Wave PhysicsPHYS 2023

Tim Freegarde

03 Waves on taut strings2020-21

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:

3

Plucked guitar string

• displace string as shown

• at time t = 0, release it from rest

• …What happens next?

4

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.

5

Waves on long, taut strings

6

Derivation of the wave equation

7

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

8

Travelling wave solutions

9

Differentiation as the limit

10

Wave velocity

11

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

12

General solutions in linear systems

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:

• the general solution is therefore:

for the case of the taut string

13

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

14

Plucked guitar string – 1

15

Plucked guitar string – 2

16

Plucked guitar string

x

17

Plucked guitar string

?xL

?

18

Plucked guitar string

xL

x x L-x

L+x

19

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

20

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.

Wave PhysicsPHYS 2023

Tim Freegarde