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Physics 202, Lecture 20

Today’s Topics   Wave Motion

  General Wave   Transverse And Longitudinal Waves   Wave speed on string   Reflection and Transmission of Waves

  Wave Function   Sinusoidal Waves   Standing Waves

General Waves   Wave: Propagation of a physical quantity in space over time q = q(x, t)

  Examples of waves: Water wave, wave on string, sound wave, earthquake

wave, electromagnetic wave, “light”, quantum wave….  Mechanic wave: Propagation of small motion (“disturbance”) in a medium.

 Physical quantity to be propagated: displacement.

Recall: Displacement is a vector.

Transverse and Longitudinal Waves   If the direction of mechanic disturbance

(displacement) is perpendicular to the direction of wave motion, the wave is called transverse wave.

  If the direction of mechanic disturbance (displacement) is parallel to the direction of wave motion, the wave is called longitudinal wave.

 see demos.

  In general, a wave can be a combination of the above modes.

  The definition can be extended to other (non-mechanic) waves.   e.g Electromagnetic waves are always transverse.

Electro-Magnetic Waves are Transverse

x

y

z

E

B c

Two polarizations possible

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Seismic Waves

Longitudinal

Transverse

Transverse

Transverse

Wave On A Stretched Rope   It is a transverse wave See demos.

  The wave speed is determined by the tension and the linear density of the rope:

lmTvΔΔ

≡= µµ

;

Reflection and Transmission of Waves Wave Function   Waves are described by wave functions in the form:

y(x,t) = f(x-vt)

y: A certain physical quantity e.g. displacement in y direction

f: Can any forms

x: space position. Coefficient arranged to be 1

t: time. Its coefficient v is the wave speed v>0 moving right v<0 moving left

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An Exercise to Explain Wave Speed   A wave is described by function y=f(x-vt).

  At time t1 in position x1, how large is the quantity y?   y= f(x1-vt1) = y1

  At a later time t2=t1+Δt, what is y in position x2=x1+vΔt?

  y2= f(x2-vt2) = f(x1+vΔt -v(t1+Δt))= f(x1-vt1) = y1

  How to interpret the result?   Between t1 and t1+Δt, the value y0 has

transmitted from position x1 to x1+vΔt  speed = (x1+vΔt - x1 )/(t1+Δt - t1 ) =v i.e v>0 moving right; v<0 moving left;

Linear Wave Equation  Linear wave equation

2

2

22

2 1ty

vxy

∂∂

=∂∂

certain physical quantity

Wave speed

 Sinusoidal wave

)22sin( φπλπ

+−= ftxAy

A:Amplitude

f: frequency φ:Phase

General wave: superposition of sinusoidal waves

v=λf k=2π/λ ω=2πf

λ:wavelength

Parameters For A Sinusoidal Wave   Snapshot with fixed t: wave length λ=2π/k   Snapshot with fixed x: angular frequency =w frequency f =ω/2π Period T=1/f Amplitude =A   Wave Speed v=ω/k v=λf, or v=λ/T   Phase angle difference

between two positions Δφ =-kΔx

Snapshot:Fixed t

Snapshot:Fixed x

Standing Waves  When two waves of the same amplitude, same

frequency but opposite direction meet standing waves occur.

€

y1 = Asin(kx −ωt)

€

y2 = Asin(kx +ωt)

€

y = y1 + y2 = 2Asin(kx)cos(ωt) Points of destructive interference (nodes)

€

kx = nπ; x =nπk

=nλ2

; n =1,2,3...

 Points of constructive interference (antinodes)

€

kx = (2n −1) π2

; x = (2n −1) λ4

; n =1,2,3...

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Standing Waves (cont)

 Nodes and antinodes will occur at the same positions, giving impression that wave is standing

Standing Waves (cont)  Standing waves with a string of given length L are

produced by waves of natural frequencies or resonant frequencies:

€

λ =2Ln

; n =1,2,3...

€

f =vλ

=nv2L

=Tµ

n2L

Forced (driven) Oscillation   If in addition there is a driving force with its own

frequency ω: F0cos(ωt), the equation becomes:

  This equation can be solved analytically. At large t, the solution is:

with

  At large t, the frequency is determined by driving ω   When ω=ω0, amplitude is maximum resonance

)cos(02

2

tFdtdxbkx

dtxdm ω+−−=

)cos( φω += tAx 2220

2

0

)2()(

/

mb

mFA+−

=

ωω

Resonance Amplitude

2220

2

0

)2()(

/

mb

mFA+−

=

ωω