WAVES
CONTENT STATEMENT 3.1 Nos. 1 - 13
Useful Web Address for additional information:
www. ph.ed.ac.uk
Select “Resources for Schools”
Select “Wave Phenomena” in the Advanced Higher
paragraph
Select “Hyperphysics Project”
Waves Page 1
WAVE PHENOMENA
WHAT IS WAVE MOTION?
When a wave is moving energy is transferred from one position to another with no net
transport of mass.
What Does This Mean?
Consider the example of water waves where each water particle moves at right angles
to the direction of travel of the wave. This is illustrated below:
Direction of the water particles
Direction of the wave
As the wave travels each water particle is displaced by the same distance
perpendicular to the direction of the wave.
The result is that the water particles themselves do not travel with the wave.
What is seen is the surface of the water bobbing up and down (in the absence of any
wind or tide).
DEFINITIONS
Amplitude (a)
wavelength (

Amplitude (a)
This is the maximum displacement of the medium from the undisturbed position.
Wavelength (
This is the distance between one point on a wave and the same point on the next wave
as along as there is a constant phase distance between the waves.
Frequency (f)
This is the number of waves passing a given point in one second.
Speed (v)
Distance travelled by the wave in one second. (s / t)
Waves Page 2
WAVE EQUATION
v = f 
where:
v is the speed of the wave measured in ms-1
f is the frequency of the wave measured in Hertz (Hz)
is the wavelength of the wave measured in metres (m)
INTENSITY OF A WAVE
By definition the intensity of a wave is directly proportional to its amplitude squared.
i.e.
I  a2
where
I is the intensity measured Watts per metre squared (Wm-2)
a is the amplitude measured in metres (m)
Waves Page 3
TRAVELLING WAVE
A travelling or progressive wave consists of disturbance moving from a source to
surrounding places as a result of which energy is transferred from one point to another
with no net transfer of mass.
There are two types of travelling wave:
1.
Transverse Wave
Direction of Travel
Direction of Disturbance
2.
Longitudinal Wave
Direction of Disturbance
Direction of Travel
This is shown using a slinky.
Waves can be either mechanical or electromagnetic
Mechanical waves are disturbances that occur in materials that have mass and
elasticity. e.g. water waves
Electromagnetic waves are disturbances that consist of varying electric or magnetic
fields. e.g. light and microwaves
Waves Page 4
Wave Superposition
The principle of superposition states:
If a number of waves are travelling through a medium, then the resulting disturbance
at a point in the medium is the algebraic sum of the individual disturbances produced
by each wave at that point.
As a consequence of this, waves can pass through one another without being affected
in any way. For example if two stones are dropped into a pool of calm water, two sets
of circular waves are produced. These two waves pass through each other, but at any
particular point in time the disturbance at that point is the algebraic sum e.g. if a
crest of one wave meets a trough of another wave then the water will remain calm at
that point.
Sine and Cosine waveforms are the simplest mathematical forms of a periodic wave.
A periodic wave is one that repeats itself at regular intervals.
It is possible to show that any periodic waveform can be made up by the superposition
of an infinite series of sine or cosine waves. These are called FOURIER SERIES.
Do not panic – this is just name – you will not be asked to explain
For example a “saw tooth” wave can be expressed as a series of individual sine
waves as follows:
y(t) = -(1/)Sint - (1/2)Sin2t -(1/3)Sin3t - … this is an example of a Fourier
Series
The more terms that are added together the closer the waveform gets to being a “saw
tooth”
Waves Page 5
Example of Producing One Waveform From An Infinite Sum Of Sine Wave
-A Fourier Series.
Please accept the following as the truth…you will not be doing Fourier Series in mathematics.
a
1
C
A
2
3
B
Diagram 1 is a square wave of
period x0 and amplitude “a” This
can be formed by adding sine
waves together. The wave could,
very approximately, be
represented by a sine wave (dotted
line on diagram 1) of the same
period, but slightly larger
amplitude. (So that the average
amplitude is roughly the same for
each.)
By adding a little to the sine wave
in regions A and B and by taking
away a little in region C a better
approximation could be obtained.
This is done by adding a small
amplitude sine wave with period
x03 as illustrated in diagram 2.
The result of adding the two sine
waves from diagrams 1 and 2 is
shown in diagram 3.
If the additional sine wave had a period of x02 (this means more waves in the same
time compared to the wave of period x03) then this would not lead to a better
approximation in the above example as this would add on in region A and take away
in region B…not what is required.
The addition of an even smaller amplitude sine wave with a period of x05 does
improve the approximation. Mathematically this is represented by the infinite series:
(4a) [Sin(2x/x0) +(1/3)Sin3(2x/x0) + (1/5)Sin5(2x/x0) +…] the Fourier series for
the square wave.
This ability to produce complex waveforms from sine or cosine waves is exploited in
analogue musical synthesisers. In principle the different frequencies are derived from
quartz crystal-controlled oscillators, the amplitudes suitably adjusted and the signals
added together by a summing operational amplifier (V0 = (V1+V2)Rf / R1)…remember
these from Higher!!
Waves Page 6
PHASE DIFFERENCE
y
A
x
y
B
x
Waves A and B are identical in that they are both sinusoidal, have the same amplitude
and the same frequency. BUT they are out of phase. This means that corresponding
points of the wave are not at the same positions at the same time.
As a result of this it can be said that if…
 Wave A can be described as
y = aSin
 Then Wave B can be described as
y = aSin ( - ) for a wave travelling in the positive direction
y = aSin ( + ) for a wave travelling in the negative direction
Where  = phase angle
If  = 900
then y = aCos
0
If  = 180
then y = -aSin
If  = a multiple of 3600 then Sin (  ) = Sin and the waves are said to be in
phase.
Waves Page 7
Phase Difference
0
/2

3/2
2
Separation of Points
0
/4
/2
3/4

In general:
Phase difference
=
Separation of points
2

=
a constant
Therefore:

x
=

=
2

giving
2 x

Where:
 is the phase angle (difference)measured in radians
x is the separation between two points on a wave measured in metres
 is the wavelength measured in metres
Waves Page 8
DERIVATION OF THE WAVE EQUATION FOR A TRAVELLING WAVE
The wave equation describes the motion of a particle in the medium through which
the wave is travelling.
As an example we shall look at the motion of a particle in a transverse wave. We shall
assume that the displacement of the particle varies with time. That means that it
exhibits S.H.M. (Simple Harmonic Motion)
Definition of SHM is that the displacement of the particle varies with time. See later
notes.
y
a
x
-a
y
=
aSin
As the y displacement of a particle depends on its x position and time t, then  must
be a function of x and t.
i.e.  =
f(x,t)
Waves Page 9
The diagram below illustrates the waveform at a particular instant in time (solid line)
and the waveform an instant later (dotted line). To do this we are assuming that the
wave is moving in a positive direction.
y

a

a
x
A
x
A
B
B
The particles A and B and their associated rotating vectors are shown in the above
diagram at time t = 0 which is their initial positions.
After a time of t the vectors will have rotated through an angle  given by
 = t
For an alternate method please see two pages on.
The y displacement for A is given as
y = aSint
The particle at B, at a displacement x from the origin will “lag” A by a phase angle 
and its y displacement is given by
y = aSin(t - )
*
When the displacement x of particle B from the origin equals one wavelength then the
phase angle  equals 2. See page 8
From page 8:
In general the phase difference between points equals the point separation
multiplied by 2/.

=
Waves Page 10
2x

Substitute this value into * gives
y = a Sin (t - 2x)

y = a Sin (2ft - 2x)

and as  = 2f this gives
hence
y = a Sin 2(ft - x )

Where:
 y is the y displacement of the particle in the medium(in the transverse direction)
 a is the amplitude of the wave
  is the wavelength of the wave
 x is the displacement of the particle from the origin
 f is the frequency of the wave
 v is the velocity of the wave
 t is the time
Please note:
The minus sign indicates that the wave is travelling in the positive x direction.
If the wave were travelling in the negative x direction then the equation would
be:
y = a Sin 2(ft + x )

Important Information
For the purpose of the exams you must be able to explain how the above
relationship represents a travelling wave. You are NOT required to derive it.
You will be given the equation and will have to state what each letter represents.
(see list above)
Please Note:
 v = f is the equation for the velocity of the wave as it moves through the
medium
 Using the above equation we can calculate the velocity of a disturbed particle as
the wave travels through a medium using dy/dx where y = a Sin 2(ft + (x/) )
 2ft illustrates that SHM is involved
 2x/ is the phase difference
Waves Page 11
Alternative Method

=
d
dt
 d
=
 d dt

=
t + c
For A when t = 0,  = 0 and so c =0
Why?
If t = 0, then t = 0. Also as  is the angle through which the vector will rotate after
time zero, this means that at time zero  = 0
Therefore as y = aSint and t=  - c, this means that c = 0.
For B when t = 0,  = - and therefore c = -
Why?
If t = 0, then t = 0. Also as  is the angle through which the vector will rotate after
time zero, this means that at time zero  = - as that is the angle that B lags behind A.
Therefore as y = aSin(t- ) and t=  - c, this means that c = -.
Hence 
Waves Page 12
=
t - 
In the example on page 10

There was a phase difference between waveforms of the same travelling wave but
at different displacements in the direction of motion.
Did you know?
Many a.c. generators have coils three coils that rotate in a magnetic field. The result
of this is that there are three completely separate (a.c. voltage outputs) waveforms,
each 1200 out of phase with each other.
Worked Example
A travelling wave has a wavelength of 60 mm. A point P on the wave is 75 mm away
from the origin and point Q is 130 mm away from the origin.
(a) What is the phase difference between P and Q?
(b) Which of the following statements best describe this phase difference
almost completely out of phase
roughly ¼ cycle out of phase
almost in phase?
See page 8
(a)
Separation of points = 130 - 75 = 55mm = 0.055m = x
Phase difference = 2x /  = 2 x 0.055 = 5.76 radians
0.060
(b)
P and Q are separated by 55 mm which is approximately one wavelength.
Therefore they are almost in phase.
Also note that 5.76 radians = 3300 (2 radians = 3600) and 3300 is close to
3600 hence we can make the claim that they are almost in phase.
Waves Page 13
Stationary Waves (aka Standing Waves)
Stationary waves result from the superposition (interference) of two waves of equal
frequency (amplitude) and speed travelling in opposite directions.
A stationary wave moves neither to the right or to the left and the wave “crests”
remain fixed while the particle displacements increase and decrease in unison.
Example
Look at the superposition of an incident wave with its reflected wave. To keep the
example simple the amplitude of the incident and reflected waves will stay the same.
y1 = aSin2(ft - x/)
y2 = aSin2(ft + x/)
Reflector
The superposition of these two waves at some point “x” from the origin is given by
y = y1 + y2
y = aSin2(ft - x/) + aSin2(ft + x/)
y = 2aSin2ft Cos (2x/)
(see further explanation on following page)
Why?
Remember SinA + Sin B = 2Sin ½(A+B) Cos ½(A-B)
Please note that the above is for information and is not required in the exams.
Waves Page 14
Explanation of the Terms In The Equation y = 2aSin2ft Cos (2x/)
2a Cos (2x/) is a constant for a given value of “x”. It defines the amplitude of the
particular oscillating particle.
This amplitude will vary with x, from a value of zero (at a node) to the value of 2a
(at an antinode). This is shown in the diagram below.
Sin2ft (=Sin t) shows that the particle is oscillating with Simple Harmonic Motion
(S.H.M.)
Do not forget that:
Nodes are positions of zero or minimum amplitude and occur every half a wavelength
Antinodes are positions of maximum amplitude (displacement either positive or
negative from zero or minimum amplitude) and occur every half a wavelength.
Examples of Antinodes
Examples of Nodes
2a
x
-2a
Examples of Antinodes
Waves Page 15
EXAMPLES OF STANDING WAVES
1.Microwaves
Source
Detector
Reflector
As the detector is moved towards the reflector then as was seen in Higher the readings
on the detector will go from minimum to maximum to minimum to maximum etc
It is the position of the detector with respect to the reflector will determine the size of
the nodes and antinodes.
The biggest contrast between nodes and antinodes is seen when the detector is placed
next to the reflector
This is because next to the reflector the reflected wave is the same magnitude as the
incoming wave (They are 1800 out of phase due to the phase change on reflection). As
the reflected wave travels away from the reflector it gets smaller in size (as it loses
energy) and so will not exactly cancel out the incoming wave. This best illustrated
using the graph below.
reading
Position from detector
Waves Page 16
2. A Stretched Elastic Cord
Pulley
Signal
Generator
Mechanical
Oscillator
Hanging Mass
(300 / 400 g)
The mechanical oscillator produces standing waves in the elastic cord by using waves
reflected from the pulley.
The reflected waves and the transmitted waves “pass over” each other creating a
standing wave.
Waves Page 17
3. Sound Waves
To be carried out if there is sufficient time.
Hollow Tube (can be made of cardboard)
Voltmeter
Microphone
Signal
Generator
This tube is known as Kundt’s Tube
As the microphone is moved down the tube the reading on the voltmeter goes from
positive reading to approx. zero to a negative reading to approximately zero to a
positive reading etc
The sound waves are reflected off of the ends of the hollow tube. You get a maximum
on reflection at the end of an open tube. This is due to the resonance effects at the rim.
(This is not covered in this course) This explains why open ended organ pipes can
give a different note to organ pipes which are covered at the end.
The echoes of sound waves from a wall could produce standing waves as there is a
change in phase when waves are reflected off of a wall. This effect is the same as the
one we have already observed using microwaves.
Waves Page 18
The Doppler Effect
The Doppler Effect is the change in frequency observed when a source of sound
waves is moving relative to an observer.
A good example of this is an ambulance’s siren as it drives past.
In general more sound waves are received per second when the source of sound
waves is moved towards the observer and so the frequency heard by the observer
is increased.
Similarly less sound waves are received per second when the source of sound
waves is moved move away from the observer and so the frequency heard by the
observer is decreased.
The Doppler Effect can happen because of one of two events. However consider first
the picture if the….
The Sound Source and The Observer Are Stationary
The diagram below shows the wave fronts of the sound waves that have been emitted
from a stationary source of sound S, as they spread out towards the stationary
observer at point O. The sound will travel at the speed appropriate to the medium
through which they are travelling. e.g. 340ms-1 in air
Please note the diagram below is trying to illustrate is a series of concentric circles,
whose radii increase by the same amount each time.
S
Waves Page 19
O
Looking more closely at what happens between the source S and the observer O. The
source of the sound waves has a frequency f and velocity v. The diagram below shows
only part of the wave fronts.
v is the distance covered in one second if the
waves are travelling at v ms-1
(distance = velocity x time)
O
S
f waves every one second, travelling
at v ms-1
Waves Page 20
Now if…
The Sound Source Is Moving Towards A Stationary Observer
Remember sound will radiate out in all directions (as shown in the previous diagram),
but to simplify the diagram below only the sound waves arriving directly at the
observer have been drawn.
v- vs is the distance covered in one second if
the waves are travelling at v ms-1 and the
source is travelling at vs ms-1
S
O
vs
f waves every one second, travelling
at v ms-1
Waves Page 21
Derivation Of The Doppler Effect Formula With A Moving Source
Please note this derivation is required for the exams.
If the source S is moving towards the observer O, with a velocity vs, the f waves per
second are compressed into a smaller distance (v – vs).
Why?
 Distance equals velocity x time and in this case we are using time equal to one
second.
 Sound waves travel a distance of v every one second.
 The source will move a distance, vs, towards the observer every one second.
Hence the observed wavelength  is given by

=
distance covered by the waves
number of waves
=
v - vs
fs
Remember…this is per second.
Using  = v/f
v
f
=
v - vs
fs
=
fs v
(v - vs)
Giving
f
Where:
f is the apparent frequency measured in Hertz
fs is the frequency of the source of the sound (Hz)
v is the velocity of the sound waves in the medium through which it is travelling(ms-1)
vs is the velocity of the source of the sound waves.(ms-1)
Hence if the source is moving towards the observer the apparent frequency is
greater than the true frequency.
e.g. If fs = 1000Hz, v = 340 ms-1 and vs = 40 ms-1 then…
f = 1000 x 340 = 1133 Hz
300
So f  fs
Waves Page 22
If the source is moving away from the observer then the observed wavelength will be
bigger than the true wavelength. Therefore

=
v + vs
fs
Giving :
f
=
fs v
(v + vs)
In general, for a moving source of sound waves, the observed frequency will be
written as:
f
=
fs v
(v + vs)
Where
f is the observed frequency measured in Hertz (Hz)
fs is the frequency of the source (Hz)
v is the velocity of sound in the medium (ms-1)
vs is the velocity of the source through the medium (ms-1)
Please note that whether you use + or - will be determined by the particular problem.
Read them and use your common sense…it will work honestly !!
If the source of sound is approaching the observer with a constant velocity, then the
apparent pitch of the note heard does not alter as the note gets closer. It remains
constant but higher than the true pitch.
Waves Page 23
And if…
The Observer Is Moving Towards A Stationary Source of Sound Waves
Remember sound will radiate out in all directions (as shown in the previous diagram),
but to simplify the diagram below only the sound waves arriving directly at the
observer have been drawn.
v+ vo is the distance covered in one second if
the waves are travelling at v ms-1 and the
observer is moving towards the source at
vo ms-1
O
S
vo
f waves every one second, travelling
at v ms-1
If the observer is moving then the wavelength is unchanged and will equal v/f where v
is the velocity of the sound in the medium.
Why?
 There are f waves sent every one second by the stationary source.
 These f waves occupy a distance equal to v metres.
If the observer is moving towards the source of the sound waves with a velocity vo
then the velocity of the waves relative to the observer will be v + vo
Think about it… if the observer is going towards the sound source then the sound
waves will reach the observer in less time. Hence the apparent velocity must be
larger.
Waves Page 24
Derivation Of The Doppler Effect Formula With A Moving Observer
Please note this derivation is required for the exams.
In general frequency =
no. of waves
time
Therefore the number of waves received in a time t is given by
No. of waves =
ft
=
(v/)t
If the observer is moving towards the source at a velocity vo then an extra (vo/)t
waves will be heard in that time.
The apparent velocity of the sound waves reaching the observer will be
(v + vo).
Hence the observed or apparent number of waves reaching the observer in time t is
given by
No. of waves =
=
No. of waves =
t
(v/)t + (vo/)t
(v + vo) t

(v + vo)

Remember
  = v/f
 no of waves/time is the frequency
Hence the apparent frequency is given by
f
f
=
=
v + vo

giving
fs (v + vo)
v
Also if the observer is moving away from the source of the sound waves then the
observed frequency will be smaller and given as:
f
=
Waves Page 25
fs (v - vo)
v
In general, for a moving observer, the observed frequency will be written as
f
=
fs (v  vo)
v
Where
f is the observed frequency measured in Hertz (Hz)
fs is the frequency of the source (Hz)
v is the velocity of sound in the medium (ms-1)
vo is the velocity of the observer through the medium(ms-1)
A thought for today
 The Doppler Effect also occurs for electromagnetic radiation and can be seen as
the “Doppler Red Shift”. However the above equations cannot be used when
calculating the velocity at which a star is moving away from the earth. This is
because the speed of light is the same for all observers and does not depend on the
medium through which it is travelling. The Doppler Red Shift is one of the main
pieces of evidence to show that the universe is expanding … because it shows that
the stars are moving away.
 The Doppler Effect also occurs for reflected waves but once again the formula is
different. This effect is used in:
speed monitoring to catch speeding motorists. To do this, the police use
microwaves that reflected off of moving vehicles to determine the speed of the
vehicle. The true speed of the vehicle is then calculated using the Doppler effect
and can be used whether the vehicle is moving away from or towards the radar
gun. If this calculation is not done then the measured speed of the vehicle will not
be the true value.
detection of blood clots. This done by including the Doppler effect in calculations
when measuring the blood flowing in body by using ultrasound reflections from
the arteries.
ships’ radar for tracking. This done by using a rotating aerial to sweep
continuously a narrow beam of microwaves through 3600. The pulses reflected
from land, buoys and other ships are then displayed on a CRO called a “plan
position indicator”(PPI). Calculations whenever the ship or other ships in the area
are moving must include the Doppler effect to give accurate positioning.
Waves Page 26
Examples
Use speed of sound in air = 340 ms-1
1. A car is being driven towards a stationary pedestrian at 28.5 ms-1. The car driver
sounds his horn, which produces a note with a frequency of 470 Hz. Calculate the
frequency of the note heard by the pedestrian.
What situation are we looking at here?
Is the…
 source moving away from the stationary observer?
 observer moving towards a stationary source?
 observer moving away from a stationary source?
 source moving towards the stationary observer?
2. The siren of a fire engine is Doppler-shifted from 560 Hz to 464 Hz as it drives
away at constant speed from a stationary observer. Calculate how fast the fire
engine is travelling.
What situation are we looking at here?
Is the…
 source moving away from the stationary observer?
 observer moving towards a stationary source?
 observer moving away from a stationary source?
 source moving towards the stationary observer?
Waves Page 27
3. A car travelling away from a stationary source of sound waves hears a note at 285
Hz. If the source is emitting sound waves at 320 Hz, calculate how fast the car is
travelling.
What situation are we looking at here?
Is the…
 source moving away from the stationary observer?
 observer moving towards a stationary source?
 observer moving away from a stationary source?
 source moving towards the stationary observer?
4. A car travelling at a constant speed passes a stationary pedestrian as the driver
sounds the horn. As the car approaches the pedestrian hears the horn sounding at
634 Hz. Once the car has driven past the sound is heard as 578 Hz.
Calculate the speed of the car and the frequency of the sound waves emitted by the
car horn.
What situation are we looking at here?
Is the…
 source moving away from the stationary observer?
 observer moving towards a stationary source?
 observer moving away from a stationary source?
 source moving towards the stationary observer?
Waves Page 28