traveling waves & wave equation sending a pulse down a string t = tension µ = mass/length
TRANSCRIPT
Traveling Waves & Wave Equation
Sending a pulse down a string
T = tension µ = mass/length
x x + Δx
TT
θ
θ+Δθ
ymass =
μ ⋅Δx
Wave Equation: F =ma
Fx =T cosθ+Δθ( )−T cosθ( ) ≈0
Fy =T sinθ+Δθ( )−Tsinθ( )≈T⋅Δθ
For small angles:
T ⋅Δθ = μ⋅Δx( )∂2y∂t2
Force massaccceleration
ΔθΔx
≈∂∂x
tanθ =∂2y∂x2
∂2y∂x2 =
μT
∂2y∂t2
Wave Equation: Solutions
Any function y(x,t) will satisfy wave eq as long as x and t appear inthe argument in the combination:x±vt
y x,t( ) =y x−vt( )=y η( )
η=x−vt
∂y∂x
=∂y∂η
∂η∂x
=∂y∂η
⋅1
∂y∂t
=∂y∂η
∂η∂t
=∂y∂η
⋅ −v( )
∂2y∂x2 =
∂2y∂η2
∂2y∂t2 =∂2y
∂η2 ⋅ −v( )2
∂2y∂x2 =v2 ∂2y
∂t2
Sample Traveling Wave
f x,t( )=1
1+ x−t( )2
t =0
Transverse traveling waves
k =2πλ
ω =2πT
v =ωk
Traveling Wave
Traveling Wave
Wave Equation Examples
∂2E∂x2 =
1c2
∂2E∂t2
∂2I∂x2 =
1c2
∂2I∂t2
Electric (E) or magnetic (B) field propagating in space from an oscillating charge (Light)
Current (I) or voltage (V) propagating in a coax cable
c - is the speed of light
The quantum-mechanical probability amplitude to find a particle at a certain location in space also satisfies a wave equation - Schroedinger’s Equation
Standing Waves
Reflected pulse
But if we have a incident sinusoidal travelingwave and a reflected sinusoidal wave:
coskx−ωt( ) −coskx+ωt( )
=2sinkx⋅sinωt
cosA +B( )=cosA ⋅cosB −sinA ⋅sinB
cosA −B( )=cosA ⋅cosB +sinA ⋅sinB
Standing Waves Trig
coskx−ωt( ) −coskx+ωt( )=2sinkx⋅sinωt
cosA −B( )−cosA +B( ) =2sinA ⋅sinB
Standing Waves
k=nπL
=2πλ
n=1
ω=kv
f =ω2π
v =Tμ
Trig Review
cosA +B( )=cosA ⋅cosB −sinA ⋅sinB
cosA −B( )=cosA ⋅cosB +sinA ⋅sinB
cosA +B( )+cosA −B( ) =2cosA ⋅cosB
a=A +B
b=A −B
A =a+b
2
B =a−b2
cosa( )+cosb( ) =2cosa−b
2⎛ ⎝
⎞ ⎠ ⋅cos
a+b2
⎛ ⎝
⎞ ⎠
Beats
cosa( )+cosb( ) =2cosa−b
2⎛ ⎝
⎞ ⎠ ⋅cos
a+b2
⎛ ⎝
⎞ ⎠
cosω1t( )+cosω2t( ) =2cosω1 −ω2
2t⎛
⎝ ⎞ ⎠ ⋅cos
ω1 +ω2
2t⎛
⎝ ⎞ ⎠