Basic Wave Concepts: Understanding Oscillations and Waves in One and Two Dimensions, Slides of Microwave Engineering and Acoustics

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BASIC WAVE CONCEPTS

REVIEW SINGLE OSCILLATOR:

The oscillation functions you’re used to describe how one

quantity (position, charge, electric field, anything ...)

changes with a single variable, TIME.

ψ( t ) = A cos( ω t + φ)

T =

2 π

f

(Cyclic) frequency, f (or ν), dimension: [time

• ]

Oscillations in time

Angular frequency, ω, dimension: [time

• ]

Period, T , dimension: [time]

Amplitude A , or ψ 0

, dimension: [whatever]

Phase, ω t +φ, dimensionless

Phase constant, φ, dimensionless

Remember the conversions between A, B, C, D

forms - see Main Ch. 1.

ψ( t ) = A cos( ω t + φ)

ψ( t ) = B

p

cos( ω t ) + B

q

sin( ω t )

ψ( t ) = Ce

i ( ω t )

• C * e

− i ( ω t )

ψ( t ) = Re De

i ω t

[ ]

Equivalent representations …….

-A

A

2 π

k

wavelength

1

-0.25 0.25 0.75 1.25 1.

time

position

space

At a FIXED TIME,^ ψ( x )^ =^ A^ cos(^ kx^^ +^ φ)

ψ( x ) = A cos( kx + φ)

λ =

2 π

k

Periodic variations in space

Wave “vector”, k , dimension: [length

• ]

(wave number is 1/λ)

Wavelength, λ, dimension: [length]

Amplitude A , or ψ 0

, dimension: [whatever]

Phase, kx +φ, dimensionless

Phase constant, φ, dimensionless

ψ( x , t ) = A cos( ω t + kx + φ)

Traveling wave

Standing wave

ψ ( x , t ) = A cos (^) ( kx )cos( ω t )

ψ ( x , t ) = A cos( ω t ± kx + φ)

Traveling waves - functions of ω t±kx

Disturbance propagates … what speed?

Look at one particular feature (constant phase)

λ =

2 π

k

; T =

2 π

ω

dx

dt

= 

ω

k

=

λ

T

ω dt ± kdx = 0

d (^) (ω t ± kx ) = 0

v = dx/dt or phase velocity is

velocity of one particular

feature.

If we had ψ = A cos(ω t - kz ), it would be v = dz/dt.

Focus on the green circle that marks a

maximum (a particular phase)

phase velocity, v phase

= ω /k = λ /T , 

dimensions [length. time

• ]

Another velocity - material/field velocity

When a wave or disturbance propagates, the

particles of the MEDIUM do not propagate, but

they move a little bit from their equilibrium

positions

material velocity, v mat

=∂ ψ /∂t , 

dimensions [ψ. time

• ]

Material/field velocity is speed of the “waving

thing” in the medium. If ψ has dimensions of

length, this is a velocity as we normally think of

it. But waves don’t necessarily need a medium in which to propagate and ψ

might well represent something more abstract like an electric field. If you

think of a better name, let me know!

Material/field velocity v mat

=∂ψ / ∂ t

dimensions [ψ. time

• ]

If the material/field velocity is perpendicular to

the phase velocity, the wave is “ transverse ”.

Examples?

If the material/field velocity is parallel to the

phase velocity, the wave is “ longitudinal ”.

Examples?

Combinations of the above are possible.

Examples?

Show wave machine

Another velocity – group velocity

There is another way to make something that has

the dimensions of a velocity:

Group velocity, v group

=∂ ω /∂k , 

dimensions [length. time

• ]

This describes the propagation of a feature in a

“wave packet” or superposition of waves of

different frequencies. We will come back to this

concept later.

Other waveforms (e.g.) sawtooth, pulses etc., can

be written as superpositions of harmonic waves of

different wavelengths and/or frequencies …

Fourier series and Fourier integrals (transforms)

ψ( x , t ) = A cos( ω t − kx + φ)

ψ( x , t ) = B

p

cos( ω t − kx ) + B q

sin( ω t − kx )

ψ( x , t ) = Ce

i ( ω t − kx )

• C * e

− i ( ω t − kx )

ψ( x , t ) = Re De

i ( ω t − kx + φ)

[ ]

Same conversions between A, B, C, D forms as for

oscillations - see Main Ch1, Ch9.

This DE results when:

Newton’s law is applied to a string under tension

Kirchoff’s law is applied to a coaxial cable

Maxwell’s equations are applied to source-free

media … and many other cases …

∂

2

∂ x

2

ψ ( x , t ) =

1

v

2

∂

2

∂ t

2

ψ ( x , t )

How do these functions arise?

PROVIDED ω/ k = v , a constant, they are solutions

to the differential equation:

(non-dispersive wave equation)