Physics 202, Lecture 21

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

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:

l

mTv

!

!"= µ

µ ;

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

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+!t, what is y in position

x2=x1+v!t?

%! y2= f(x2-vt2) = f(x1+v!t -v(t1+!t))= f(x1-vt1) = y1

$! How to interpret the result?

%! Between t1 and t1+!t, the value y0 has

transmitted from position x1 to x1+v!t

#!speed = (x1+v!t - x1 )/(t1+!t - t1 ) =v

i.e v>0 "#moving right; v<0 "#moving left;

Linear Wave Equation

"!Linear wave equation

2

2

22

21

t

y

vx

y

!

!=

!

!

certain

physical quantity

Wave speed

"!Sinusoidal wave

)22

sin( !"#

"+$= ftxAy

A:Amplitude

f: frequency ":Phase

General wave: superposition of sinusoidal waves

v=#f

k=2$/#%

&=2$f #:wavelength

Parameters For A Sinusoidal Wave

"! Snapshot with fixed t:

wave length #=2$/k

"! Snapshot with fixed x:

angular frequency =w

frequency f =&/2$%

Period T=1/f

Amplitude =A%

"! Wave Speed v=&/k

# v=#f, or

# v=#/T

"! Phase angle difference between two positions

!" =-k!x

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 "#t)

!

y2 = Asin(kx +"t)

!

y = y1 + y2 = 2Asin(kx)cos("t)

"!Points of destructive interference (nodes)

!

kx = n"; x =n"

k=n#

2; n =1,2,3...

"!Points of constructive interference (antinodes)

!

kx = (2n "1)#

2; x = (2n "1)

$

4; n =1,2,3...

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:

!

" =2L

n; n =1,2,3...

!

f =v

"=nv

2L=

T

µ

n

2L

Forced (driven) Oscillation

"! If in addition there is a driving force with its own

frequency &: F0cos(&t), 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 &%

%! When &=&0, amplitude is maximum# resonance

)cos(02

2

tFdt

dxbkx

dt

xdm !+""=

)cos( !" += tAx 222

0

2

0

)2()(

/

m

b

mFA

+!

=

""

Resonance Amplitude

222

0

2

0

)2()(

/

m

b

mFA

+!

=

""