13. Wave Motion13. Wave Motion13-1. Three Wave Characteristics

1. Wave Length λ2. Period T or Frequency f = 1/T3. Speed of wave v

13-2. Types of Waves

Transverse WaveLongitudinal Wave

13-3. One-Dimensional Transverse Traveling Wave

y = f (x) at t = 0

ymax : Amplitude

• General form of wave, y (x,t)

y = f (x – vt) : traveling to the right

y = f (x + vt) : traveling to the left

Same phase ( ) vtxt,x −=φ( ) ( )ttvxxtt,xx δ+−δ+=δ+δ+φ

tvxvtx δ⋅−δ++=

( ) vtxt,x +=φ=

dtdxv =⇒ : wave velocity,

phase speed (velocity)

• Wave- Energy ⇒ Propagate - Media

13-4. Sinusoidal Traveling Waves

( )

−

λπ

= vtxsinAy 2

Wave length λWave velocity v Period T

vT = λ ⇒ T = λ / v , v = λ / T

λ

−λπ

= tT

xsinAy 2

π

−λπ

= tT

xsinA 22

Convenient to define

fT

π=π

≡ω 22λπ

≡2k

( )tkxsinAy ω−=⇒

fk

v λ=ω

=

Generalized wave function form

( )φ+ω−= tkxsinAy

Phase constant

Sinusoidal Waves on Strings

( )tkxsinAy ω−=

Transverse speed

( )tkxcosAty

dtdyv

ttanconsxy ω−ω−=

∂∂

≡=

=

Transverse acceleration

( )tkxsinAt

vdt

dva y

ttanconsx

yy ω−ω−=

∂

∂=

=

=

2

( ) Avmaxy ω=

( ) Aamaxy

2ω=

13-5. Superposition and Interference of waves

• if two or more traveling waves are moving through a medium, the resultant wave function at any point is the sum of the wave functions of the individual waves.

: Superposition principle

• Two traveling waves can pass through each other without being destroyed or even altered.

⇒ Interference Phenomena (간섭현상)

One of the wave characteristics

13-6. The Speed of Transverse Waves on Strings

θ⋅≅θ=∑ FsinFF 22r

Rmv2

=

µ: the mass per unit length (선밀도)

θ⋅µ=∆⋅µ= 2Rsm

22

222 vR

vRF µθ=⋅θ⋅µ

=θ⋅

µ=

Fv : speed of transverse waves on strings

F = Mg

M

Mgµ

=µ

=MgFv

13-7. Reflection and Transmission of Waves

Reflection Rigid End Free End

The reflected pulse remains same without inversion.

Inverted reflected pulsebut the shape remains same.

Transmission

vA

vB

AA

Fvµ

=

BB

Fvµ

=

BA vv >

Inverted Reflected pulse but no inversion in the Transmitted pulse.

vBvA BA vv <

No inversion in both the Reflected and Transmitted pulse.

13-8. Energy Transmitted by Sinusoidal Waves on Strings

( )tkxsinAy ω−=

∆m : Simple Harmonic Motion

Total Energy 22212

21 AmkAE ω==

Total Energy for ∆m 22212

21 AmkAE ω∆==∆

( ) 2221 AxE ω∆µ=∆

2221 A

dtdx

dtdEP ω

µ==Power P

vAP 2221 µω=

13-9. Sound WavesLongitudinal Wave

λPressure

( ) ( )tkxcosst,xs max ω−=

Pressure relative to its equilibrium

( )tkxsinPP max ω−∆=∆

maxmax svP ωρ=∆

ρ : density of mediumv : the wave speedωsmax : the maximum longuitudinal

speed of the medium

( )2π−ω−ωρ=∆ tkxcossvP max

90° phase off

AtmvP s

⋅∆∝

Axm ⋅∆ρ=

( )tkxsinsdtdsv maxs ω−ω==

( )tkxsinstxP max ω−ω

∆∆

ρ∝

( )tkxsinsv max ω−ωρ=

13-10. Doppler EffectRelative motion between the source and the observer causes a higher or lower frequency than the original frequency.

Moving Observerfrequency

λ=

vf

vvvfvvf 00 +

=λ+

=′

Observer moving toward source

Moving Source

λ′=′ vf

vT=λ

( )fvvTvv s

s−

=−=λ′

−

=λ′

=′svv

vfvf

Source moving toward observerIn general

±=′

s

o

vvvvff

m