The principle of superposition

From Mastering Physics:


The principle of superposition states:

If two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves y1(x,t)+y2(x,t), where y1(x,t) is the wave described in Part A and y2(x,t) is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

y_s(x,t) = y_e(x)y_t(t)


This form is significant because:

 y_e(x) called the envelope, depends only on position

 y_t(t) depends only on time


Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of ye(x).



COMBINING TWO WAVES TO CREATE A STANDING WAVE:

y1(x,t)=Asin(kx−ωt).

y2(x,t)    = Asin(kx+ωt)
y1(x,t)= 
  y1(x,t)=

Consider the sum of two waves y1(x,t)+y2(x,t)

Their sum can be written as follows:
ys(x,t)=ye(x)yt(t).

ye(x), yt(t) =

2Asin(kx),cos(ωt)