Wave motion

Waves

Condition: elastic media

-Taut string, spring (slinky)-Water surface, fluid-Solid state -Gas-Etc.

Wave pulse

x

f(x)t=0

vr

s

f(x-vt)

vr

s

f(x) t=0f(x+vt)

x

Traveling wave pulse

Wave – like motion of a worm

== xλπ2

sinA0)ty(x,

−= )vtx(2

sinAt)y(x,λπ

λ

2πk =Angular wave number:

ωttT2π

tλ/v2π

vtλ

2π ===

vr

( )ϕ+−= ωtkxsinAt)y(x,

max

vλ 1

T & Tλ

v =⋅→== ff

kω

v =

=A

SuperpositionLinearity

Reflection of wave pulse

Open endFixed end

inverted

uninverted

Standing waves

2λ

Standing waves

( )kx-ωtAcost)(x,y1 = ( )kxωtAcost)(x,y2 +=

Standing wave: t)(x,yt)(x,yt)y(x, 21 +=

( ) ( )kxcosωtAcos2t)y(x, =

( ) ( ) /kxsinωtAsin2t)y(x, : / =ϕ

Harmonics

fixed

L2λ1 =

Lλ L2λ

2 22 =→=⋅

3L

2λ L2λ

3 33 ⋅=→=⋅

4L

2λ L2λ

4 44 ⋅=→=⋅

λ

v=f

2Lv

λ

v

11 ==f

12

2 22Lv

2λ

vff ⋅=⋅==

13

3 32Lv

3λ

vff ⋅=⋅== 1n n ff ⋅=

L

L2λ1 =

22L

λ2 =

32L

λ3 =

2Lv

λ

v

11 ==f

12

2 22Lv

2λ

vff ⋅=⋅==

13

3 32Lv

3λ

vff ⋅=⋅==

Open ends

1n n ff ⋅=

Open endFixed end

L

4Lλ L4λ

11 =→=

34L

λ L4λ

3 22 =→=⋅

4Lv

λ

v

11 ==f

4Lv

3λ

v

22 ⋅==f

54L

λ L4λ

5 33 =→=⋅

4Lv

5λ

v

33 ⋅==f

1n 1)-n2( ff ⋅=

Ultrasound

Beat frequency

Doppler-effect

Sound source: at restObserver: moving at v

±=s

o vv

1ff

Sound source: moving at v Observer: at rest

=

s

o

vv

1

ff

m

±=

s

2

s

1

o

vv

1

vv

1ff

mSound source: moving at v2Observer: moving at v1