Wave PhysicsPHYS 2023

Tim Freegarde

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Coming up in Wave Physics...

• local and macroscopic definitions of a wave

• transverse waves on a string: • wave equation

• travelling wave solutions

• other wave systems: • electromagnetic waves in coaxial cables

• shallow-water gravity waves

• sinusoidal and complex exponential waveforms

• today’s lecture:

3

Wave Physics

• a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points

Local/microscopic definition:

• a time-dependent feature in the field of an interacting body, due to the finite speed of propagation of a causal effect

Macroscopic definition:

• speed of propagation is derived

• speed of propagation is assumed

static

dynamic

particles (Lagrange) fields (Euler)

equilibrium

SHM

eg Poisson’s equation

WAVES

4

Wave Physics

• a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points

Local/microscopic definition:

• speed of propagation is derived

• What is the net force on the penguin?

• For an elastic penguin, Hooke’s law gives

• If the penguin has mass , Newton’s law gives

• rest position

• displacement• pressure• elasticity• density

• separation

• where

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Waves on long strings

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Solving the wave equation

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• shallow waves on a long thin flexible string

• travelling wave

• wave velocity

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Travelling wave solutions

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• use chain rule for derivatives

where

• consider a wave shape at which is merely translated with time

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General solutions

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• wave equation is linear – i.e. if

are solutions to the wave equation, then so is

arbitrary constants

• note that two solutions to our example:

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Particular solutions

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• fit general solution to particular constraints – e.g.

x

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Plucked guitar string

x

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Wave propagation

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• transverse motion of taut string

• travelling wave:

• e-m waves along coaxial cable• shallow-water waves• flexure waves• string with friction

• general form• sinusoidal• complex exponential

• standing wave• damped

• soliton

• speed of propagation• dispersion relation• string motion from initial conditions

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Wave equations

use physics/mechanics to write partial differential wave

equation for system

insert generic trial form of solution

find parameter values for which trial form is a solution

• waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points

• propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position

• but note that not all wave equations are of the same form

e.g.

Electromagnetic waves

• Radio

• Gamma radiation

• Light

aliexpress.com

NASA/DOE/Fermi LAT Collaboration

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Waves along a coaxial cable

b

x x+δx

r

-δQI

δQ

I

x

x

a

V(x) V(x+δx)

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Waves along a coaxial cable

b

x x+δx x

x

a

-δQ

r

δQ

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Waves along a coaxial cable

b

x x+δx x

x

a

r

I

I

Water waves

• Severn bore

• Kelvin ship wake

• Ocean waves

www.fluidconcept.co.uk/Images/Uploads/capetown1-400-279.jpg

theguardian.com

© Jason Hawkes / Getty Images

• Tsunami

© Reuters / Mainichi Shimbun

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Shallow-water waves

xx x+δxx-δx

h(x)

δx

v2v1

volume = h(x) (δx+ε2-ε1) δy

ε2ε1

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Shallow-water waves

xx x+δxx-δx

h(x)

δx

v1

volume = h(x) (δx+ε2-ε1) δy

x

hhgtv

121

dd

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Velocities of waves on a string

xx

• phase velocity• (group velocity)• transverse string velocity