A transverse wave that is propagated through a wire, is described through this function: y(x,t) = 0.350sin(1.25x + 99.6t) SI
Consider the point of the wire that is found at x= 0:
a) What's the time difference between the two first arrivals of x = 0 at the height y = 0.175m?
b) How much distance does the wave cover during that time?
v = λ*f
v = Δx/Δt
sinx = sinφ => x = 2kπ + φ OR Χ = 2kπ + π - φ
The Attempt at a Solution
a) First up, the oscillation function for x = 0 is: y(0,t) = 0.350sin(99.6t)
For y = 0.175m => ... 0,5 = sin(π/6) = sin(99,6t) => 99.6t = 2kπ + π/6 OR 99.6t = 2kπ + 5π/6
And here's I find the problem. I don't remember how to solve these (it's been a while), so while I know that I should put k = 0, get a result, then k = 1, get a result, and then find the difference between the two, I don't know which formula to pick.
For example, for k = 0 we have: t = 5,25 * 10-3s from the first, and t = 0,026s from the second.
Likewise, for k = 1 we have: t = 0.068s & t = 0.089s
The book's answer is t = 21 ms, which I get if I find the difference between the first set (0,026 - 0,00525 gives 0,7 ms), or the difference between the second set (0.089 - 0.068 gives us a perfect 0,021). Problem is, I don't know why. I don't remember the theory behind this is what I'm saying. Why can't I find the difference between the results of just one formula, one for k = 0, and the other for k = 1, eg 0,068 - 0,00525.
b) That's an easy one. v = λ*f <=>... <=> v = 79.68 m/s | v = ΔX/ΔT <=> ... <=> Δx = 1.68m
Any help is appreciated!
PS: I then tried something different, and while I didn't get proper results, I'd like to know why it's wrong. Say, from putting x = 0 & y = 0.175 into the main function, if solved through arcsin, I can get t ~ 0,0052. So, considering that x goes from Position of Balance, to = +A, then to PoB, then to -A, then back to PoB and so on and so forth, couldn't I find the "wanted time" by this logic:
Δt = (T/4 - t) + T/4 + T/4 + T/4 + t
Obviously it's wrong, but I want to know why.