Superposition - Waves - Lecture Slides, Slides of Microwave Engineering and Acoustics

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WAVE PACKETS & SUPERPOSITION

k →

1

σ (^) x σ (^) x

x →

2

∂ x

2

ψ ( x , t ) =

v

2

2

∂ t

2

ψ ( x , t )

Non dispersive wave equation

So far we’ve discussed single-frequency waves,

but we did an experiment with a pulse … so let’s

look more formally at superposition. You must

also review Fourier discussion from PH421.

So far, we know that this equation results from

• application of Newton’s law to a taut rope
• application of the Maxwell equations to a

dielectric medium

(What do the quantities represent in each case?)

Example: Non dispersive wave equation

A general solution is the superposition of solutions of all possible

frequencies (or wavelengths). So any shape is possible!

ψ ( x , t ) =

A

ω

cos

ω v

x cos ω t + B

ω

cos

ω v

x sin ω t

+ C

ω

sin

ω v

x cos ω t + D

ω

sin

ω v

ω x^ sin^ ω t

ψ ( x , t ) = Re L ( k ) e

i (^) ( kx − vkt ) ⎡ ⎣

• Re M ( k ) e

i (^) ( kx + vkt ) ⎡ ⎣

{ dk }

2

∂ x

2

ψ ( x , t ) =

v

2

2

∂ t

2

ψ ( x , t )

Example 1:

Wave propagating in rope (phase vel. v ) with fixed boundaries

at x = 0, L. (Finish for homework)

Which superposition replicates the initial shape of the

wave at t = 0? How can we choose coefficients A ω

, B

ω

C

ω

, D

ω

to replicate the initial shape & movement of the

wave at t = 0?

ψ (^) ( x , (^0) ) = given; what is it?

ψ ( x , t ) =

A

ω

cos

ω v

x cos ω t + B

ω

cos

ω v

x sin ω t

+ C

ω

sin

ω v

x cos ω t + D

ω

sin

ω v

ω x^ sin^ ω t

∑

∂ψ (^) ( x , t )

∂ t t = 0

= given; what is it?

Example 2:

Class activity - build a Gaussian wave packet

ψ ( x , t ) = A

0

e

−

( x − vt )

2

2 σ (^) x

2

x

x →

ψ ( x , 0 ) = A

0

e

−

x

2

2 σ (^) x

2

This shape can propagate in a rope, right? Therefore it must be

a solution to the wave equation for the rope. It’s obviously not a

single-frequency harmonic. Then which superposition is it?

How is this problem different from the fixed-end problem we

recently discussed?

Gaussian wave packet

ψ ( x , t ) = A

0

e

−

( x − vt )

2

2 σ (^) x

2

ψ ( x , 0 ) = dk L ( k ) e

ikx

−∞

∞

∫

Which superposition replicates the initial shape of

the wave at t = 0?

ψ ( x , 0 ) = A

0

e

−

x

2

2 σ (^) x

(^2) How can we choose coefficients

L ( k ) to replicate the initial shape

of the wave at t = 0?

(note we dropped the “Re” - we’ll

put it back at the end)

Build a Gaussian wave packet

L ( k ) =

dx A 0

e

−

x

2

2 σ (^) x

2

e

− ikx

−∞

∞

∫

dy e

vy e

− uy

2

−∞

∞

∫ =^

π

u

e

v

2

4 u

L ( k ) =

π

x

2

e

(^ i k )

2

4

1

2 σ (^) x

2

⎛

⎝

⎜

⎞

⎠

⎟

v → ik ; u →

x

2

L ( k ) =

x

2

e

−

σ (^) x

2 k

2

2

k →

x

Build a Gaussian wave packet

ψ ( x , 0 ) = dk L ( k ) e

ikx

−∞

∞

∫

ψ ( x , 0 ) = dk

x

2

e

−

σ (^) x

2 k

2

2 e

ikx

−∞

∞

∫

ψ ( x , t ) = dk

x

2

e

−

σ (^) x

2 k

2

2 e

i (^) ( kx − kvt )

−∞

∞

∫

ψ ( x , t ) = A

0

e

−

( x − vt )

2

2 σ (^) x

2

Now put back the time dependence … remember

each k -component evolves with its own velocity …

which just happens to be the same for this non-

dispersive case

Build a Gaussian wave packet - summary

ψ ( x , t ) = dk

x

2

e

−

σ (^) x

2 k

2

2 e

i (^) ( kx − kvt )

−∞

∞

∫

ψ ( x , t ) = A

0

e

−

( x − vt )

2

2 σ (^) x

2 Initial pulse - shape

described by Gaussian

spatial function

Written as a “sum” of

“sinusoids” of different

wavelengths (and freqs)

• Important to be able to deconstuct an arbitrary

waveform into its constituent single-frequency or

single-wavevector components. E.g. R and T values

depend on k. Can find out how pulse propagates.

• Especially important when the relationship between k

and ω is not so simple - i.e. the different wavelength

components travel with different velocities

(“dispersion”) as we will find in the QM discussion

Build a Gaussian wave packet - check

ψ ( x , t ) = dk

x

2

e

−

σ (^) x

2 k

2

2 e

i (^) ( kx − kvt )

−∞

∞

∫

ψ ( x , t ) = A

0

e

−

( x − vt )

2

2 σ (^) x

2

dy e

vy e

− uy

2

−∞

∞

∫ =^

π

u

e

v

2

4 u

ψ ( x , t ) =

π

x

2

x

2

e

− (^) ( x − vt )

2

4

σ (^) x

2

2

• Other shapes than sinusoids can propagate in systems
• Other shapes are superpositions of sinusoids
• Fourier integrals/series for unconstrained/constrained

conditions

• Fourier transform of a narrow/wide Gaussian is a wide/

narrow Gaussian

• Group velocity, phase velocity
• Mathematical representations of the above

WAVE PACKETS & SUPERPOSITION -

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