2.3Wave Motion

8/9/2019 2.3Wave Motion

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Saman AravindaBsc (Hons Physical Science) Edexcel Physics

Wave motion

Waves carry energy from one place to another. Think about the energy released

when waves break onto a beach or the immense energy carried along by a tidal

wave or tsunami. These waves make the material in which they are travellingvibrate as they pass through it. Experiments with the slinky spring and the water

tank will have shown you that vibrations make waves. The individual particles of the

material do not move along, they simply vibrate as the wave energy is passed to

them.

For example a fishing float or a cork will simply bob up and down as a water wave

passes it – the float does not move along in the direction that the wave is moving.Waves in the sea are uite complicated especially in shallow water. !ecause of the

friction between the water and the sea bed the water particles move in circles and

that is why waves "break" as the

depth of the water gets less andless.

# wave motion is the transmission

of energy from one place to

another through a material or avacuum. Wave motion may occur

in many forms such as waterwaves, sound waves, radio waves

and light waves, but the waves

are basically of only two types$

(a)Transverse waves % the oscillation is at right angles to the direction of

propagation of the wave &Figure '&a((. Examples of this type are water

waves and most electromagnetic waves.

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(b) Longitudinal waves % the oscillation is along the direction of propagation of the wave &Figure '&b((. #n example of this type is sound

waves.

!asic definitions$

Wavelength: the distance between any two successive corresponding points on the

wave, for example that between two maxima or two minima (λ)

Displacement

:

the distance from the mean, central, undisturbed (y)

Amplitude: the maximum displacement (a)

Frequency: the number of vibrations per second made by the wave (f)

Period: the time taken for one complete oscillation (T= 1f)

Phase: a term related to the displacement at !ero time (")

!asic Wave euationThe velocity, freuency and wavelength of a wave are given by the formula$

Wave velocity (v )=frequency of waves( f ) x wavelength of waves( λ)

Example Problems

'. )f a water ripple has a wavelength of * cm and a freuency of + - what

is its wave speed///////////////////////////////////////////////

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*. )f a note played on a guitar has a freuency of 001 - what is its

wavelength &2elocity of sound 3 441 m5s(///////////////////////////////////////////////

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Wave euations

Euation of a sinusoidal wave

The graph represents a sinusoidal wave with displacement y at time t ,

vibrating at a freuency f and amplitude a. The motion can be described by

the euation$

We can understand how this euation is constructed by introducing 6

&omega( , the angular velocity &units rad s%' (.

7ubstituting for *8f, our euation then becomes,

The diagram shows how the value of the function &y( is calculated from the

radius &a % red( of the circle sweeping out an angle 9 &theta(.

#ngle swept out 3 angular velocity x time of sweep

9 3 6t

From simple trigonometry, the value of y' &green( is eual to asin9.

The angle swept out at time t' is 6t' where 6 is the angular velocity. This is

a measure of the rotation of the a%vector in radians per second.

7o the value y' at time t' is given by$

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We will consider here the motion of a sine wave &Figure *(, since this type is

the most fundamental. owever it can be shown that any other wave maybe built up from a series of sine waves of differing freuency.

We can express a wave

travelling in the positive xdirection by the euation$

Positive x direction: y=a sin(wt –kx )

and for one travelling in the opposite direction$

Negative x direction : y=asin (wt +kx)

Where k is a constant and ω 3 *πf.

The sign gives the direction of the motion. We can separate each euation

into two terms$

&a( # term showing the variation of displacement with time at a particular

place % for example, when x 3 1 y 3 a sin &ωt(, that is, the variation of

displacement with time at the particular place x 3 1.

&b( # term showing the variation of displacement with distance at a

particular time % for example, when t 3 1 y 3 a sin &kx(, that is, thevariation of displacement with distance at a particular time t 3 1.

#n alternative form of the euation can be proved as follows.

7ince the period T 3 '5f where f is the freuency and ω 3 *πf we have

ω 3*π 5T.

#lso when t 3 1 y 3 1 at x 3 1, λ 5*, λ...and so on, and so k 3 *πλ. The

euation may therefore be written$

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y=asin 2 p(t /T + x / l)

Example problem

# certain travelling wave has freuency &f( of *11 -, a wavelength &λ( of *m and

amplitude &a( of 1.1* m.

:alculate the displacement &y( at a point 1.4m from the origin at a time 1.1's after

-ero displacement at that point. &#ns. 1.1'+ m(

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7uperposition7uperposition is when two waves are superimposed on each other and add up. The

phenomenon is described by the ;rinciple of 7uperposition, which states$

When two waves are travelling in the same direction and speed, at any

point on the combined wave the total displacement of any particle equalsthe vector sum of displacements of the waves.

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Look carefully at the diagram. The blue and red displacements add up

algebraically.

Above the line is positive. elo! the line is

negative.

"ence a red displacement# above the line# on top of ablue displacement $of e%ual magnitude& belo!

the line# !ill cancel out. This produces a point

on the hori'ontal a(is.

A red displacement# above the line# on top of a blue

displacement# also above the line# !ill produce a

displacement above the line e%ual to their sum.

A red displacement# belo! the line# belo! a blue displacement# also

belo! the line# !ill produce a displacement belo! the line e%ual to their

sum.

;hase differenceThe phase difference of two waves is the

hori-ontal distance a similar part of one

wave leads or lags the other wave.

;hase difference is measured in fractions of a

wavelength, degrees or radians.

)n the diagram &above(, the phase difference is ¼ λ.

This translates to 90o &< of 4+1o( or π/2 &< of *8(.

7tanding waves# stationary or standing wave is one in which the amplitude varies from

place to place along the wave. Figure ' is a diagram of a stationary wave.=ote that there are places where the amplitude is -ero and, halfway

between, places where the amplitude is a maximum> these are knownasnodes &labelled =( and antinodes &labelled #( respectively.

&7ee Figure '(

# node is a place of -ero amplitude

#n antinode is a place of maximum amplitude

)




The distance between successive nodes, and successive antinodes, is half a

wavelength. &?5*(

The amplitude of the points on a stationary wave varies along the wave. )n

Figure ' the amplitude at point ' is a', that at point * is a* and that at point4 is a4. The displacement &y( at these points varies with time.

#ny stationary wave can be formed by the addition of two travelling waves

moving in opposite directions.

# wave moving in one direction reflects at a barrier and interferes with theincoming wave.

@athematical treatment of the formation of a standing

wave from two travelling waves:onsider two travelling waves ' and *. Aet the displacements at time t and

position x be y' and y*.

y' 3 a sin &6t % kx( &say right% left(

y* 3 a sin &6t B kx( &say left% right(

Therefore$

y1+ y 2=a sin(wt −kx)+a sin(ωt +kx)=2a sin (ωt ) .cos(kx)= Asin(wt )

=ote that this expression is composed of two terms$

*




&a( sin &ωt( % this shows a varying amplitude with time at a particular place.

&b( cos &kx( % this shows a varying amplitude with position at a particular

time.

When x 3 1, λ 5* ... # is a maximum and we have an antinode>

When x 3 ?50, 4λ 50, Cλ 50 ... # is a minimum and we have a node.

=otice that the maximum value of # is *a.

Student investigationThe standing waves on a string may be studied using an experiment due to @elde

and shown in Figure *&a(. # 1.C m long rubber cord of 4 mm* cross%sections is

clamped at one end, stretched to twice its length and fixed to a vibrator. Thevibration generator is connected to a signal generator which can give a range of

freuencies between '1 - and '11 -.

Dbserve the effects of slowly increasing the vibrator freuency from '1 - to '11- and explain what you see. The experiment is best done in a darkened laboratory

with the cord illuminated with a stroboscope.

7tanding waves may also be investigated using the *. cm wavelength microwave

euipment used by most schools &Figure *&b((.

The transmitter is set up facing a vertical metal plate about C1 cm away. )f a probe

detector connected to a meter is moved along the line between transmitter andplate a series of nodes and antinodes can be found. The distance between

successive nodes or antinodes is half the wavelength of the microwaves.

+




The ;hysics of sound in tubesThis part of the ;hysics of sound is the basis of all wind instruments, fromthe piccolo to the organ. !asically the ideas are very simple but they can

become complex for a specific musical instrument. For that reason we will

confine ourselves to a general treatment of the production of a note from auniform tube.

,




The stationary waves set up by the vibrations

of the air molecules within the tube are dueto the sum of two travelling waves moving

down the tube in opposite directions. Dne of these is the initial wave and the other its

reflection from the end of the tube.

#ll air%filled tubes have a resonant freuency

and if the air inside them can be made tooscillate they will give out a note at this

freuency. This is known as the fundamental

freuency or first harmonic.

igher harmonics or overtones may also be

obtained and it is the presence of theseharmonics that gives each instrument itsindividual uality. # note played on a flute will

be uite unlike one of exactly the same pitchplayed on a bassoon

# harmonic is a note whose freuency is anintegral multiple of the particular tube"s or string"s fundamental freuency.

Tubes in musical instruments are of two types$

&a( Dpen at both ends, or&b( Dpen at one end and closed at the other.

The vibration of the air columns of these types of tube in their fundamental

mode are shown in Figure '. =otice that the tubes have areas of no vibrationor nodes at their closed ends and areas of maximum vibration or antinodes

at their open ends.

#ntinodes also occur at the centre of a tube closed at both ends in this

mode.

=odes are areas where the velocity of the molecules is effectively -ero butwhere there is a maximum variation in pressure, while the reverse is true for

antinodes.

7ome of the higher harmonics for the different tubes are shown in Figure *.

1-




=otice that a closed tube gives odd%numbered harmonics only, while the

open tube will give both odd and even%numbered.

=otice that although the sound waves in the tubes are longitudinal it isconventional to represent them as transverse vibrations for simplicity.

owever two examples of what are really going on is given for

completeness.

=orth :urry village church organ

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)f the velocity of sound is denoted by v and the length of a tube by A, thenfor a tube closed at one end the fundamental freuency is given by$

unda!entalfrequency (f )=v /wavelength=v /4 "

For a tube open at both ends the fundamental freuency is given by$

unda!ental frequency (f )=v /wavelength=v /2 "

End correctionsThe vibrations within the tube will be transmitted to the air Gust outside the

tube, and the air will then also vibrate. )n accurate work we must also allow

for this effect, by making an end correction &Figure 0(.

This means that we consider that the tube is effectively longer than itsmeasured length by an amount d, that is$

The true length 3 A H d. The euation for aclosed tube then becomes$

f = v

4 [ "+d ]

@easurement of the velocity of soundThe velocity of sound in air may be found uite simply by using theresonance of a column of air in a tube. #n open% ended tube is placed in a

glass cylinder containing water, as shown in Figure 0, so that the watercloses the bottom end of the tube.

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# tuning fork of known freuency is sounded over theupper end, the air in the tube vibrates and a note is

heard. The length of the air column is adGusted byraising the tube out of the water until a point is found

where resonance occurs and a loud note is produced.#t this point the freuency of the tuning fork is eualto the resonant freuency of the tube.

)n its fundamental mode the wavelength # is four

times the length of the air column &A(, that is$ λ3 0A

7ince velocity 3 freuency x wavelength the velocity of

sound may be found. For accurate determinations thefollowing precautions should be taken$

&a( The temperature of the air should be taken, sincethe velocity of sound is temperature%dependent, and

&b( The end correction should be allowed for. This may

be done by finding the resonance for the second harmonic with the sametuning fork.

Example problems

'. # '.C m tube is open at both ends. What is$

&a(The freuency of the fundamental &first harmonic( &''1 -(

&b(The wavelength of the first overtone &second harmonic(

Take the speed of sound in the air in the tube to be 441 ms%'. &0. m(

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2ibrating strings

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)f a string stretched between two points is plucked it

vibrates, and a wave travels along the string. 7incethe vibrations are from side to side the wave is

transverse. The velocity of the wave along the stringcan be found as follows.

2elocity of waves along a stretched

string

#ssume that the velocity of the wave v depends upon

&a( the tension in the string &T(,&b( the mass of the string &@( and

&c( the length of the string &A( &see Figure '(.Therefore$ v 3 kTx@yA-

7olving this gives x 3 I , - 3 I , y 3 % I .The constant k can be shown to be eual to ' in this case and we write m as

the mass per unit length where m 3 @5A. The formula therefore becomes$

velocity of waves on a stretched string=√ T

!

7ince velocity 3 freuency x wavelength

requency of a vi#rating string= λ √ [T /! ]

Example problems

'. :alculate the fundamental freuency for a string 1.0C m long, of mass 1.C

gm5metre and a tension of JC =. &041 -(

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*. What is the tension needed to give a note of 'k- using a string of length 1.C m

and mass 1.JC gm. &4 k=(

The ;hysics of vibrating strings# string is fixed between two points. )f the centre of the string is plucked

vibrations move out in opposite directions along the string. This causes atransverse wave to travel along the string. The pulses travel outwards along

the string and when they reaches each end of the string they are reflected&see Figure *(.

The two travelling waves then interfere with each other to produce astanding wave in the string. )n the fundamental mode of vibration there are

points of no vibration or nodes at each end of the string and a point ofmaximum vibration or antinode at the centre.

=otice that there is a phase change when the pulse reflects at each end of

the string.

The first three harmonics for a vibrating string are shown in the following

diagrams.

&a( #s has already been shown> for a string of length A and mass per unit

length m under a tension T the fundamental freuency is given by$

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requency (f )=( 12 " )√T

!

&b( First overtone or second harmonic$

requency (f )=( 1 " )√ T

!

&c( 7econd overtone or third harmonic$

requency (f )=( 32 " )√T

!

# string can be made to vibrate in a selected

harmonic by plucking it at one point &the

antinode( to give a large initial amplitude andtouching it at another &the node( to prevent vibration at that point.

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2.3Wave Motion