Mathematical treatment of the formation of a standing wave

Let the displacements of two travelling waves 1 and 2 at time t and position x be y_{1} and y_{2}.

y_{1} = a sin (wt - kx) (say right-left)

y_{2} = a sin (wt + kx) (say left-right)

Therefore:

Note that this expression is composed of two terms:

(a) sin (wt) - this shows a varying amplitude with time at a particular place.

(b) cos (kx) - this shows a varying amplitude with position at a particular time.