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