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

4. Electromagnetic waves
vertical component of force

5. Electromagnetic waves
vertical component of force
delay may be due to propagation speed of force (retarded potentials)
electric field = force per unit charge (q2)

6. Gravitational waves vertical component of force
delay due to propagation speed of force
gravitational field = force per unit mass (m2)
centre of mass motion  quadrupole radiation
delay may be due to propagation speed of force (retarded potentials)
electric field = force per unit charge (q2)
vertical component of force

7. Gravitational waves vertical component of force
delay due to propagation speed of force
gravitational field = force per unit mass (m2)
centre of mass motion  quadrupole radiation
coalescing binary stars:
neutron stars, ~1.4 solar mass
separation few tens of km
several rotations per second
stars coalesce after minutes
detector is laser interferometer several km in size

8. 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

9. 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

10. 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

11. Plucked guitar string displace string as shown
at time t = 0, release it from rest
…What happens next?

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. Waves on long strings

14. 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

15. 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

16. 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:

17. 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

18. Plucked guitar string x

19. Plucked guitar string x L ? ?

20. Plucked guitar string x L x x
L-x
L+x