Back in high school, as part of a summer science program, I ran an experiment to determine if you really can tell how ripe a watermelon is by thumping it. I only got to thump and taste-test eight melons, so I don’t have great data, but as I recall, damping and natural frequency were predictive—high damping suggested an overripe, mushy melon.
There is, naturally, peer-reviewed literature on this.
Abbaszadeh, R., Moosavian, A., Rajabipour, A., & Najafi, G. (2013). An intelligent procedure for watermelon ripeness detection based on vibration signals. Journal of Food Science and Technology. doi:10.1007/s13197-013-1068-x. http://link.springer.com/10.1007/s13197-013-1068-x
Apparently, kNN on the FFT of laser Doppler vibrometer data (they got to fire lasers at watermelons and call it research?!) works well. However, they have a small test set, and they’re also the group responsible for this masterpiece:
Abbaszadeh, R., Rajabipour, A., Mahjoob, M., Delshad, M., & Ahmadi, H. (2013). Evaluation of watermelons texture using their vibration responses. Biosystems Engineering, 115(1), 102–105. doi:10.1016/j.biosystemseng.2013.01.001. http://linkinghub.elsevier.com/retrieve/pii/S1537511013000032
…which shoved the whole spectrum (hundreds of data points) into stepwise regression to predict quality, with only a few dozen melons to fit on. They claimed an R-squared of 0.9986 on their test set, which seems completely implausible.
Zeng, W., Huang, X., Müller Arisona, S., & McLoughlin, I. V. (2014). Classifying watermelon ripeness by analysing acoustic signals using mobile devices. Personal and Ubiquitous Computing, 18(7), 1753–1762. doi:10.1007/s00779-013-0706-7. http://link.springer.com/10.1007/s00779-013-0706-7
Built a system which extracts recording of the thump, extracts various acoustic features, and shoves it into an SVM. Then (because of course) built it into an Android app and released it on Google Play.