The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t)
To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t.
A separate wave on the same wire is: (3) y= 0.5 sin (6x + 4t)
b. For y= 0.5, solve for x to get (4) x(t) then take a (partial) derivative of x(t) to get the rate of change of x which is the velocity of the wave.
c. From parts a and b, state how you can tell whether a wave is moving toward +x or toward -x direction.
a. In a general equation, y= ym sin (kx - wt), if we let y= ym then ym cancels and we get:
sin (kx - wt) = 1 from which kx - wt = Arcsin 1 and x= (w/k)t + (Pi)/(2k)
The derivatives of wave equations are determined. A separate wave on the same wire are determined. With good explanations and calculations, the problems are solved.