Describe constructive and destructive interference in terms of the combining of crests and troughs and the reesulting amplitude.

Answer 1

T in degrees
add A= sin wt + sin (wt + T)

if T = 0
A = 2 sin wt

if T = 90
A = sin wt + sin (wt+90)
= 2 sin .5(2 wt +90) cos .5(-90)
= 2 sin (wt + 45) cos 45
= 2 (.707) sin (wt+45)
phase changed, amplitude .707 times the constructive 2 A but still sqrt 2 or 1.41 times just one A

If T = 180
A = sin wt + sin (wt + 180)
= 2 sin .5(2 wt +180) cos .5(-180)
= 2 sin (wt + 90) cos 90
but cos 90 = 0, result is nothing, destructive.

Answer 2

In other words, when the phase angle is zero, the crests add to give 2 A
and when the crest is 180 degrees off, the trough lines up with a crest and gives zero.
in between is in between.