1. Wave Physics
PHYS 2023
Tim Freegarde

2. Coming up in Wave Physics...
today’s lecture:
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

3. Wave Physics Local/microscopic definition:
a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points
speed of propagation is derived
particles (Lagrange)
fields (Euler)
static
dynamic
equilibrium
eg Poisson’s equation
SHM
WAVES
Macroscopic definition:
a time-dependent feature in the field of an interacting body, due to the finite speed of propagation of a causal effect
speed of propagation is assumed

4. Wave Physics Local/microscopic definition:
a collective bulk disturbance in which what happens at any given position is a delayed response to the disturbance at adjacent points
speed of propagation is derived
What is the net force on the penguin?
rest position
For an elastic penguin, Hooke’s law gives
separation
displacement
If the penguin has mass , Newton’s law gives
pressure
elasticity
density
where

5. Waves on long strings

6. Solving the wave equation
shallow waves on a long thin flexible string
use physics/mechanics to write partial differential wave equation for system
travelling wave
insert generic trial form of solution
wave velocity
find parameter values for which trial form is a solution

7. Travelling wave solutions
consider a wave shape at
which is merely translated with time
use physics/mechanics to write partial differential wave equation for system
where
insert generic trial form of solution
use chain rule for derivatives
find parameter values for which trial form is a solution

8. General solutions wave equation is linear – i.e. if
use physics/mechanics to write partial differential wave equation for system
are solutions to the wave equation, then so is
insert generic trial form of solution
arbitrary constants
find parameter values for which trial form is a solution
note that two solutions to our example:

9. Particular solutions
fit general solution to particular constraints – e.g.
use physics/mechanics to write partial differential wave equation for system
insert generic trial form of solution x
find parameter values for which trial form is a solution

10. Plucked guitar string x

11. Wave propagation transverse motion of taut string
use physics/mechanics to write partial differential wave equation for system
e-m waves along coaxial cable
shallow-water waves
flexure waves
string with friction
travelling wave:
general form
sinusoidal
insert generic trial form of solution
complex exponential
damped
standing wave
soliton
speed of propagation
find parameter values for which trial form is a solution
dispersion relation
string motion from initial conditions

12. Wave equations
waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points
use physics/mechanics to write partial differential wave equation for system
propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position
insert generic trial form of solution
e.g.
find parameter values for which trial form is a solution
but note that not all wave equations are of the same form

13. Electromagnetic waves
aliexpress.com
NASA/DOE/Fermi LAT Collaboration
Light
Radio
Gamma radiation

14. Waves along a coaxial cable
V(x)
V(x+δx) r a I x
-δQ δQ x
x+δx

15. Waves along a coaxial cabler a x
-δQ δQ x
x+δx

16. Waves along a coaxial cabler a I x x
x+δx

17. Water waves Ocean waves Severn bore Kelvin ship wake Tsunami
theguardian.com
© Jason Hawkes / Getty Images
© Reuters / Mainichi Shimbun
Ocean waves
Severn bore
Kelvin ship wake
Tsunami

18. Shallow-water waves ε1 ε2 h(x) volume = h(x) (δx+ε2-ε1) δy v1 v2 δx

19. Shallow-water waves h(x) volume = h(x) (δx+ε2-ε1) δy v1 δx x-δx x x+δx

20. Velocities of waves on a stringx x
phase velocity
(group velocity)
transverse string velocity