### lecture 2

```Wave Physics
PHYS 2023
Tim Freegarde
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
2
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
equilibrium
eg Poisson’s equation
dynamic
SHM
WAVES
3
Electromagnetic waves
• vertical component of force
4
Electromagnetic waves
• vertical component of force
• delay may be due to propagation speed of force (retarded potentials)
• electric field = force per unit charge (q2)
5
Gravitational waves
a

• vertical component of force
• delay due
may to
bepropagation
due to propagation
speed ofspeed
force of force (retarded potentials)
• electric
field =
force
per unit
(q2)(m2)
gravitational
field
= force
percharge
unit mass
• centre of mass motion  quadrupole radiation
6
Gravitational waves
m1m2
• coalescing
binary stars:
• neutron
  G~1.4 solar 3mass
vertical component
of force
F tstars,
at  r c 
4
r
• separation few tens
of 0km
several
per second
• delay due to propagation•speed
of rotations
force
• stars coalesce after minutes
• gravitational field = force per unit mass (m2)
• detector is laser interferometer several km in size
• centre of mass motion  quadrupole radiation
7
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
equilibrium
eg Poisson’s equation
dynamic
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
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
• 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
• where
• density
9
Wave equations
• 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
use physics/mechanics to
write partial differential wave
equation for system
insert generic trial form of
solution
e.g.
• but note that not all wave equations are of the same
form
find parameter values for
which trial form is a solution
10
Plucked guitar string
• displace string as shown
• at time t = 0, release it from rest
• …What happens next?
11
Wave equations
• 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
use physics/mechanics to
write partial differential wave
equation for system
insert generic trial form of
solution
e.g.
• but note that not all wave equations are of the same
form
find parameter values for
which trial form is a solution
12
Waves on long strings
13
Solving the wave equation
• shallow waves on a long thin flexible string
• travelling wave
• wave velocity
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
14
Travelling wave solutions
• consider a wave shape
at
which is merely translated with time
where
use physics/mechanics to
write partial differential wave
equation for system
insert generic trial form of
solution
• use chain rule for derivatives
find parameter values for
which trial form is a solution
15
General solutions
use physics/mechanics to
write partial differential wave
equation for system
• wave equation is linear – i.e. if
are solutions to the wave equation,
then so is
insert generic trial form of
solution
arbitrary constants
• note that two solutions to our example:
find parameter values for
which trial form is a solution
16
Particular solutions
• fit general solution to particular constraints – e.g.
x
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
17
Plucked guitar string
x
18
Plucked guitar string
yx,0t 
L
?
?
x
19
Plucked guitar string
yx, t 
y  x 
x
y  L  x 
y  x 
x
L+x
L
L-x
x
y  L  x 
20
```