PAP 113

Jan. 27, 2006

Superposition of waves

Resultant (or total) wave = sum of partial waves

Valid for linear waves. Not valid for nonlinear waves.

Partial waves are all the elemental waves.

Example:

Boundary (x=0)

incident wavetransmitted wave

reflected wave

x<0: incident wave and reflected waves are elemental waves (their sum is the total wave)

x>0: transmitted wave is total wave

Superposition of waves ( ) ( )20

2

20

2

21xvtx

xvtx

eeyyy+

−−

−

+=+=

t=0 t>0

t>>0

Example:

Pulse = interference of sinusoidal waves of different frequencies

Amplitude Amplitude

frequencytime

Short pulse Large frequency content (it is a sum of many sinusoidal waves with large difference in frequencies)

Long pulse Small frequency content (it is a sum of fewer sinusoidal waves with smaller difference in frequencies)

1≈ΔΔ ft Δt=the duration of the pulse

Δf=the frequency span

Amplitude Amplitude

frequencytime

Δt Δf

Example:

Pulse = interference of sinusoidal waves of different frequencies

Femtosecond laserA femtosecond laser generates pulses with pulse width of about 1-1000 fs

1 femtosecond=1 fs=10-15 s

Fast laser pulses can be used to:

1) Probe fast chemical reactions. Such measurements gave Ahmed Zewail the Nobel prize in Chemistry in 1999.

2) Make ultraprecise optical clocks. Such inventions gave John Hall and Theodor W. Hänsch the Nobel Prize in Physics 2005.

Interference of Waves

• Sound waves interfere– Constructive interference occurs when the path difference between two

waves’ motion is zero or some integer multiple of wavelengths• path difference = nλ

– Destructive interference occurs when the path difference between two waves’ motion is an odd half wavelength

• path difference = (n + ½)λ

Standing Waves

• When a traveling wave reflects back on itself, it creates traveling waves in both directions

• The wave and its reflection interfere according to the superposition principle

• With exactly the right frequency, the resultant (or total) wave will appear to stand still– This is called a standing wave

Example: Standing waves between two fixed boundaries

)cos()cos(),( tkxAtkxAtxs ωω ++−= −+

s(x=0,t)=0 s(x=L,t)=0

)sin(sin2),( txL

nAtxs ωπ⎟⎠⎞

⎜⎝⎛= +

s(x=0,t)=0 gives A++A-=0

s(x=L,t)=0 gives k=nπ/L

Allowed wavelength is quantized:

λn=2L/n where n=1,2,3,…

Cavity = a space where the wave is confined

Allowed wavelengths are quantized:

λn=2L/n ; n=1,2,3,….

densitymassTension

Lnvf

nL

nn

n

2

2

==

⇓

=

λ

λ

Allowed frequencies are quantized:

fn=nv/2L (all other frequencies will die out)

Example: Standing waves between two fixed boundaries

Standing Waves

• A node occurs where the two traveling waves have the same magnitude of displacement, but the displacements are in opposite directions– Net displacement is zero at that point– The distance between two nodes is ½λ

• An antinode occurs where the standing wave vibrates at maximum amplitude

Standing Waves on a String

• Nodes must occur at the ends of the string because these points are fixed

nodeantinode

Standing Waves

• The red arrows indicate the direction of motion of the parts of the string

• All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion

Standing Waves on a String

• The lowest frequency of vibration (b) is called the fundamental frequency

1ƒ ƒ2n

n Fn

L μ= = vsE nn

2max

2

21 μω=

nn fπω 2=

Standing Waves on a String – Frequencies

• ƒ1, ƒ2, ƒ3 form a harmonic series– ƒ1 is the fundamental and also the first

harmonic– ƒ2 is the second harmonic

• Waves in the string that are not in the harmonic series are quickly damped out– In effect, when the string is disturbed, it

“selects” the standing wave frequencies

Standing waves near a fixed and an open boundary

s(x=0,t)=0 s(x=L,t)=±2A+sin(ωt)

)sin(2

sin2),( txL

mAtxs ωπ⎟⎠⎞

⎜⎝⎛= +

s(x=0,t)=0 gives A++A-=0

s(x=L,t)=±2A+sin(ωt) gives sin(kL)=±1

Allowed wavelength is quantized:

λm=4L/m ; m=1,3,5…

( ) ( ) ( ) ( )tkxAAtkxAAtkxAtkxAtxs

ωωωω

sin)sin(cos)cos()cos()cos(),(

−+−+

−+

−++=++−=

mL

mL

mLk

m

m

m

4

22

2

=

⇓

=

⇓

=

λ

πλπ

π

vL

mvf

mL

nn

n

4

4

==

⇓

=

λ

λ

Allowed frequencies are quantized:

fn=mv/4L (all other frequencies will die out)

Allowed wavelength is quantized:

λn=4L/m ; m=1,3,5…

Standing waves with one open boundary

Example: Standing waves on a string (one

end fixed, the other oscillating)

y(x=L,t)=0y(x=0,t)=2Asin(ωt)

x=0 x=L

Allowed wavelength is quantized:

λn=4L/m ; m=1,3,5…

Allowed wavelength is quantized:

λn=4L/m ; n=1,3,5…

or

L=mλn/4 ; n=1,3,5…

Example 18.7 in Serway&Jewett

Beats: interference in time

• Beating is the periodic variation in amplitude at a given point due to superposition of waves with slightly different frequencies.

• Beats=temporal interference• Waves have slightly different frequencies and the time between

constructive and destructive interference alternates• The beat frequency equals the difference in frequency between the

two sources:

2 1ƒ ƒ ƒb = −