Traveling Waves
x x + Δx
TT
θ
θ+Δθ
ymass =
μ ⋅Δx
Wave Equation
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 v2 =
Tμ
Wave Equation ∂2y∂x2 =
μT
∂2y∂t2
v2 =Tμ
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
tension
mass/lengthvelocity
Wave Equation ∂2y∂x2 =
μT
∂2y∂t2
v2 =Tμ
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
Solutions to Equation
k =2πλ
ω =2πT
v =ωk
The disturbance y(x,t) performs simple harmonic motion aboutequilibrium in t (for fixed x) or x (for fixed t).
Wave Equation
∂2E∂x2 =
1c2
∂2E∂t2
∂2I∂x2 =
1c2
∂2I∂t2
Other quantities satisfying wave equation
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
Transverse traveling waves
k =2πλ
ω =2πT
v =ωk
Sample Traveling Wave
f x,t( )=1
1+ x−t( )2
t =0
Change the x-axis scale
Sample Traveling Wave
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Traveling Wave
Traveling Wave
Traveling Wave
y x,t( ) =cosx−t( )
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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
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