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
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