Transverse Wave on a String
Wave speed depends on properties of medium:
 Forastring,willsee v  T /  o T isstringtension
o  ismass/unitlength(i.e.lineardensity)
 tosee,apply2ndlawtoelementofstring(red)inframemovingleftwithcrest
atspeedv
o Inthisframe:stringelement s nearcrestmovesaroundcirculararcof
radiusR(don’tactuallyknowR)
 Define  ashalfofanglesubtendedbyarc s ofradiusR:
o Propertiesofstringelementarethen:
 Length: s  R  2  Mass: m    R  2  ForceonelementisduetotensionTatends
o Radialcomponentofforceis FR  2T sin   2T forsmall  (inradians)
2
 Centripetalaccelerationofelementis acent  v / R  Newton’s2ndlawsays FR  m acent 2
o So: 2T    R  2  v / R  Cancel  and R (didn’tknowanyway)toget v  T /  o Says:forwaveonstring, v :
 Increaseswithtension→tighterstring,fasterwave
 Decreaseswithlineardensity→heavierstring,slowerwave
2
 Assumptions:
o Small amplitude so that small part of crest approximates circular arc
o Tension does not change as wave passes
 BUT:
o Did not assume specific shape
o v  T /  applies to any small (linear) wave on a string
REFLECTION & TRANSMISSION OF WAVES (Important for standing waves)
 Whathappenswhenwaves(allkinds)meetboundaryinmedium?
o Fixedendofstring(oraircolumn)
o Loose(orfree)endofstring
o Changeinpropertiesofmedium
Start by looking at single pulse on string (BUT results are general)
 Fixedboundary
o Reflectedpulse:invertedandsame
amplitude
 Notransmittedpulse
o Understandby3rdlaw
 Pulseappliesupwardforceon
(fixed)boundary
 Boundaryappliesdownwardforceonstring
o Forsinusoidalwave:resultis180°(i.e.πradian)phaseshift
 Freeboundary(looseend)
o Reflectedpulse:notinvertedand
sameamplitude
 Notransmittedpulse
 Pulseencountersdensermedium(largermass/unitlength)
o Reflectedpulse:inverted(180°
phaseshift)andsmalleramplitude
o Transmittedpulse:samesenseas
incident
 Pulseencounterslessdensemedium
o Reflectedpulse:notinvertedand
smalleramplitude
o Transmittedpulse:samesenseas
incident
ENERGY CARRIED (TRANSMITTED) BY A SINUSOIDAL WAVE ON STRING
1 2 1
2 2
E

kA

m

A
 Recall:forSHM, mech 2
2
 Aswavepasses,eachelementofmass
m   x movesinSHM
1
2 2
E

o Soforthatelement: mech 2  x A 1
2 2

E




A  Somechanicalenergyinonecompletewavelengthis: mech 2
 Thetimeittakesfortheenergyinalength
 ofthewavetomovepastagivenpointon
thestringis t   / v  Rateofenergytransfer≡Power=energypassingfixedpointperunittime
 So:
Emech
t
1
 2 A2
 2
 /v
1
  v 2 A2
2
P
1
P

 v 2 A 2  Result
2
o derivedforatransversesinusoidalwaveonastringBUT:
2
o dependenceoftransmittedPon v ,  ,and A2 isgeneral.