String waves by hcj


									String Waves

Throughout the history of physics two models have competed for being the
fundamental model of matter, these are the particle and the wave models. To
this point our study has only considered particles. Now we consider waves,
these come about as a result of vibrations.

Consider a medium such as water, air, or a string. Stretch a string between
two horizontal poles, we say the string is in equilibrium. If we pluck the
string then we call this a disturbance of the medium. A simple kind of wave
called a "pulse" is created. The pulse moves down the string and so
everywhere the pulse goes that portion of the string acquires kinetic and
elastic energy. In general, a wave is defined as a disturbance in a medium
that transports both energy and momentum. This definition is in line with
everyday usage, if you threaten to make a disturbance you may be told not to
make waves.

String waves are an example of transverse waves because the string moves
up and down at right angles to the horizontal motion of the wave. (There are
also longitudinal waves, e.g. sound, where the medium and the wave move
along the same direction but we will focus on the transverse case.)

So far everything has been qualitative-- now we make things more
quantitative. Suppose you take a snapshot of a pulse on a string. Assume the
string lies along the x-axis when it is in equilibrium. The disturbance from
equilibrium at a point x of the string is a distance y from equilibrium. But
the value of y can also change with time, t. The wavefunction is a way of
describing the disturbance mathematically,

                              y = y(x,t).

In class we consider a snapshot of a wave that moves down a string. From
analyzing this we find traveling waves have a wave function that can always
be written

                      y = y(x - vt) or y(x + vt).
Later we will study standing waves, where the wave function does not look
like the above form.

The most important kind of wave is the sinusoidal traveling wave where,

                               y(x,t) = A sin(kx - t),
                                          A = amplitude
                                     k = wave number = 2/
 = angular frequency = 2f
                                        f = frequency = 1/T
                                            T = period

Each piece of the string at x is in SHM, all with the same amplitude and

The speed of a wave is easily found from the wave relation,

                                        v = f .

(This follows because one wavelength of the wave takes a time of one period
to pass a point of the string, therefore, v = /T)

For the special case of string waves, the wave speed can also be shown, via
Newton's second law, to be given by,

                                 v = (FT/), where

FT is the tension in the string and  is the linear mass density of the string,
i.e. M/L , where M and L are the mass and length of the string,
respectively. A quick check of the units makes this relation plausible.

Another speed is the up and down speed of the string, called the transverse

              vy = ∂y/∂t , where ∂ indicates partial differentiation.
What happens if two waves, y1 & y2, are generated on a string? The
principle of superposition shows how to add or superimpose the waves to
get the total wave function, y, for the string,

                              y= y1 + y2 .

Wave addition also goes by the name of wave interference.

Two extreme forms of wave interference are constructive and destructive
interference. These will be illustrated in class.

Another topic is the behavior of a wave when it encounters a different
medium. For example, if one ties two different ropes together then there is a
boundary at the knot where the two meet. In such a situation an incident
pulse is partly reflected and is partly transmitted when the incident pulse
encounters the knot. As will be described in class, a “phase” change can
occur for the reflected wave but it never occurs for the transmitted wave.

Finally, we show that a sinusoidal wave transmits energy at the rate,

                               P  2 A2
EXAMPLES[in class]

To top