Net vibration at point P:

1 2 cos( )y y y A t

Where: 2 2 21 2 1 22 cosA A A A A

2 1 2 1

2( ) ( )r r

)2cos()2cos(

)2sin()2sin(

222

111

221

111

r

Ar

A

rA

rA

tg

Coherent waves

Two coherent waves must have

parallel vibration directions

identical frequency

constant phase difference

2 1 2 1

2( ) ( )r r

phase difference

=

2k Interference constructively

(2 1)k Interference destructively

21max AAAA

|| 21min AAAA

Path length difference

2 1 2 , 0,1,2,...k k if

2 1r r =

k Interference constructively

(2 1)2

k

Interference destructively

§2-6 standing wavesTraveling waves The energy of traveling waves

What are standing waves?

Standing waves are special cases of wave interference.

The superposition of two coherent waves with identical amplitude, frequency and common vibration axis but opposite propagating direction is standing wave.

x

yy1 and y2 are identical

y1

1 2y y y

The propagating directions are the same.

y is Not standing wave!

x

y y1

y2

(2 1) , 1, 2,3,...k k 2 1A A

1 2y y y

y is Not standing wave!

x

y

u

ustanding wave

xy

Characteristics of standing wave

1)Every element in the medium vibrates steadily. There is no traveling of waveform.

2) The amplitude of standing wave is A(x).

3) The positions of maximum amplitude are called antinodes.

antinode

antinode

4) The positions of zero amplitude are called nodes.

node

Mathematical expression of standing wave

Suppose there are two coherent waves with identical amplitude, frequency and common vibration axis but opposite propagating directions.

x

yy1

)2

cos(1 xtAy Set the initial phase is 0.

u

y2

u

)2

cos(2 xtAy

The standing wave is the superposition of these two coherent waves.

)2

cos()2

cos(21 xtAxtAyyy

using interference of wave, we get:

2 2 2 2 11 2 1 2 2 12 cos 2net

r rA A A A A

1 2A A A 1 2 0

2r x 1r x

Where:

2 2 2 2 11 2 1 2 2 12 cos 2net

r rA A A A A

2 22 1 cos 2

xA

2 22 2cos 2x

A

2 24 cos 2x

A

2 cos 2net

xA A

The initial phase of the superposed wave:

)2cos()2cos(

)2sin()2sin(

222

111

221

111

r

Ar

A

rA

rA

tg

take 1 2 ,A A A 1 2 0, 2 ,r x 1r x

0

22 cos cosAy tx

The mathematical expression of standing wave is:

standing wave equation

22 cos cosAy tx

Amplitude of standing wave, A(x)

Simple harmonic vibration

Standing wave equation does not satisfy:

) ,() ,( xtytuxtty Therefore, standing wave is different from traveling wave.

22 cos cosAy tx

1)Every element of the medium in standing wave is doing simple harmonic vibration at identical frequency ω.

2) But the amplitude of every element might be different at various position.

Since every element of the medium vibrates steadily, the disturbance does not propagate in standing wave.

Amplitude2

2 cosA A x

max 2A A when2

cos 1x

antinode

Node: 0A when2

cos 0x

, ( 0, 1, 2...)2

x k k

1( ) ,( 0, 1, 2...)

2 2x k k

node

x

y

antinode

Adjacent antinodes are separated by a distance of λ/2.

node

Adjacent nodes are also separated by λ/2.

The distance between adjacent antinode and node is λ/4.

We can measure the distance between two adjacent nodes to determine the wavelength of λ.

Phase of standing wave2

2 cos cosAy tx

If time is varying, for all the elements in the medium their phases are identical, that is ωt.

The displacement, velocity and phase of the elements locating in the two sides of one node are opposite.

But the displacement, velocity and phase of the elements locating between two nodes are with the same sign.

§2-7 half-wavelength lossIf the incident wave reflects on a certain interface, the phase of the reflected wave is opposite to that of the incident wave.Such phenomena is called half-wavelength loss.

Half-wavelength loss

An electromagnetic wave undergoes a phase change of 180o on reflection from a medium of higher reflection index than the one in which it is traveling.

How can we get the reflected wave with half-wavelength loss?

incident

Wave with opposite vibration

But the reflected wave with half-wavelength loss is not this opposite wave, but this wave with a phase difference of π.

The conditions which will cause half-wavelength loss:

1) The reflection point is the fixed end of the medium.

2) When the wave propagates from a wave thinner medium to a wave denser medium, the reflected wave has half-wavelength loss.

Wave denser medium: the medium with a larger reflection index n.

Wave thinner medium: the medium with a smaller reflection index n.

Air, n=1

Water, n=1.33

thinner

denser

The reflected wave has half-wavelength loss

glass, n=1.52

Water, n=1.33

denser

thinner

The reflected wave has no half-wavelength loss

Only reflection may have half-wavelength loss possibly. Refraction never has such phenomena.

Example 2-3-1

The wave equation of one wave propagating along x axis can be written as:

)(2cos1 x

T

tAy

The reflection occurs at x=0 and the reflection point is one node. Find: 1) the wave equation of the reflected wave. 2) the wave equation of the superposition of these two waves. 3)the position of the nodes and antinodes.

Solution:

1) Because the reflection point is one node, the reflected wave has half-wavelength loss.

The wave equation of the reflected wave can be expressed as:

2 cos[2 ( ) ]t x

y AT

)(2cos1 x

T

tAy The incident wave equation: