Physics CPhysics CPhysics CPhysics C

Chapter Chapter Chapter Chapter 18 18 18 18

From serway book From serway book From serway book From serway book

Prepared by Prepared by Prepared by Prepared by

Anas A.Anas A.Anas A.Anas A. AlkanoaAlkanoaAlkanoaAlkanoa

Prepared by Prepared by Prepared by Prepared by

Anas A.Anas A.Anas A.Anas A. AlkanoaAlkanoaAlkanoaAlkanoa

M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,

Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,

Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).

Chapter threeSuperposition & Standing Waves

18.1 Superposition and Interference

18.2 Standing Waves18.2 Standing Waves

18.3 Standing Waves in a String Fixed at Both Ends

18.4 Resonance

18.5 Standing Waves in Air Columns

18.7 Beats: Interference in Time

18181818....1 1 1 1 Superposition and Interference:Superposition and Interference:Superposition and Interference:Superposition and Interference:

Superposition principle:Superposition principle:Superposition principle:Superposition principle:

If two or more traveling waves are moving through a medium, the

resultant value of the wave function at any point is the algebraic sum of

the values of the wave functions of the individual waves.

Linear waves:Linear waves:Linear waves:Linear waves: are generally characterized by having amplitudes much

smaller than their wavelengths.

Linear waves:Linear waves:Linear waves:Linear waves: are generally characterized by having amplitudes much

smaller than their wavelengths.

Nonlinear wavesNonlinear wavesNonlinear wavesNonlinear waves : are often characterized by large amplitudes.

The figure shows the superposition of two pulses:The figure shows the superposition of two pulses:The figure shows the superposition of two pulses:The figure shows the superposition of two pulses:** The wave function for the pulse

moving to the right is y1 ,

and the wave function for the pulse

moving to the left is y2.

** The pulses have the same speed.

** When the crests of the pulses

coincide (Fig. b,c), the resulting wave

** When the crests of the pulses

coincide (Fig. b,c), the resulting wave

given by y1 + y2 has a larger

amplitude than that of the individual

pulses.

** The two pulses finally separate and

continue moving in their original

directions(1)

Interference:Interference:Interference:Interference: The combination of

separate waves in the same region of

space to produce a resultant wave.

** When the displacements caused

by the two pulses are in the same

direction, we refer to their

superposition as constructiveconstructiveconstructiveconstructive

(2)

direction, we refer to their

superposition as constructiveconstructiveconstructiveconstructive

interferenceinterferenceinterferenceinterference....

** When the displacements caused

by the two pulses are in opposite

directions, we refer to their

superposition as destructivedestructivedestructivedestructive

interferenceinterferenceinterferenceinterference.

Superposition of Superposition of Superposition of Superposition of

Sinusoidal Waves:Sinusoidal Waves:Sinusoidal Waves:Sinusoidal Waves:

ا�شتقاق الرياضي

(3)

الرياضي في التصوير

1- The resultant wave function y also is sinusoidal and has the same

frequency and wavelength as the individual waves

Remarks

2- If the two waves are in phase that is, the phase constant

then the amplitude of resultant

wave maximum and equal to 2A as shown in Fig(3a)(((( constructiveconstructiveconstructiveconstructive interference)interference)interference)interference)

πππφ n2,...,4,2,0 ±±=

3- If the two waves are out of phase that is the phase constant

then the amplitude of resultant

wave is zero as shown in Fig(3b)( destructive interference) ( destructive interference) ( destructive interference) ( destructive interference)

πππφ )12(,...,3, +±±= n

4- Finally, when the phase constant has an arbitrary value other

than 0 or an integer multiple of rad Fig. 3c, the resultant wave

has an amplitude whose value is somewhere between 0 and 2A .π

Interference of Sound WavesInterference of Sound WavesInterference of Sound WavesInterference of Sound Waves

Relationship between path difference and phase angle is given by Relationship between path difference and phase angle is given by Relationship between path difference and phase angle is given by Relationship between path difference and phase angle is given by

differencepathdifferencephaseλπ2=

For example :For example :For example :For example :

In constructive interference In constructive interference In constructive interference In constructive interference

Generally

For example :For example :For example :For example :

In constructive interference In constructive interference In constructive interference In constructive interference

differencepathnλππ 2

2 = λndifferencepath =

In destructive interference In destructive interference In destructive interference In destructive interference

differencepathnλππ 2

)12( =+2

)12(λ+= ndifferencepath

For example, the interference in sound wavesFor example, the interference in sound wavesFor example, the interference in sound wavesFor example, the interference in sound waves

Or

18181818....2222 Standing WavesStanding WavesStanding WavesStanding WavesWe can analyze such a situation by considering wave functions for

two transverse sinusoidal waves having the same amplitude,

frequency, and wavelength but traveling in opposite directions in the

same medium:

Adding these two functions gives the resultant wave function y :

This expression reduces to

(1)

** Equation (1) represents the wave function of a standing wave.

** Notice that Equation (1) does not contain a function of tkx ω−

** We see that Equation (1) describes a special kind of simple

harmonic motion.

** The amplitude of the simple harmonic motion of a given element

depends on the location x of the element in the medium.

The maximum amplitude of an element of the medium has aThe maximum amplitude of an element of the medium has a

minimum value of zero when x satisfies the condition sinkx = 0, that

is, when ,...3,2, πππ=kx

.2

λπ=kBecause then x values is given by 2

...,2

3,

2

2,

2

λλλλ nx ==

These points of zero amplitude are called nodesThese points of zero amplitude are called nodesThese points of zero amplitude are called nodesThese points of zero amplitude are called nodes

The positions in the medium at which this maximum

displacement occurs are called antinodesantinodesantinodesantinodes.The antinodes are located at positions for which the coordinate xsatisfies the condition that is, when1)sin( ±=kx

The distance between adjacent antinodes is equal to 2

λ

The distance between adjacent nodes is equal to2

λ

Standing waves through time

18.3 Standing Waves in a String Fixed at Both Ends

Consider a string of length L fixed at both ends, as shown in Figure (1)

Standing waves are set up in the string by a continuous superposition

of waves incident on and reflected from the ends.

The boundary condition y = 0 at the ends of string, the two ends are called nodes. the two ends are called nodes.

the string having a number of natural patterns of oscillation, called

normalnormalnormalnormal modesmodesmodesmodes, each of which has a characteristic frequency.

Only certain frequencies of oscillation are allowed is called quantizationquantizationquantizationquantization.

(1)

From this figure we can determine the wavelength for any n ,From this figure we can determine the wavelength for any n ,From this figure we can determine the wavelength for any n ,From this figure we can determine the wavelength for any n ,

n

Ln

2=λ Where n = 1 , 2 , 3,…(Wavelengths of normal modes)

Frequencies of normal modes as functions of wave speed and length of

string is given by

Frequencies of normal modes as functions of string tension and linear

mass density

These natural frequencies are also called the quantized frequenciesquantized frequenciesquantized frequenciesquantized frequencies

Where the index n refers to the nth normal mode of oscillation.

The figure shows oneoneoneone ofofofof thethethethe normalnormalnormalnormal

modes of oscillation of a string fixed

The lowest frequency f1, which corresponds to

n = 1, is called the fundamentalfundamentalfundamentalfundamental frequencyfrequencyfrequencyfrequency

and is given by

modes of oscillation of a string fixed

at both ends. Except for the nodes,

which are always stationary, all

elements of the string oscillate

vertically with the same frequency

but with different amplitudes of

simple harmonic motion.

Frequencies of normal modes that exhibit an integer-multiple

relationship such as this form aaaa harmonicharmonicharmonicharmonic seriesseriesseriesseries, and the normal

modes are called harmonicsharmonicsharmonicsharmonics....

1) The fundamental frequency f1 is the frequency of the first harmonicthe first harmonicthe first harmonicthe first harmonic

RemarksRemarksRemarksRemarks

2) The frequency f2 = 2 f1 is the frequency of the second harmonicthe second harmonicthe second harmonicthe second harmonic

3) The frequency fn = nf1 is the frequency of the nnnnth harmonic.th harmonic.th harmonic.th harmonic.

Now using the boundary conditions we get Now using the boundary conditions we get Now using the boundary conditions we get Now using the boundary conditions we get

0),0( =ty At the first boundary condition at x=At the first boundary condition at x=At the first boundary condition at x=At the first boundary condition at x=0 0 0 0 for any timefor any timefor any timefor any time

If we substituted x = 0 in the general form of the wave function of the

standing waves then 0)cos())0(sin(2),0( == tkAty ω

To meet the second boundary condition 0)cos())(sin(2),( == tLkAtLy ω

Then 0)sin( =kL At πnLK n = Where n = 1, 2, 3,…Then 0)sin( =kL At πnLK n = Where n = 1, 2, 3,…

And because λπ2=nK we find that n

LornL n

n

22 == λπλπ