phy 113 c general physics i 11 am – 12:15 pm tr olin 101...

10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 1

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)(:solutiongeneral

:equationaldifferenti

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

ωwhere

sin/)φω(cos)(:solutiongeneral

sin:equationaldifferenti

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 ωsin/)φω(cos)(

w

) 00 amplitudesin)( XtXtx

0

022

00 /

/mXF

mk

mkmFX


The wave equation

Wave variable

What 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 sophistocatedanalysis:


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

μwhere 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

λφλ

π2sin),( 0

phase factor


cTT

txytxy

λφλ

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:fieldsmagneticandelectriccoupled todueLight wave:Example

:density andpressureair with in waveSound:Example

:h mass/lengtandion with tensstringaon Wave:Example

occurs.avein which wmediumand/or processon thedependsspeedWave

8 smc

pcp

TcT

c


Example from webassign:


cTT

txytxy

λφλ

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:speede transversmaximum

φλ

π2cos2),(

λφλ

π2sin),(

yT

Ttxy

Tttxy

cTT

txytxy

phy 113 c general physics i 11 am – 12:15 pm tr olin 101...

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