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⎛

⎝ ⎞ ⎠