Describe constructive and destructive interference in terms of the combining of crests and troughs and the reesulting amplitude.
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.
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.
Constructive interference occurs when two waves with the same frequency and amplitude align precisely, resulting in their crests and troughs combining together. When the crests of the waves align, they reinforce each other, leading to an overall increase in amplitude. Similarly, when the troughs align, they also reinforce each other, resulting in a greater amplitude.
On the other hand, destructive interference occurs when two waves with the same frequency and amplitude are out of phase, meaning that their crests and troughs meet at opposite points. In this scenario, the crests of one wave align with the troughs of the other wave, causing cancellation or reduction of their amplitudes. As a result, the waves subtract from each other, resulting in a lower overall amplitude.
To understand this concept practically, imagine two identical water waves approaching each other. If their crests align, the water molecules will pile up, producing a higher peak. This is constructive interference. Conversely, if a crest aligns with a trough, the water molecules will be displaced in opposite directions, resulting in a smaller wave or even complete cancellation. This is destructive interference.
In summary, constructive interference occurs when the crests and troughs combine, resulting in an increased overall amplitude, while destructive interference occurs when the crests and troughs meet out of phase, leading to a reduced or canceled amplitude.