Transcript Electric Potential - McMaster Physics and Astronomy

Waves
Chapter 16: Traveling waves
Chapter 18: Standing waves, interference
Chapter 37 & 38: Interference and diffraction
of electromagnetic waves
Wave Motion
A wave is a moving pattern that moves along a
medium. For example, a wave on a stretched string:
v
time t
Δx = v Δt
t +Δt
The wave speed v is the speed of the pattern.
Waves carry energy and momentum.
Transverse waves
The particles move
up and down
The wave moves this way
If the particle motion is perpendicular to the direction
the wave travels, the wave is called a “transverse wave”.
Examples: Waves on a string; light & other
electromagnetic waves.
Longitudinal waves
The particles move
back and forth.
The wave moves this way
Example: sound waves in gases
Even in longitudinal waves, the particle velocities
are quite different from the wave velocity.
Quiz
B
A
C
wave motion
Which particle is moving at the highest speed?
A) A
B) B
C) C
D) All move with the same speed
Non-dispersive waves: the wave always keeps the
same shape as it moves.
For these waves, the wave speed is constant
for all sizes & shapes of waves.
Eg. stretched string:
tension
v=
mass/unit length
=
T

Here the wave speed depends only on tension and
mass density .
Reflections
Waves (partially) reflect from any boundary
in the medium:
1) “Soft” boundary:
Reflection is upright
light string, or free end
Transmitted is upright
Reflections
2) “Hard” boundary:
heavy rope, or fixed end.
Reflection is inverted
Transmitted is upright
The math: Suppose the shape of the wave at t = 0, is
given by some function y = f(x).
y
v
at time t = 0: y = f (x)
vt
at time t :
y
y = f (x - vt)
Note: y = y(x,t), a function of two variables;
For non-dispersive waves:
y (x,t) = f (x ± vt)
+ sign: wave travels in negative x direction
- sign: wave travels in positive x direction
f is any (smooth) function of one variable.
eg. f(x) = A sin (kx)
Quiz
y
v
x
A wave y=f(x-vt) travels in the positive x direction. It
reflects from a fixed end at the origin. The reflected
wave is described by:
A)
B)
C)
D)
E)
y= - f(x-vt)
y= - f(x+vt)
y= - f(- x-vt)
y= - f(- x+vt)
something else
Travelling Waves
The most general form of a traveling sine wave
(harmonic waves) is y(x,t)=A sin(kx ± ωt +f )
amplitude
“phase”
y(x,t) = A sin (kx ± wt +f )
Wavenumber k =
phase constant f
2

The wave speed is
angular frequency
ω = 2πf
v=
w
k
= f
This is NOT the speed of the particles on the wave
For sinusoidal waves, the shape is a sine function, eg.,
(A and k are constants)
f(x) = y(x,0) = A sin(kx)
A
y
x
-A
One wavelength 
Then y (x,t) = f(x – vt) = A sin[k(x – vt)]
or
y (x,t) = A sin (kx – wt)
with w = kv
Phase constant:
f(x) = y(x,0) = A sin(kx+f)
A
<- snapshot at t=0
y
x
-A
To solve for the phase constant f, use y(0,0):
eg: y(0,0)=5, y(x,t)=10sin(kx-ωt+f)
Principle of Superposition
When two waves meet, the displacements add algebraically:
yobserved ( x, t ) = y1 ( x, t )  y2 ( x, t )
So, waves can pass through each other:
v
v
Light Waves (extra)
Light (electromagnetic) waves are produced by
oscillations in the electric and magnetic fields:
2E
2E
=  o o 2
2
x
t
2B
2B
=  o o 2
2
x
t
For a string it can be
shown that (with F=T):
These are ‘linear wave equations’, with
v=c=
1
 o o
= 2.99792458108 m / s
2 y  2 y
=
2
x
F t 2