I vaguely remember the following conversation from back when I was a PhD student.
Student A: What's a Bessel function?
Student B (waving his arms about): It's a wavy thing - goes like this, doesn't it?
Me: Sounds vaguely familiar - I think we did it in third-year.
Student A: But what IS it?
Me: Something to do with waves on a membrane, isn't it?
Student A: Does it have a formula?
Me and student B: Don't know...
And so it went on. We couldn't really help each other much, save for the fact it had something to do with waves. Now, I've been prompted to recall this conversation since I'm about to have to teach it in a Master's paper. Hopefully, I'll be able to do a better job at getting the message across than my teachers did. So what's a Bessel function about?
They crop up when we consider waves on a membrane. An example would be the vibration of the skin of a drum. Hit the drum and the skin vibrates - and different parts vibrate with different amplitudes. Formally, we are talking about solving the wave equation in two dimensions (a drum skin being a two-dimensional object). The lowest mode of vibration of the drum has the largest displacement of the membrane at the centre, and the smallest (zero) around the circumference, where it is held in place. In other words, the skin is vibrating the most in the centre, and the least around the edge. The Bessel function (formally, the zero-th order Bessel function of the first kind) tells us the amplitude of the vibration of the skin at a given radius from the centre of the drum.
The Bessel function does indeed look 'wavy', with the waves tailing off in size as distance increases. On a drum the modes of vibration are quantized - that is, the edge of the drum must correspond to a zero on the Bessel function, so that only certain frequencies are allowed. This is why a drum (or any musical instrument) has a characteristic fundamental tone but overlaid with higher harmonics.
Unfortunately I can't give you a nice clean mathematical formula for a Bessel function (there isn't one); instead, if you look in the text books, they are most likely just to show the plot of one. (See here for example.) Maybe this is where my student friend A was having problems; something for me to bear in mind when I present it to my students.