Wave Physicsphyweb.phys.soton.ac.uk/quantum/lectures/waves/waves17...Plucked guitar string L x y x,t...

Wave PhysicsPHYS 2023

Tim Freegarde

2

Wave Physics

WAVE EQUATIONS & SINUSOIDAL SOLUTIONS

wave equations, derivations and solution

sinusoidal wave motions

complex wave functions

WAVE PROPAGATION

Huygens’ model of wave propagation

interference

general wave phenomena

Fraunhofer diffraction

longitudinal waves

BEHAVIOUR AT INTERFACES

continuity conditions

boundary conditions

SUPERPOSITIONSlinearity and superpositions

Fourier series and transforms

FURTHER TOPICSwaves from moving sources

operators for waves and oscillations

waves in three dimensions

http://www.avcanada.ca/albums/displayimage.php?album=topn&cat=3&pos=7

further phenomena and implications

3

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• Huygens reflection, refraction and diffraction• reflection and transmission at interfaces• string motion from initial conditions

4

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

5

Plucked guitar string

x

6

Plucked guitar string

? xL

?

• boundary conditions:

7

Plucked guitar string

xL

txy ,

x x L-xL+x

xy

xy

xLy

xLy

• boundary conditions:

Acoustic waves

Ophélie Gaillard, Bach, Prelude, Cello suite n. 1

9

Spectrum

• periodic waveform can be expressed as a superposition of harmonics

amplitude or intensity

• commonly represented by spectrum

10

Spectrum

• periodic waveform can be expressed as a superposition of harmonics

• commonly represented by spectrum• often plotted on log-log or lin-log scales

11

Spectrum

• periodic waveform can be expressed as a superposition of harmonics

• commonly represented by spectrum• often plotted on log-log or lin-log scales• may be plotted with frequency axis upwards...• ... with marks for integer ... • ... and time on the horizontal axis

12

Boundary conditions• for a boundary at :

• transverse waves on a string: fixed point

• sound waves in organ pipe:

• water waves:

• thermal waves:

• electromagnetic waves:

wall

since

lagged end

conductor

open

stopped

•

thermal reservoir

13

Open organ pipe• first six harmonics lie on major chord

J S Bach Toccata & Fugue in D minor BWV565

14

Stopped organ pipe• even harmonics are absent

•

• register key raises by a twelfthCLARINET:

15

Oboes and saxophones• even harmonics are present

•

• register key raises by a twelfthCLARINET:

• register key raises by an octaveSAXOPHONE:

Wind instruments

•

17

The bell

• cyclic boundary conditions