formulas. How do they differ? Why?).
2) More complex idea: Use computer programs to show wave patterns. Using almost any modern computer, you can download tons of free programs that will allow you to process, analyze, edit, morph, and manipulate various audio sources. In addition, some programs allow you to generate sounds using sine waves, cosine waves, square waves, etc. I think it would be a novel project to map certain combinations of waves by generating them and graphing the results via an
oscilloscope. This allows you the ability to make predictions based on the mathematics of sound and then test those predictions using this program. Or, you may find that playing around with certain combinations of waves produces distinct and reproducable results. Several such occurrences could form the beginnings of a brand new hypothesis which further merges mathematics and sound waves.
(example: Interference patterns from all waves with frequencies that are a prime number from the range of 20Hz to 20,000Hz (human range of hearing) define equations such that...) [feel free to PM me if you'd like help getting programs that would be needed for this]
I must say though, that Modest's suggestion is appealing because it covers all the subjects in one project that is fairly cheap, safe, and simple to implement.
