Chapter 14
Superposition & Standing waves
The Principle of Superposition
If two or more traveling waves are moving through a medium, the resultant wave function at any point is the algebraic sum of the wave functions of individual waves.
Two traveling waves can pass through each other without being destroyed or even altered.
Interference of Waves
Two waves traveling to the right
According to the principle of superposition, the resultant wave is:
1 siny A kx t 2 siny A kx t
1 2 sin siny y y A kx t kx t
2cos sin2 2
y kx t
Amplitude Phase angle
Interference of Waves1. Some source of disturbance; 2. A medium;3. Some physical mechanism through which particles of the
medium can influence one another.
In phase =0,constructive interference
Out of phase =, destructive interference
Other phase =/3, y falls to somewhere between the extremes
Standing Waves
Two waves, one traveling to the right and one to the left
According to the principle of superposition, the resultant wave is:
1 siny A kx t 2 siny A kx t
1 2 sin siny y y A kx t kx t
2 sin cosy A kx t
The function of a standing wave
Standing Waves
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.
Standing Waves in Strings
the wavelength of the nth mode of vibration
2
2
2
n
nn
L
nand
L n
nf
L
The wavelength of the nth mode of vibration
Standing Waves in Strings
Standing Waves in Strings
The fundamental frequency of vibration is adjusted by pressing and releasing the finger.
The length of string is changed!
A two-dimensional standing wave on a disk
the fundamental mode A higher harmonic standing wave on a disk with two nodal lines crossing at the center.
Beats: Interference in Time
The displacement that each wave produced at a fixed point
According to the principle of superposition, the resultant displacement:
1 1cos 2y A f t 2 2cos 2y A f t
1 2 1 2cos 2 cos 2y y y A f t f t
1 2 1 22 cos cos2 2
f f f fy A t t
Amplitude varies in time
Beats: Interference in Time
1 2 1 22 cos cos2 2
f f f fy A t t
1 22 cos2
f fA t