Wave PhysicsPHYS 2023

Tim Freegarde

2

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:

• energy in wave motions

• standing wave solutions

3

Sinusoidal waves

• conveniently manipulated in differential equations• allow standing wave solutions• only real travelling wave solutions in dispersive systems• useful terminology for dissipative systems• complete set of orthogonal solutions in linear systems

• occur when source is rotating or performing SHM• detected by resonant detectors

• convenient for energy/power calculations

44

Sinusoidal waves

0tt

0tt at ,

• wavenumber2

k

x

y

000 sin, tkxytxy

• spectroscopists’

1~ wavenumber

00~2sin txy

00

2sin txy • wavelength

kxtytxy sin, 0

55

Sinusoidal waves

0xx

0xx at ,

• angular frequency 2

t

y

000 sin, kxtytxy

• frequency

1 00 2sin kxty

00

2sin kxty • period

kxtytxy sin, 0

6

Complex wave functions

φ

tkxtxy sin, tkxi expRe txi ,expRe

• simple harmonic motion • circular motion

7

Dispersion in dissipative systems

xx x+δxx-δx

2

1 0, txy

y

W

Wδx

xM v

8

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

9

Energy of waves on a string

xx

h(x)

δx

vδy

• kinetic:

• potential:

10

Velocities of waves on a string

xx

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

11

Sumatra-Andaman earthquake 2004

• magnitude 9.15; 275,000 perished

Tsunami Inundation Mapping EffortsNOAA/PMEL - UW/JISAO

• 1200 km along India/Burma plate subduction zone• slip of 15 m sideways, several metres vertically• formed ridges 1.5 km high, trench kms wide

26 Dec 2004

• 30 km3 water displaced

12

Sumatra-Andaman earthquake 2004

• magnitude 9.15; 275,000 perished

NOAA

• 1200 km along India/Burma plate subduction zone• slip of 15 m sideways, several metres vertically• formed ridges 1.5 km high, trench kms wide• 30 km3 water displaced

• ocean waves ~ 60 cm• 25 m high near shore

13

Deep water waves

xx x+δxx-δx

h(x)

δx

v2v1

volume = h(x) (δx+ε2-ε1) δy

ε2ε1

14

Deep water waves

xx x+δxx-δx

h(x)

δx

v1

volume = h(x) (δx+ε2-ε1) δy

1515

BeatingTWO DIFFERENT FREQUENCIES

tttt2

cos2

coscoscos 212121

•

16

Group velocity

• 2 sinusoidal components:• 10 sinusoidal components:• spreading of wavepacket

• this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation

• generally: the group velocity = speed of energy propagation= speed of information propagation

•

17

Dispersion and group velocity

• time-dependent Schrödinger equation(in 1-D)

• wavefunction of atomic electron

• flexural waves (Kirchhoff plate theory)

18

Kelvin ship waves

19

Hull speed

20

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

How’s the course so far?https://www.isurvey.soton.ac.uk/25366