Waves in One dimension
A wave medium is an object, each of whose points can undergo
simple harmonic motion (SHM).
Examples:
1. A string. Each little piece of string can oscillate in a plane
perpendicular to the string’s length.
2. The free surface of a body of water. Each little piece of surface
can oscillate up and down.
3. The air in a room. Each little group of air molecules can
oscillate back and forth in any direction.
5. The electric and magnetic fields in vacuum.
Whenever a wave medium is disturbed, the material
composing the medium will oscillate. The oscillations will
occur for many points on the material. In general, the
oscillations at two different locations will be very different, but
in some special cases the oscillations at two different locations
are simply related to one another. In this case, we say that a
simple wave exists on the material.
.
Suppose a simple wave exists on a taut string.
Suppose one little piece of a string is oscillating up and down
according to the equation y (t )  A cos( t )
Let’s put the x-axis along the undisturbed length of string and
put the coordinate origin at the equilibrium location of this
piece of string. Then we’d say that
y ( x  0, t )  A cos( t )
where y(x,t) denotes the displacement above or below
equilibrium of the piece of string whose equilibrium location is
x.
Nearby pieces of string are also oscillating. Let’s assume that
all points are oscillating with the same period T and amplitude
A. Pieces of string that are close to together must have nearly
the same phase.
Why?
Suppose the phase changes by 2 radians every time we
move along the x-axis a distance . This is called the
wavelength of the wave.
The wave number k is defined to be the change in phase of the
wave per meter moved along the x-axis (at a fixed time). For
example, if the phase changes by 8.0 radians in a distance
x  2.0 m the wave number is k = (8.0 radians)/2 m = 4.0
radians/m. A property of simple waves is that the wave number
doesn’t change over time.
Problem: What is the relation between the wave number k and
the wavelength ?
Returning to our string, we know how the little bit of string near
x = 0 is moving.
y ( x  0, t )  A cos( t )
The phase of the wave at x = 0 is therefore  t
Suppose we also know the wave number k. Then moving a
distance x along the x-axis away from the origin should change
the phase by kx.
phase of wave at position x   t  k x
If all parts of the string are oscillating with the same amplitude,
this means that
y ( x, t )  A cos( t  kx)
Question: If the wave is represented by the amplitude function
y ( x, t )  A cos( t  kx)
In what direction is the wave moving?
Does the wave look like this?
Or like this?
The height of a bit of string at position x changes with time according to
 y ( x, t )  A cos(kx   t )

  A sin(kx   t )
t
t
This is the vertical velocity of the piece of string. Pick a particular instant of
time, say t = 0. Then
v y ( x)   A sin(k x)
Looking at bits of string to the right of x = 0 at this instant,
the vertical velocity points downward and increases in magnitude
as we move to the right.
The height of a bit of string at position x changes with time according to
 y ( x, t )  A cos(kx   t )

  A sin(kx   t )
t
t
This is the vertical velocity of the piece of string. Pick a particular instant of
time, say t = 0. Then
v y ( x)   A sin(k x)
Looking at bits of string to the right of x = 0 at this instant,
the vertical velocity points downward and increases in magnitude
as we move to the right.
As we saw before, this
means that the wave moves
to the left.
What amplitude function would you use to represent a wave (with
amplitude A, frequency , and wave number k) moving to the right?
What amplitude function would you use to represent a wave (with
amplitude A, frequency , and wave number k) moving to the right?
y ( x, t )  A cos( t  kx)
What amplitude function would you use to represent a wave (with
amplitude A, frequency , and wave number k) moving to the right?
y ( x, t )  A cos( t  kx)  A cos(kx   t )
> with(plots):
> k:=1; omega:=1;
> animate(cos(k*x-omega*t),x=
10..10,t=1..20,frames=50);
Then animate cos(k*x+omega*t)
Another way to see the same thing is to look at a particular
wave crest. Use the amplitude function y ( x, t )  A cos(k x -  t ).
The wave crests occur where the phase is 0,  2 ,  4 ,  6 etc.
Pick a particular wave crest where the phase is 2 m for some
integer m.
This wave crest is at location x at time t when k x -  t
 2 m. If the wave crest is at position x  x at time t  t , then
k  x  x  -   t  t   2 m so that k x  k x -  t   t  2 m
But k x -  t  2 m so that k x   t  0
This means that as t gets bigger, x gets bigger.
As time increases, the x-position of the wave crest
increases. In other words, the wave moves to the right.
In the same way, a wave described by the amplitude
function y ( x, t )  A cos( kx   t ) has wave crests at
positions x at times t satisfying kx   t  2 m.
In t time the wave crest moves a distance x satisfying
k x  t  0. So as time increases the x-coordinate of the
wave crest becomes smaller: the wave is moving to the left.
y ( x, t )  A cos(kx   t )
y ( x, t )  A cos(kx   t )
wave moving in + xˆ direction
wave moving in -xˆ direction
Let’s define the wave vector:
k  a vector of magnitude k (the wave number) which points
in the direction the wave is moving
so that
 k x if the wave is moving to the right
ˆ
k  x  k   x x  
k x if the wave is moving to the left
then no matter which direction the wave is moving in,
the amplitude function can always be written as

y ( x, t )  A cos k  x   t

How far does a wave crest move in a time of one period?
In one period the wave moves a distance of one wavelength
distance moved in any time t
speed of a wave crest 
t
distance moved in time T 


T
T
This is called the phase velocity of the wave
v

T
In one period the wave moves a distance of one wavelength
distance moved in any time t
speed of a wave crest 
t
distance moved in time T 


T
T
This is called the phase velocity of the wave
v
v

T


T

1/ f
f
In one period the wave moves a distance of one wavelength
distance moved in any time t
speed of a wave crest 
t
distance moved in time T 


T
T
This is called the phase velocity of the wave
v
v

T


T

f
1/ f
 2     
vf 


 k   2  k
Another way to see the same thing:
We saw earlier that for a wave described by the amplitude
function y ( x, t )  A cos( k x   t ), the position of a wave
crest changes by x in a time t where k x   t  0
Therefore,
x

 phase velocity 
t
k
Summary:
1. Waves can be either transverse (where the oscillating
quantity moves perpendicular to the wave) or longitudinal
(where the oscillating quantity moves parallel to the wave)
2. Waves may be either traveling waves or standing waves.
3. For a simple one-dimensional wave at any fixed time, the
phase of the oscillations at two points varies linearly
with distance:  = k x where k 
2

4. The wave vector k is defined to have magnitude k
and to point in the direction of wave motion. If the
displacement from the coordinate origin is denoted
x, then any simple wave has the general amplitude function
y ( x, t )  A cos(k  x   t   )
5. Speed of a wave crest  phase velocity 

k
f