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.

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 8 ?

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

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?

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?

> with(plots): > k:=1; omega:=1; > animate(cos(k*x-omega*t),x= ,t=1..20,frames=50); Then animate cos(k*x+omega*t)

Let’s define the wave vector:

How far does a wave crest move in a time of one period?