WAVE MOTION

1. Wave motion is any kind of self-repeating (periodic, or oscillatory) motion that transports
energy from one point to another. Waves are of two basic kinds:
(a) Longitudinal Waves: the oscillation is parallel to the direction of wave travel.
Examples: sound, spring, "P-type" earthquake waves.
(b) Transverse Waves: the oscillation is perpendicular to the direction of wave travel.
Examples: radio or light waves, string, "S-type" earthquake waves.
2. Waves transport energy, not matter. Taking the vibration of a string as an example, each
segment of the string stays in the same place, but the work done on the string at one end is
transmitted to the other end. Work is done in lifting the mass at the other end below.
0
3. The height of a wave is called the amplitude. The average power (or intensity) in a wave
is proportional to the square of the amplitude. So if ( ) sin( ) is a wave of some
kind, then
a t =a ωt−kx
2
0 0 is the amplitude and .
4. A sound source placed at the origin will radiate sound waves in all directions equally. These
are called spherical waves. For spherical waves the amplitude 1 and
a I a
r
∝
∝ 2
so the power 1 .
We can easily see why this is so. Consider a source of sound and draw two spheres:
r
∝
1 1 1
1 2
2
1 1
Let be the total radiated power and I the intensity at , etc. All the power (and energy)
that crosses also crosses since none is lost in between the two. We have that,
4
P r
r r
π r I = P
2
2 1 2
1 2 2 2 1 2 2 2
2 1
and 4 r I P. But P P P, and so I r or I 1 .
I r r
π = = = = ∝
( ) ( )
( )
( )
0 5. We have encountered waves of the kind , sin in the previous lecture.
Obviously 0,0 0. But what if the wave is not zero at 0, 0? Then it could be
represented by , sin m
y x t y kx t
y xt
y x t y kx
= −ω
= = =
= ( − )
( )
( )
, where is called the phase and
is called the phase constant. Note that you can rewrite , either as,
a) , sin ,
or as, b)
m
t kx t
y x t
y x t y k x t
k
ω φ ω φ
φ
φ
ω
− − −
= ⎡⎢⎣ ⎛⎜⎝ − ⎞⎟⎠− ⎤⎥⎦
( , ) sin .
The two different ways of writing the same expression can be interpreted differently. In
(a) has effectively been shifted to whereas in (b) has been
m y x t y kx t
x x t
k
φ
ω
ω
φ
= ⎡⎢⎣ − ⎛⎜⎝ + ⎞⎟⎠⎤⎥⎦
− shifted to . So
the phase constant only moves the wave forward or backward in space or time.
6. When two sources are present the total amplitude at any point is the sum of the two
separate a
t φ
ω
+
( ) ( ) ( ) 1 2
2
1 2
mplitudes, , , , . Now you remember that the power is
proportional to the of the amplitude, so ( ) . This is why
happens. In the following we shall see why. J
y x t y x t y x t
square P y y interference
= +
∝ +
( ) ( ) ( ) ( ) 1 1 2 2
ust to make things easier, suppose the two
waves have equal amplitude. So lets take the two waves to be :
, sin and , sin
The total amplitudes is:
m m y x t = y kx−ωt−__________φ y x t = y kx−ωt−φ
( ) ( ) ( )
( ) ( )
( ) ( )
1 2
1 2
, , ,
sin sin
Now use the trigonometric formula, sin sin 2sin 1 cos to get,
2
m
y x t y x t y x t
y kx t kx t
B C B C B C
ω φ ω φ
= +
= ⎡⎣ − − + − − ⎤⎦
+ = + × −
( ) ( ) ( )
( )
1 2
2 1
, sin sin
2 cos sin .
2
Here is the difference of phases, and
m
m
y x t y kx t kx t
y kx t
ω φ ω φ
φ
ω φ
φ
φ φ φ φ
= ⎡⎣ − − + − − ⎤⎦
=⎡⎢⎣ ⎛⎜⎝Δ ⎞⎟⎠⎤⎥⎦× − − ′
Δ = − ′ =( ) 1 2
2 1
2
is the sum. So what do
2
we learn from this? That if , then the two waves are in phase and the resultant
amplitude is maximum (because cos0 1). And that if , then the two waves a
φ
φ φ
φ φ π
+
=
= = + re
out of phase and the resultant amplitude is minimum (because cos / 2 0). The two
waves have interfered with each other and have increased/decreased their amplitude in
these two extreme cas
π =
es. In general cos will be some number that lies between
2
-1 and +1.
⎛Δφ ⎞
⎜ ⎟
⎝ ⎠
x′
y
P
f (x′)
O
y ′
O′
vt
7. There was no time in the lecture to prove it, but you can look up any good book to
find a proof for the formaula that the speed of sound in a medium is: v where
is the bulk modulus and
B
B
ρ
=
ρ is the density of the medium.
( ) ( )
8. A pulse is a burst of energy (sound, electromagnetic, heat,...) and
could have any shape. Mathematically any pulse has the form , v . Here
is any function (e.g
y x t f x t
f
= −
The speed of a pulse.
. sin, cos, exp,....). Note that at time 0, ( ,0) ( ) and the
shape would look as on the left in the diagram below. At a late time t, it will look just the
same, but shifted to the right. In ot
t= yx =f x
her words at time , ( , ) ( ) where v .
Fix your attention on any one point of the curve and follow it as the pulse moves to the
right. From v constant it follows that v 0 , or v
t y x t f x x x t
x t dx d
dt
= ′ ′= −
− = − = = . This is called the
because we derived it using the constancy of phase.
x
dt
phase velocity