Standing waves
A stationary or standing wave is one in which the amplitude varies
from place to place along the wave. Figure 1 is a diagram of a stationary wave. Note that
there are places where the amplitude is zero and, halfway between, places where the
amplitude is a maximum; these are known as
nodes
(labelled N) and
antinodes (labelled A) respectively. (See
Figure 1)
The distance between successive
nodes, and successive antinodes, is half a wavelength. (
l/2)
The amplitude of the
points on a stationary wave varies along the wave. In Figure 1 the amplitude at point 1 is
a
1, that at point 2 is a
2 and that at point 3 is a3. The displacement (y) at
these points varies with time.
Any stationary wave can be formed by the addition of
two travelling waves moving in opposite directions.
A wave moving in one direction
reflects at a barrier and interferes with the incoming wave.
Mathematical
treatment of the formation of a standing wave from two travelling waves
Consider
two travelling waves 1 and 2. Let the displacements 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.
When x = 0,
l/2 ... A is a maximum and
we have an antinode;
When x =
l/4, 3
l/4, 5
l/4 ... A is a minimum and we have a
node.
Notice that the maximum value of A is 2a.
