Question: Finding Roots of Periodic Function

I want to find the first positive and first negative root of a bessel function

j2 := (x, a) -> k(x)^2*a^2*(diff((diff(1000000000*sin(1/1000000000*k(x)*a)/(k(x)*a), x))/(k(x)*a), x))/(k(x)*a)

I couldn't use the built in bessel function with two variables, so I had to enter it by hand.

I want to find the values of x where j2 is zero as a function of a:

f := (a) -> RootOf(j2(x, a), x, 0 .. 2)

plot(f(a), a = 2 .. 2.25, y = -3 .. 3)

I can plot j2 and get the function I expect. I can see the root in the interval 0-2 for some value of a. The root changes pretty drastically with a, so I've tried setting the interval 0..2/a, which does a good job of keeping the bounds about only 1 root, but still gives an empty plot. I don't know if that is acceptable, but Maple doesn't give an error. I've also tried setting the bounds to very short intervals so that I know the range is acceptable for all values.

If I use a guess value instead of an interval it will find roots over a couple of short segments, but leaves most of the range empty. I tried to vary the guess value with a as well to no avail.

I've tried to use both index and label, but with no results. I'm not sure I'm using them correctly as the documentation on those parameters is pretty poor.

Thanks for any help you can provide!