waves in one dimension. a wave medium is an object, each of whose points can undergo simple harmonic...

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

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

where y(x,t) denotes the displacement above or below equilibrium of the piece of string whose equilibrium location is x.

( ) cos( )y t A t

( 0, ) cos( )y x t A t

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

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 ?

2.0x m

Returning to our string, we know how the little bit of string near x = 0 is moving.

The phase of the wave at x = 0 is therefore

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.

If all parts of the string are oscillating with the same amplitude, this means that

( 0, ) cos( )y x t A t

t

phase of wave at position x t k x

( , ) cos( )y x t A t kx

Question: If the wave is represented by the amplitude function

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

( , ) cos( )sin( )

This is the vertical velocity of the piece of string. Pick a particular instant of

time,

y x t A kx tA kx t

t t

say t = 0. Then

v ( ) sin( )

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.

y x A k x

The height of a bit of string at position x changes with time according to

( , ) cos( )sin( )

This is the vertical velocity of the piece of string. Pick a particular instant of

time,

y x t A kx tA kx t

t t

say t = 0. Then

v ( ) sin( )

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.

y x A k x

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?


( , ) cos( ) cos( )y x t A t kx A 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 ( , ) cos( - ).

The wave crests occur where the phase is 0, 2 , 4 , 6 etc.

Pick a particular wav

y x t A k x t

e crest where the phase is 2 for some

integer .

m

m

This wave crest is at location at time when -

2 . If the wave crest is at position at time , then

- 2 so that - 2

But - 2 so that 0

x t k x t

m x x t t

k x x t t m k x k x t t m

k x t m k x t

This means that as gets bigger, gets bigger.

As time increases, the x-position of the wave crest

increases. In other words, the wave moves to the right.

t x

In the same way, a wave described by the amplitude

function ( , ) cos( ) has wave crests at

positions at times satisfying 2 .

In time the wave crest moves a distance satisfying

y x t A kx t

x t kx t m

t x

k x

0. So as time increases the x-coordinate of the

wave crest becomes smaller: the wave is moving to the left.

t

ˆ( , ) cos( ) wave moving in direction

ˆ( , ) cos( ) wave moving in - direction

y x t A kx t

y x t A kx t

+ x

x

Let’s define the wave vector:

a vector of magnitude (the wave number) which points

in the direction the wave is moving

k k

so that

if the wave is moving to the rightˆ

if the wave is moving to the left

k xx

k x

k x k x

then no matter which direction the wave is moving in,

the amplitude function can always be written as

( , ) cosy x t A t k x

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 tspeed of a wave crest

tdistance moved in time T

This is called the of the wave

v

T Tphase velocity

T





v

T Tphase velocity

T

v1/

fT f





v

T Tphase velocity

T

v1/

fT f

2v

2f

k k

Another way to see the same thing:

We saw earlier that for a wave described by the amplitude

function ( , ) cos( ), the position of a wave

crest changes by in a time where 0

Therefore,

y x t A k x t

x t k x t

phase velocityx

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

2with distance: = where

4. The wave

k x k

vector is defined to have magnitude

and to point in the direction of wave motion. If the

displacement from the coordinate origin is denoted

, then any simple wave has the general amplitude functi

kk

x

on

( , ) cos( )

5. Speed of a wave crest phase velocity

y x t A t

fk

k x

waves in one dimension. a wave medium is an object, each of whose points can undergo simple harmonic...

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