note - chapter 15

8/11/2019 Note - Chapter 15

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If we shake the free end of a string in a repetitive, or periodic motion, then each particle in the string alsoundergoes periodic motion as the periodic transverse

wave propagates along the string.

B. Longitudinal Waves : The direction of motion of the particles in the medium is parallel to the direction of propagation of the wave. Examples include:(1) sound

Consider a long tube filled with a fluid and a piston at theleft end. If we push the piston just once to the right andthen to the left, then a longitudinal wave pulse travels tothe right along the fluid as shown in the figure below.



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If we now push the piston to the right and to the left insimple harmonic motion along a line parallel to the lengthof the tube, then a periodic longitudinal wave propagatesalong the fluid parallel to the tube.

This motion forms regions in the fluid where the pressureand density are higher or lower than the equilibriumvalues. The regions in the fluid of increased pressure anddensity are called compressions , and the regions ofreduced density and pressure are called rarefactions.



4

Some waves have both transverse and longitudinalcomponents as in the case of waves on the surface of aliquid.

A periodic wave propagates a distance equal to thewavelength ! in a time interval corresponding to its

period T. Thus the speed of wave propagation v equals

speed of wave propagation =

dis tan cetime

=

"

T = " f



5

15.3 Mathematical Description of a Wave

A sinusoidal wave has a waveform equal to that of a sine

(or cosine) wave.

The Traveling Wave

The disturbance of a traveling wave is always a functionof vt x ± .(i) If x and vt have the same sign , then the wave travels inthe direction of decreasing x.(ii) If x and vt have opposite signs , then the wave travelsin the direction of increasing x.

A. Consider a transverse sinusoidal waveform in a ropeof the kind shown in the figure above. The general



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functional form of the displacement y ( x,t ) of a piece ofrope at position x and time t is

t xk At x y != msin,

where A is the amplitude , k is the wave number , and ! isthe angular frequency as described in chapter 14. Also,remember from chapter 14 that

!

"=

2k T f

!=!="

22

so that one may write the displacement y( x,t ) as

( ) !"#$

%&

!"#$

%&

'(=

T t x

At x y m2sin,

To find the speed v of wave propagation, one can simplysay that the wave propagates a distance equal to onewavelength ! in a time interval equal one period T.Hence,

time

cedisv

tan=

f T

v !=

!=



8

note carefully that the maximum value of the transversespeed of a piece of rope equals (setting the cosine termequal to one)

Av y !=

max,

which is the same result obtained in chapter 14 where wesaid after using conservation of mechanical energy that

A Amk vkAmv !=="=

max22

max21

21 same result!

We know that a piece of rope will attain its maximumspeed when it is moving through the equilibrium position.That is, a piece of rope that has a transverse wave

propagating through it undergoes simple harmonicmotion!

Be very careful here!!! Do not confuse the speed of propagation of the wave f v != with the transverse speedv y of a piece of rope!!!

As for the acceleration of a piece of rope at position x andtime t , then

dt

vd a y

y

)(= (holding x constant)

}sin{ t kx Aa y !""!"!"=



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( )t kx Aa y !"!"= sin2

ya y

2!"=

This result is identical to the result obtained in chapter 14. Note in addition that the maximum value of themagnitude of the acceleration of a piece of rope occurs atthe amplitudes, i.e.,

Aa y2

max, !=

By the way, the wave function ( ( t xk At x y !"= sin, satisfies an equation called a “wave equation”, given by

2

2

22

2 1

t

y

v x

y!

!=

!

!

Any function of the form ( )vt x f m satisfies the waveequation. We will learn in chapter 32 that when anelectromagnetic wave propagates through vacuum, both

the electric field E

r

and the magnetic field B

r

obey a waveequation quite similar to the last equation above.



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15.4 Speed of a Transverse Wave on a String

The speed v with which a wave propagates in a rope of

mass per unit length µ and under tension F is given by

v =

Tension

µ

Derivation:

Consider a small segment " x and mass " m of a very long string under tension F. The tension (force) at each end ofthe string is tangent to the string at the point ofapplication.



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The string to the right of the small segment exerts a force F 2 on the segment. The string to the left of the small segment exerts a force F 1 on the segment. There is a net

vertical force on the segment of string, but not a nethorizontal force since the motion of the string is verticalonly.

Apply Newton’s Second Law of motion:

F y"=

#m( )$

2 y

$ t 2

but

F y" = F 2 y # F 1 y= F tan $

2 # F tan $ 1

where ! 2 is the angle that F 2 makes with the horizontal,and ! 1 is the angle that F 1 makes with the horizontal. Butthe tan( ! ) = slope of the tangent line to the string (curve)

and this is also equal to" y

" x. Hence

F y" = F tan # 2 $ F tan #

1= F % y

% x&

' ( )

* +

x+ , x$ F % y

% x&

' ( )

* +

x

note that in the limit as " x # 0 ,



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so that

F y" = F #

2 y

# x2

$

%

&

'

(

) * x( )

Therefore,

F y" = #m( )$

2 y

$ t 2

= F $

2 y

$ x2

%

&

'

(

)

* # x( )

use " m = µ " x , where µ is defined as the mass per unitlength of the string, to obtain

µ " x( )#

2 y

# t 2

= F #

2 y

# x2

$

%

&

'

(

) " x( )

" 2 y

" x2

=

µ

F

" 2 y

" t 2

Finally, comparing this last expression with the general

expression for a wave equation, 2

2

22

2 1

t

y

v x

y!

!=

!

! yields,

" y

" x

#

$ %

&

' (

x + ) x

* " y

" x

#

$ %

&

' (

x

="

2 y

" x2 ) x( )



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v =

F

µ

15.6 Wave Interference, and the SuperpositionPrinciple

If two or more waves are traveling in a medium andinterfere, the resultant wave is found by adding thedisplacements of the individual waves.

Interference of Waves Use transparencies to discuss concept of phase difference.

A. Constructive Interference of 2 waves of the samefrequency:



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Constructive interference will result if the phasedifference ! between the 2 interfering waves is

,...6,4,2,0 !!!="

B. Destructive Interference of 2 waves of the same frequency:

Destructive interference will result if the phase difference! between the 2 interfering waves is ,...5,3, !!!="



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15.8 Normal Modes of a String

The string has various natural patterns of vibration, called

normal modes , and each normal mode has a characteristicfrequency.

What are the allowed frequencies of vibration?

L = 1"

2 , n = 1

L = 2 "

2

#

$

%

&

'

( , n = 2

L = 3 "

2

#

$

%

&

'

( , n = 3

L = 4 "

2

#

$

%

&

'

( , n = 4



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In general, L = n "

2

#

$

%

&

'

( where n = 1, 2, 3, 4, …. , and so

" =

2 L

n

and the allowed frequencies of vibration f =

v"

are equal

to

f n = n v2 L

"

#

$

%

&

' ( ,...4,3,2,1=n

Becauseµ

=Tension

v , one can express the natural

frequencies of vibration of a stretched string as

µ=

Tension Ln

f n 2 ( ,...4,3,2,1=n

The lowest allowed natural frequency of vibration iscalled the fundamental frequency . Any integer multiple ofthe fundamental frequency is called a harmonic . Thus,



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!1 f fundamental frequency

µ=!

tension L

f 2

11

!=

12 2 f f second harmonic (or first overtone)

!=

13 3 f f third harmonic (or second overtone)

and so on. All the even and odd harmonics are present.

15.7 Mathematical Representation of Standing WavesConsider two waves of the same frequency ! and sameamplitude A traveling in opposite directions .

wave 1: wave 2:( t xk A y !"= sin

1 ( t xk A y !+= sin2

Note that wave 1 travels to the right, while wave 2 travelsto the left.

What will the resultant wave be? The resultant wave will be a “standing wave” or “blinking wave” as I like to call

them. Applying the superposition principle yields

21 y y y +=

( ) ( )t kx At kx A y !++!"= sinsin



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( ( t kxt kx A y !++!"= sinsin

one learns in trigonometry that

( ( ( ( (kxt t kxt kx cossincossinsin !"!=!"

( ( ( ( (kxt t kxt kx cossincossinsin !+!=!+

so that the resultant wave is

( ( t kx A y != cossin2 a “standing wave”

Note carefully that this is NOT a traveling wave becausey is not a function of vt x ± .

The sine term in the standing wave describes the spatialdependence , while the cosine term describes the timedependence of the standing wave.

Questions:

1. At positions or values of x will the disturbance be zeroat all times ? These positions are called nodes .

( 0sin = xk !!!!= n xk ,...,3,2,,0 where ( ,...3,2,1,0=n



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!=

"

!n x

2

2

!= n x (nodes)

that is, when ,...2,2

3,,

2

1,0 !!!!= x

2. At what points x will the disturbance be a maximum atall times ? These positions are called antinodes .

( 1sin ±=kx

2,...,2

5

,2

3

,2

!!!!=

n xk

where ( ,...7,5,3,1=n

2

2 !=

"

! n x

4

!= n x (antinodes)



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3. At what times t will the disturbance be zero for allvalues of x?

( 0cos =! t

2,...,

2

5,

2

3,

2

!!!!="

nt where ( ,...7,5,3,1=n

2

2 !=

! nt

T where T = period

4

T nt =

that is, zero disturbance for all positions of x at times

,...4

5,

4

3,

4

T T T t =

The figure below shows the shape of a string at 2 differentinstants:



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15.5 Energy in Wave Motion Waves transport energy from one place to another. To

produce a wave, we have to exert a force on the wave

medium, thus doing work on the system . As the wave propagates, each portion of the medium does work on theadjoining portion. In this way, a wave transports energyfrom one region of space to another.

A. Waves Propagating Through an Elastic Medium

Consider a transverse wave traveling from left to right ona string. Now consider a specific point on the string. Thestring to the left of this point exerts a force (in the y-direction) to the right of it given by F y = " F tan # , where

tan " =

# y

# x , so that F y = " F

# y# x

. This force does work on

the portion of string to the right of it and thereforetransfers energy to it. The corresponding power (rate ofdoing work) at this point in the string is P = F y vy , that is,the transverse force multiplied by the transverse velocity.



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2. The energy transported by a wave is proportional to thesquare of the frequency and to the square of its amplitude.This is true of mechanical waves .

3. The maximum value of the instantaneous rate of energytransfer is

Pmax imum

= µ F " 2 A 2

4. The average power P

transported by the wave is therate of energy transferred. The average of the cosinesquared term over a whole cycle is always ! so that

Paverage=

1

2µ F "

2 A

2



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5. The intensity transported I is the average powertransported per unit cross sectional area so that

I = Paverage

Area=

µ F " 2

A2

2 Area( )

If waves spread out equally in all directions from asource, then the intensity I of the wave at a distance r from the source is inversely proportional to r 2, becausethe cross sectional area is 4" r 2. Thus

I =

Paverage

Area=

P4 " r

2

so from conservation of energy, the power output P fromthe source is equal at the surface with radius r 1 and at thesurface with radius r 2 if no energy is absorbed betweenthe two surfaces. Thus

Power = 4 " r1

2 I

1= 4 " r

2

2 I

2



B . Electromagnetic Waves Propagating throughVacuum

We will learn in chapter 32 that the intensity transported by an electromagnetic wave in vacuum is proportional tothe square of the electromagnetic field amplitude, but isindependent of the frequency of the wave.

note - chapter 15

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