phy 113 c general physics i 11 am – 12:15 pm tr olin 101 plan for lecture 16:

PHY 113 C Fall 2013 -- Lecture 16 110/22/2013

PHY 113 C General Physics I11 AM – 12:15 PM TR Olin 101

Plan for Lecture 16:Chapter 16 – Physics of wave motion

1. Review of SHM2. Examples of wave motion3. What determines the wave velocity4. Properties of periodic waves


Some comments on Simple Harmonic Motion

k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided

)φω(cos)( :solution general

:equation aldifferenti

Suppose that you know: (in standard units)

x(t)=2 cos(2pt+3p)

What is k ? k = m(2p)2

m=1 kg



k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided




x(t)=2 cos(2pt+3p)

What is the frequency of oscillations ? w=2pf=2p f=1 Hz

m=1 kg



k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided




x(t)=2 cos(2pt+3p)

What is amplitude of the displacement ? xmax=2m

m=1 kg



k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided




x(t)=2 cos(2pt+3p)

What is the maximum velocity ? v(t)=-2(2p) cos(2pt+3p) vmax=4p

m=1 kg



k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided




x(t)=2 cos(2pt+3p)

What is the maximum acceleration ? a(t)=-2(2p)2 cos(2pt+3p) amax=8p2

m=1 kg



k

mktAtx

xmk

dtxddtxdmmakxF

2

2

2

2

2

ωthatprovided




x(t)=2 cos(2pt+3p)

What is the displacement at t=0.3 s ? x(0.3)=2 cos(2p(0.3)+3p) =2(0.309)=0.618 m

m=1 kg


Some comments on driven Simple Harmonic Motion

k

)

)

)

mk

tmFtAtx

tFxmk

dtxd

dtxdmmatFkxF

2

220

02

2

2

2

0

ω where

sin/)φω(cos)( :solution general

sin :equation aldifferenti

sin

w

F(t)=F0sin(t)


Webassign question (Assignment 14)

Damping is negligible for a 0.175-kg object hanging from a light, 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.430 m?

)mktmFtAtx

2

220 ω sin/)φω(cos)(

w

) 00 amplitude sin)( XtXtx

0

022

00

//

mXF

mk

mkmFX


The wave equation

Wave variableWhat does the wave equation mean?Examples Mathematical solutions of wave equation and

descriptions of waves

2

22

2

2

xyc

ty

),( txyposition time


Source: http://www.eng.vt.edu/fluids/msc/gallery/gall.htm

Example: Water waves

Needs more sophistocated analysis:


Mechanical waves occur in continuous media. They are described by a value (y) which changes in both time (t) and position (x) and are characterized by a wave velocity c: y=f(x-ct) or y=f(x+ct)


Waves on a string:

Typical values for c:

3x108 m/s light waves

~1000 m/s wave on a string

343 m/s sound in air


Transverse wave:


Longitudinal wave:


General traveling wave –

t = 0

t > 0


iclicker question:

t=0t=1 s t=2s


Basic physics behind wave motion --

example: transverse wave on a string with tension T and mass per unit length m

y

ABT

dtydx

xm

TTdtydm

xy

xyμ

μ

θsinθsin

2

2

AB2

2

qBy

x

BBB x

y θtanθsin

2

2

0 xy

xy1

xy

xLimABx

2

2

2

2

μ xyT

ty


The wave equation:

Solutions: y(x,t) = f (x ct)

2

22

2

2

xyc

ty

string) a(for

μ where Tc

function of any shape

) )

) ) )

) ) ) 22

22

2

2

2

2

2

22

2

2

2

2

Let :Note

cuuf

tu

uuf

tuf

uuf

xu

uuf

xuf

xu

uuf

xuf

ctxu


iclicker question Is it significant to write the wave equation with the special symbols?

A. YesB. No

2

22

2

2

xyc

ty


Examples of solutions to the wave equation:

Moving “pulse”:

Periodic wave:

)20),( ctxeytxy

) )φsin),( 0 ctxkytxy

“wave vector”not spring constant!!!

ωπ2π2λπ2

fT

kc

k

cTT

txytxy

λ φλ

π2sin),( 0

phase factor


cTT

txytxy

λ φλ

π2sin),( 0

Periodic traveling waves:

Amplitude

wave length (m)

period (s); T = 1/f

phase (radians)

velocity (m/s)


Snapshot of periodic wave at t=t0

Time plot of periodic wave at x=x0

l

1/f

f l = c


φ

λπ2sin),( φ

λπ2sin),( 00 T

txytxyTtxytxy leftright

Ttxytxtxy leftright

π2cosφλπ2sin2),(y),( 0

“Standing” wave:

Combinations of waves (“superposition”)

) ) BABABA 21

21 cossin2sinsin

: thatNote


Summary of wave properties:

constant) al(fundament /102.9979

:fields magnetic and electric coupled todue Light wave :Example

: density and pressureair with in waveSound :Example

:h mass/lengt and ion with tensstring aon Wave:Example

occurs. avein which w mediumand/or process on the depends speed Wave

8 smc

pcp

TcT

c

mm


Example from webassign:


cTT

txytxy

λ φλ

π2sin),( 0

Periodic traveling waves:

Amplitude

wave length (m)

period (s); T = 1/f

phase (radians)

velocity (m/s)


Example from webassign:

0

0

0

2 :speed e transversmaximum

φλ

π2cos2),(

λ φλ

π2sin),(

yT

Ttxy

Tttxy

cTT

txytxy

p

p

phy 113 c general physics i 11 am – 12:15 pm tr olin 101 plan for lecture 16:

Documents

c fall

wave velocity c

ms wave

p cos2pt

transverse wave

wave motionwhat

longitudinal wave

c general physics i11