Standing Waves

1. PHYSICS 101-
LEARNING OBJECT 6
Standing Waves

2. Standing Wave - Definition
• When two harmonic waves that have equal amplitude,
frequency, and wavelength are moving in opposite
directions along the same medium, the resulting wave is
defined as a standing wave.
*Assume in this case that both waves have a phase
constant of zero.*

3. Standing Wave – Wave Function
• The wave moving in a direction of increasing position (x)
has a wave function of D1(x,t) = A sin(kx-ωt).
• The wave moving in a direction of decreasing position
has a wave function of D2(x,t) = A sin(kx+ωt).
• The resultant standing wave function is:
D(x,t) = D1(x,t) + D2(x,t)
= A sin(kx-ωt) + A sin(kx+ωt)
= A [sin(kx-ωt) + sin(kx+ωt)]
= 2A sin(kx) cos(ωt)
Note: wave function obtained from a
trigonometric identity.

4. Standing Wave – Wave Function (cont’d)
• To make the amplitude of a standing wave dependent on
the position, x, we can rewrite the wave function of a
standing wave,
D(x,t) = 2A sin(kx) cos(ωt)
as D(x,t) = A(x)cos(ωt)
where Amplitude = A(x) = 2A sin(kx)

5. Standing Wave – Nodes and Antinodes
• Nodes are points on a standing wave that have an
amplitude of zero.
• Antinodes are points on a standing wave that have the
maximum amplitude possible (2A/-2A).
• All standing waves have nodes and antinodes.
This is because amplitude, A(x), is a sine function, meaning
that certain points have an amplitude of zero, certain points have the
maximum amplitude possible, and the remaining points have an
amplitude that is between 0 and 2A or 0 and -2A.

6. Standing Wave – Nodes and Antinodes (cont’d)
• Nodes are points that seam to be standing still (they are
at rest). Nodes are sometimes referred to as being points
of no displacement.
• Antinodes are points that undergo the maximum
displacement possible.
• A standing wave alternates between nodes and antinodes
 node followed by antinode followed by node,
and so on.

7. Standing Wave – Nodes and Antinodes (cont’d)
• Nodes are points where the amplitude is equal to zero.
A(x) = 2A sin(kx) = 0
sin(kx) = 0
sin(2πx/λ) = 0
Therefore the distance between two successive nodes is half
a wavelength.
• Antinodes are point where the amplitude is equal to +2A.
A(x) = 2A sin(kx) = +2A
sin(kx) = +2A
sin(2πx/λ) = +2A
Therefore the distance between two successive antinodes is
also half a wavelength.
• This also results in a distance of a quarter of a wavelength
between adjacent nodes and antinodes.

8. Practice Questions
1) When does a standing wave form?
a) When interference occurs between two waves
moving in the same direction.
b) When a wave moves from one medium to
another.
c) When interference occurs between two waves
moving in different directions.
2) The amplitude of a standing wave is:
A(x) = (0.5m) sin(3.00x). What is the location, x, of the first
point where the amplitude is equal to 0.17m?

9. Solutions
Answers:
1) c) When interference occurs between two waves moving in
different directions.
2) A(x) = (0.5m) sin(3.00x)
0.17m = (0.5m) sin(3.00x)
0.17/0.5 = sin(3.00x)
0.34 = sin(3.00x)
sin-1(0.34) = 3.00x
sin-1(0.34) = x
3.00
0.1156… = x
0.12m = x