Scene 1

From moving to Stationary
•Waves we have considered so far have been PROGRESSIVE WAVES. •PROGRESSIVE WAVES : start from source & travel outwards
•Transfer energy from one place to another. 

Scene 2

From moving to Stationary
•Waves we have considered so far have been 
PROGRESSIVE WAVES. 
•PROGRESSIVE WAVES : start from source & 
travel outwards
•Transfer energy from one place to another

From moving to Stationary •Next is STATIONARY WAVES or STANDING WAVES. 

Scene 3

From moving to Stationary 
•Next is 
STATIONARY 
WAVES or 
STANDING 
WAVES

Nodes and antinodes
•Nodes : points that do not move at all
•Antinodes : points where the spring oscillates with a maximum amplitude
Sections of spring in adjacent loops are always moving in anti-phase ; they are half a cycle out of phase with one another.. 

Scene 4

Nodes and antinodes
•Nodes : points that do not 
move at all
•Antinodes : points where the 
spring oscillates with a 
maximum amplitude
Sections of spring in adjacent loops are always moving 
in anti-phase ; they are half a cycle out of phase with 
one another.

Formation of stationary waves
•String stretched with 2 fixed ends
•Pulling and releasing produced stationary wave
•Node at each end •Antinode in the middle. 

Scene 5

Formation of stationary 
waves
•String stretched with 2 
fixed ends
•Pulling and releasing 
produced stationary 
wave
•Node at each end 
•Antinode in the middle

Formation of stationary waves
•Releasing string produces 2 stationary waves travelling at OPPOSITE directions
•These get reflected at the fixed ends
•These then superimpose to produce stationary waves. 

Scene 6

Formation of stationary 
waves
•Releasing string produces 2 stationary waves 
travelling at OPPOSITE directions
•These get reflected at the fixed ends
•These then superimpose to produce stationary 
waves

formation of stationary waves
•Copy diagram from blackboard. 

Scene 7

formation of stationary 
waves
•Copy diagram from blackboard

Formation of stationary waves
•t = 0, left and right waves are in phase. => CI hence amplitude is twice of each wave
•t = T/4, each travels λ/4 to the left and right. Hence in antiphase (phase difference 180˚) => DI giving zero displacement
•t =T/2, back in phase again => CI
•t = 3T/4, antiphase => DI => Zero displacement
•t=T, CI similar as when t=0. 

Scene 8

Formation of stationary 
waves
•t = 0, left and right waves are in phase. => CI hence 
amplitude is twice of each wave
•t = T/4, each travels λ/4 to the left and right. Hence 
in antiphase (phase difference 180˚) => DI giving 
zero displacement
•t =T/2, back in phase again => CI
•t = 3T/4, antiphase => DI => Zero displacement
•t=T, CI similar as when t=0

formation of stationary waves
•Cycle repeats showing nodes and antinodes
•Separation between N and AN tells us abt the progressive waves that produces the stationary waves
•The separation is related to λ. 

Scene 9

formation of stationary 
waves
•Cycle repeats showing nodes and antinodes
•Separation between N and AN tells us abt the 
progressive waves that produces the stationary waves
•The separation is related to λ

Formation of stationary waves •Impt CONCLUSIONS:
•separation between 2 adjacent N (or AN) = λ/2
•separation between adjacent N and AN = λ/4
•λ of ANY progressive waves can be determined from separation of N or AN of standing wave (This is = λ/2). 

Scene 10

Formation of stationary 
waves  
•Impt CONCLUSIONS:
•separation between 2 adjacent N (or AN) = λ/2
•separation between adjacent N and AN = λ/4
•λ of ANY progressive waves can be determined 
from separation of N or AN of standing wave (This 
is = λ/2)

Formation of standing waves
•We can use this to determine either v or f
•Using the wave eqn: v = f λ
•NB: Stationary waves DO NOT TRAVEL hence has NO SPEED.
•It also DOES NOT transfer ENERGY between 2 points like the progressive waves. 

Scene 11

Formation of standing waves
•We can use this to determine either v or f
•Using the wave eqn: v = f λ
•NB: Stationary waves DO NOT TRAVEL hence 
has NO SPEED.
•It also DOES NOT transfer ENERGY between 2 
points like the progressive waves

Summary
Progressive waves
Stationary waves
wavelength
λ
λ
frequency
f
f
speed
v
zero. 

Scene 12

Summary
Progressive waves
Stationary waves
wavelength
λ
λ
frequency
f
f
speed
v
zero

Test yourself 1
•A stationary wave is set up on a vibrating spring. Adjacent nodes are separated by 25cm.Determine
•a)the wavelength of the stationary wave
•b) the distance from a node to an adjacent antinode. 

Scene 13

Test yourself 1
•A stationary wave is set up on a vibrating spring. 
Adjacent nodes are separated by 25cm.Determine
•a)the wavelength of the stationary wave
•b) the distance from a node to an adjacent antinode

Scene 14

Observing stationary waves
•string attached to vibration generator driven by the signal generator.
•weight to maintain tension in string
•string vibrate with small amplitude
•adjusting frequency - produces stationary waves with bigger amplitudes. 

Scene 15

Observing stationary waves
•string attached to vibration generator 
driven by the signal generator.
•weight to maintain tension in string
•string vibrate with small amplitude
•adjusting frequency - produces stationary 
waves with bigger amplitudes

observing stationary waves
•pulley and oscillator end are nodes.
•Flashing stroboscope - to reveal motion of string that appear blurred to the eye
•Frequency of stroboscope set to match the freq of the vibrations
•see string in slow motion
•This is known as Melde’s Experiment. 

Scene 16

observing stationary waves
•pulley and oscillator end are nodes.
•Flashing stroboscope - to reveal motion of string 
that appear blurred to the eye
•Frequency of stroboscope set to match the freq of 
the vibrations
•see string in slow motion
•This is known as Melde’s Experiment

Extension of melde’s experiment
•investigate the effect of changing •i) length of the string
•ii) tension in the string
•iii) thickness of the string. 

Scene 17

Extension of melde’s 
experiment
•investigate the effect of changing 
•i) length of the string
•ii) tension in the string
•iii) thickness of the string

Test yourself
•For n=2, the vibrating section of the string is 60cm. •Determine the wavelength of the wave and the separation between two neighbouring antinodes. •The freq of vibration is increased until a stationary wave with three antinodes appears on the string
•Sketch the stationary wave •What is the wavelength of the stationary wave?. 

Scene 18

Test yourself
•For n=2, the vibrating section of the string is 60cm. 
•Determine the wavelength of the wave and the 
separation between two neighbouring antinodes. 
•The freq of vibration is increased until a stationary 
wave with three antinodes appears on the string
•Sketch the stationary wave 
•What is the wavelength of the stationary wave?

Scene 19

microwaves
•Direct the microwave transmitter at a metal plate
•Then it gets reflected back towards the source
•By moving the probe receiver around will allow use to observe the positions of low and high intensities. •These represent the nodes and antinodes respectively. 

Scene 20

microwaves
•Direct the microwave transmitter at a metal plate
•Then it gets reflected back towards the source
•By moving the probe receiver around will allow use 
to observe the positions of low and high intensities. 
•These represent the nodes and antinodes 
respectively

Microwaves
•The wavelength of the microwave can be determined from the distance between the nodes using the formula c=fλ. 

Scene 21

Microwaves
•The wavelength of the microwave can be 
determined from the distance between the nodes 
using the formula c=fλ

Scene 22

an air column closed at one end
•Glass tube open at both ends is clamped so that one end dips into a cylinder of water
•adjusting height of clamp, we are changing the length of the column of air in the tube
•hold a vibrating fork at one end
•the air column is forced to vibrate and the note becomes louder. 

Scene 23

an air column closed at one 
end
•Glass tube open at both ends is clamped so that one 
end dips into a cylinder of water
•adjusting height of clamp, we are changing the 
length of the column of air in the tube
•hold a vibrating fork at one end
•the air column is forced to vibrate and the note 
becomes louder

An air column open at one end
•This phenomenon is called resonance
•The experiment described here is the resonance tube
•The length of air column must be just right
•Air at the bottom is unable to vibrate => node. 

Scene 24

An air column open at one 
end
•This phenomenon is called resonance
•The experiment described here is the resonance 
tube
•The length of air column must be just right
•Air at the bottom is unable to vibrate => node

An air column closed at one end
•The air at the open end is able to vibrate => an antinode. •length of air column must be 1/4 λ
•alternatively can be set to 3λ/4. 

Scene 25

An air column closed at one 
end
•The air at the open end 
is able to vibrate => an 
antinode. 
•length of air column must 
be 1/4 λ
•alternatively can be set to 
3λ/4

NB
•The representation of standing sound wave is misleading. •Sound wave : longitudinal wave
•but we draw it like a transverse wave
•Copy figure on board.. 

Scene 26

NB
•The representation of standing sound wave is 
misleading. 
•Sound wave : longitudinal wave
•but we draw it like a transverse wave
•Copy figure on board.

open ended columns
•Air in open ended tubes vibrate similar to that in a closed column.
•Take an open ended tube and blow gently across the top
•Should hear a note whose pitch depends on the length of the tube
•Now cover the bottom and repeat. 

Scene 27

open ended columns
•Air in open ended tubes vibrate similar to that in a 
closed column.
•Take an open ended tube and blow gently across 
the top
•Should hear a note whose pitch depends on the 
length of the tube
•Now cover the bottom and repeat

Open ended columns
•pitch produced is now an octave higher that previous note
•means freq is approximately twice the original freq
•how is this possible? •in this situation there must be an antinode at each end. •hence the node at the midpoint. 

Scene 28

Open ended columns
•pitch produced is now an octave higher that 
previous note
•means freq is approximately twice the original freq
•how is this possible? 
•in this situation there must be an antinode at each 
end. 
•hence the node at the midpoint

open ended air columns
•tube of length, l •in closed tube: standing wave is formed at 1/4 λ
•therefore wavelength = 4l
•in open tube : 1/2 λ
•therefore wavelength = 2l
•closing a tube halves the wavelength hence the freq doubles. 

Scene 29

open ended air columns
•tube of length, l 
•in closed tube: standing wave is formed at 1/4 λ
•therefore wavelength = 4l
•in open tube : 1/2 λ
•therefore wavelength = 2l
•closing a tube halves the wavelength hence the freq 
doubles

26.3.01 Standing waves in air columns X=2L Standing wave Wavelength column closed at one end colunm open at both ends Harmonic 1st harmonic (fundamental) 2nd hannonic 3rd hannonic 1st harmonic (fundamental) 2nd hannonic 3rd Fre quency f =v/4L (natural) f = v/2L (natural) 2f n n a a "GORE 0-0 Srarxling in pipe- end a lid i b} pipe are ',viü Siandinæ u.x.e The of tnnxirnurn and an open Only the ruruhn-wntal shoon. a a a X =4ßL a a n a n = node, a = antinode. 

Scene 30

26.3.01 Standing waves in air columns
X=2L
Standing wave Wavelength
column
closed
at one
end
colunm
open
at both
ends
Harmonic
1st harmonic
(fundamental)
2nd hannonic
3rd hannonic
1st harmonic
(fundamental)
2nd hannonic
3rd
Fre quency
f =v/4L
(natural)
f = v/2L
(natural)
2f
n
n
a
a
"GORE 0-0
Srarxling in pipe- end
a lid i b}
pipe are ',viü Siandinæ
u.x.e The of
tnnxirnurn and an open Only
the ruruhn-wntal shoon.
a
a
a
X =4ßL
a
a
n
a
n = node, a = antinode

musical instruments
•production of diff notes dep on creation of stationary waves
•for stringed instruments, nodes are at the two fixed ends
•string is plucked halfway hence antinode is at the midpoint.
•This is known as the fundamental mode of vibration. 

Scene 31

musical instruments
•production of diff notes dep on creation of 
stationary waves
•for stringed instruments, nodes are at the two fixed 
ends
•string is plucked halfway hence antinode is at the 
midpoint.
•This is known as the fundamental mode of 
vibration

musical instruments
•Fundamental freq is the minimum freq of a standing wave for a given system or arrangement.. 

Scene 32

musical instruments
•Fundamental freq is the minimum freq of a standing 
wave for a given system or arrangement.

musical instruments
•by changing the length of the air column, the note can be changed. •holes can be uncovered so that air can vibrate freely.
•this gives diff pattern for nodes and antinodes
•in practice, sounds are produced made up different standing waves having diff nodes and antinodes. 

Scene 33

musical instruments
•by changing the length of the air column, the note 
can be changed. 
•holes can be uncovered so that air can vibrate freely.
•this gives diff pattern for nodes and antinodes
•in practice, sounds are produced made up different 
standing waves having diff nodes and antinodes

musical instruments
•eg guitar may vibrate with 2 antinodes along its length •give a note having twice the freq
•this is described as the harmonic of fundamental
•the skill of musician is in stimulating the string or air column to produced desired mixture of freq.. 

Scene 34

musical instruments
•eg guitar may vibrate with 2 antinodes along its 
length 
•give a note having twice the freq
•this is described as the harmonic of fundamental
•the skill of musician is in stimulating the string or air 
column to produced desired mixture of freq.

(a) Displacement of air TUBE CLOSED AT ONE END (b) Pressure variation in the air First harmonic = fundarnental Third harmonic Fifth harmonic Copyright 0 2005 Pearson Prentice Han. Inc. A First Harmonic or Fundamental - f Second Harmonic or First Overtone Third Harmonic or Second Overtone - f Fourth Harmonic or Third Overtone - f. 

Scene 35

(a) Displacement of air
TUBE CLOSED AT ONE END
(b) Pressure variation in the air
First harmonic = fundarnental
Third harmonic
Fifth harmonic
Copyright 0 2005 Pearson Prentice Han. Inc.
A
First Harmonic or Fundamental - f
Second Harmonic or First Overtone
Third Harmonic or Second Overtone - f
Fourth Harmonic or Third Overtone - f

Determining wavelength and speed of sound
•recall separation of nodes and antinodes of stationary waves are separated by 1/2 λ
•we can use eqn v=f λ •one approach is the Kundt’s dust tube
•copy diagram on the board. 

Scene 36

Determining wavelength and 
speed of sound
•recall separation of nodes and antinodes of 
stationary waves are separated by 1/2 λ
•we can use eqn v=f λ 
•one approach is the Kundt’s dust tube
•copy diagram on the board

Kundt’s dust tube
•Loudspeaker send sound waves along the inside of the tube
•sound is reflected at the closed end
•when stationary wave is established- dust at antinodes vibrate violently.
•accumulates at the nodes where movement of air is ZERO. 

Scene 37

Kundt’s dust tube
•Loudspeaker send sound waves along the inside of 
the tube
•sound is reflected at the closed end
•when stationary wave is established- dust at 
antinodes vibrate violently.
•accumulates at the nodes where movement of air is 
ZERO

kundt’s dust tube
•hence we can observe the positions of nodes and antinodes clearly
•alternative method is using same arrangement as the microwave experiment in the previous slides.
•copy diagram on the board
•loudspeaker- sound waves
•reflected by vertical board. 

Scene 38

kundt’s dust tube
•hence we can observe the positions of nodes and 
antinodes clearly
•alternative method is using same arrangement as the 
microwave experiment in the previous slides.
•copy diagram on the board
•loudspeaker- sound waves
•reflected by vertical board

•microphone detects sound wave
•output displayed on the oscilloscope
•turn off the time base of the oscilloscope so that the spot no longer moves across the screen •spot moves up and down •height measures intensity of sound
•moving the mic help to detect N and A.
•max accuracy DO NOT measure separation of adjacent nodes
•but measure distance across several nodes. 

Scene 39

•microphone detects sound wave
•output displayed on the oscilloscope
•turn off the time base of the oscilloscope so that the 
spot no longer moves across the screen 
•spot moves up and down 
•height measures intensity of sound
•moving the mic help to detect N and A.
•max accuracy DO NOT measure separation of 
adjacent nodes
•but measure distance across several nodes

•resonance tube expt can also determine the λ & v
•however, need to take account of systematic error in the expt.. 

Scene 40

•resonance tube expt can also determine the λ & v
•however, need to take account of systematic error in 
the expt.

Eliminating errors
•look at how the standing waves in the tube are shown
•A is at the open end
•slightly extending beyond the tube
•reason: expt shows that the air slightly beyond the tube vibrates as part of the stationary wave. 

Scene 41

Eliminating errors
•look at how the standing 
waves in the tube are 
shown
•A is at the open end
•slightly extending beyond 
the tube
•reason: expt shows that the 
air slightly beyond the tube 
vibrates as part of the 
stationary wave

Eliminating errors
•A is at a distance c beyond the end of the tube
•c is called the end correction
•we do not know c
•it cannot be measured directly
•however we can write
•for shorter tube = λ/4 = l1 + c
•for longer tube = 3 λ/4 = l2 + c. 

Scene 42

Eliminating errors
•A is at a distance c beyond the end of the tube
•c is called the end correction
•we do not know c
•it cannot be measured directly
•however we can write
•for shorter tube = λ/4 = l1 + c
•for longer tube = 3 λ/4 =  l2 + c

eliminating errors
•show on the board
•We can make 2 measurements for l1 and l2 and obtain accurate value of λ
•end correction c is the example of systematic error. 

Scene 43

eliminating errors
•show on the board
•We can make 2 measurements for l1 and l2 and 
obtain accurate value of λ
•end correction c is the example of systematic error

summary •stationary waves are formed when two identical waves travelling at opposite directions meet and superimpose. •this usually happens when one wave is a reflection of the other
•a stationary wave has a characteristic pattern of nodes and antinodes
•a. 

Scene 44

summary 
•stationary waves are formed when two identical 
waves travelling at opposite directions meet and 
superimpose. 
•this usually happens when one wave is a reflection 
of the other
•a stationary wave has a characteristic pattern of 
nodes and antinodes
•a

summary
•a node is a point where the amplitude is always zero
•an antinode is the point of maximum amplitude
•adjacent nodes or antinodes are separated by distance equal to half wavelength
•use v=fλ to determine speed or freq. •wavelength can be found using nodes and antinodes of stationary wave pattern.. 

Scene 45

summary
•a node is  a point where the amplitude is always zero
•an antinode is the point of maximum amplitude
•adjacent nodes or antinodes are separated by distance 
equal to half wavelength
•use v=fλ to determine speed or freq. 
•wavelength can be found using nodes and antinodes of 
stationary wave pattern.