Transcript Slide 1
A
cos
A
cos 2 envelope 2
t
cos 2 carrier 2
t
2
f
The diagrams below show beats that occur when two different pairs of waves are added. For which of the two is the difference in frequency of the original waves greater? A. Pair 1 B. Pair 2 C. The frequency differences are the same. D. You need more information to answer this question.
z
1
z
2 2 But *
z
1
z
1 2
z
2
z
2
z
1 * 2
z
2 * *
x
1
iy
1
x
2
iy
2 * * *
z
1
z
2 2 * 2(
z
1 2
z
2 2 ) 2 Re 2 Re *
Same frequency – different
A
z
1
ae i
z
2
be i
z
1
z
2 2
a
2
b
2 2
ab
Re
e i
e
i
with Re
e i
e
i
Re
i
e e
i
Re
e i
cos
z
1
z
2 2
a
2
b
2 2
ab
cos
arg
z
1
z
2 tan 1 Im Re
z
1
z
1
z
2
z
2 where Im
z
1 Re
z
1
z
2
z
2
a
sin
a
cos
b
sin
b
cos
waves Simple harmonic oscillator phasors, complex expon’l not’n, addition beats Energy transfer Wave equation solutions speed waves waves waves Sound waves Doppler phase/group velocities dispersion
Einstein and Infield, in their book
The Evolution of Physics
, describe a wave in this way: "The wind passing over a field of grain, sets up a wave which spreads out across the whole field. Here again we must distinguish between the motion of the wave and the motion of separate plants, which undergo only small oscillations....
The essentially new thing here is that for the first time we consider the motion of something which is not matter, but energy propagating through matter.
“ A mechanical wave is a disturbance moving through a medium. S&B §16
WAVE EQUATION
( S&B §16) The wave equation is a
linear second-order partial differential equation
. For a disturbance , which might represent the pressure in a sound wave, or the electric field in a radio wave, the amount of disturbance varies from place to place and from time to time. That is, is a function of a spatial variable,
x
, and time,
t
. The wave equation is 2
t
2
v
2 2
x
2
linear second order partial differential equation linear
because appears to the first power (linearly) throughout the equation;
second order
because the highest (indeed, the only) derivatives which appear are ∂ 2 ;
partial
because is a function of two variables,
x
and
t
, and therefore we have to use partial rather than total derivatives.
The constant
v
is the speed of the wave.
displacement of string sideways vertical movement of a small volume of water change in pressure for sound waves in a gas electric/magnetic field for light or radio waves
T F y
T
tan
B
sin
B
T
tan
A
sin
A
for small angles
T
y
x
B
F y
ma y
y
x
A
2
y
t
2
x
2
y
t
2
T
2
y T
But
t
2
f
x
y
x
B
y
x x
B
x
lim 0
A
x
x
A
T
2
y
t
2 2
y
x
2 2
y
x
2 1
v
2 2
y
t
2 with
v
T
Most general form of solution
Where
h
and
g
are any continuous, twice-differentiable functions.
vt
Single-frequency waves
x e
vt
multiplying by 2 gives a quantity with the dimensions of radians
A
amplitude
T
f
angular frequency frequency period 1/
f
2 2 2
v
v
v
wavelength
k
wavenumber phase angle 2 (wave vector)
ae i
2
ae i
2
x
t
ae i
2
x
ft
ae i
x
t v
ae a
Ae i
t
PHASE VELOCITY
v
f
k
In a sinusoidal wave, this quantity gives the speed at which peaks and troughs (points of constant phase) move through the medium – it is called the
phase velocity
.
LINEARITY/SUPERPOSITION
As the wave equation is
linear
, we may superpose solutions and still get a solution which is a solution of the wave equation.
a
1
b
2
c
3