Read the Chapter 7 of Instant Physics (Rothman) and
write a one-page reaction to or summary of.
Reminder: there's a paper for this course.
Due: Nov. 24
Monday:
We started off considering the basic mathematical function sin(x). It starts at 0, varies between 1 and -1 and repeats itself after
the variable x goes through 2 π. We eliminated the last restriction by changing the function to sin(2 π x / λ). Now
the function repeats itself after x goes through a distance λ -- known as the wavelength.
We eliminated the "varies between 1 and -1" restriction by changing the function to A sin(2 π x / λ). Now the function
varies between -A and A -- known as the amplitude. Finally we eliminated the "starts at 0" restriction by changing the function to
A sin(2 π x / λ + φ) where φ is known as the phase constant.
Joseph Fourier introduced a process (Fourier series and/or Fourier analysis) by which other shapes can be considered to be
a sum of the sine functions
So far what we have is a "snapshot" of a wave, it does not change in time. Another addition to our mathemtical form
is A sin(2 π x / λ ± 2 π t / T + φ ) where T plays a role like λ but for time instead of
space. T is known as the period and it is the time required to go through a cycle. It has a reciprocal relationship to
the frequency (the number of cycles per second) f=1/T. If we combine λ the distance for a cycle and T the time for a
cycle we obtain v = λ / T the speed of the wave.
In a simple travelling wave, a disturbance moves say to the left or right. A standing wave results if two waves -- one
travelling to the left and another travelling to the right occopy the same space. The resulting disturbance just seems to
oscillate in space. The second wave travelling in the opposite direction from the first often results from the first wave
encountering some boundary and being reflected back. Imagine yelling across some canyon, your sound wave travels across the
canyon interacts with the cliff on the opposite side and returns as an echo.
Next we considered waves on a string -- like the lab we did. We thought about what happens to a pulse -- a more spatially
confined wave -- strikes the part of the string said to be fixed -- wighted down and not allowed to move. We argued that a
wave under such a voundary condition is not only turned back but it is flipped.
Note how B (the pulse) and C (the flipped, reflected pulse) would when added leave the string undisturbed at the fixed boundary
point.
We went on to consider a situation with two fixed boundary conditions
When the would-be wave (dashed red on the right) gets flipped and reflected, note that it adds constructively (crest meets crest)
making a big wave.
When we pluck a string we are intrducing a combination of waves with all sorts of wavelengths (Fourier), but ones that "match up"
nicely with the boundary conditions undergo constructive interference and have a big dominant effect. Others experience destructive
interference and have their effect reduced.
The wave with a wavelength twice the length of the string is called the fundamental. Waves with wavelength equal to the length
of the string, and waves with wavelength 2/3 of the length of the string, etc. also have the right conditions to be constructive.
These are the so-called harmonics. In part one of our experiment we varied the frequency to find a set of patterns -- the various
harmonics.
Next we considered what affected the speed of the wave on the string. A wave is a disturbance from equilibrium. If something is not
in equilibrium there is a force and Netwon tells us that F=ma. The speed of the wave is realted to this acceleration. The
acceleration increases if the force (in the case of the string its tension) increases. The acceleration decreases if the mass
(in the case of the string its linear mass density) increases. We just stated that v = (tension / Linear Mass density )1/2.
Think of a piano the thicker wires are associated with the low notes (frequencies). Lower speed leads to lower frequency (if the wavelength
is the same). The second part of our experiment varied the tension and thus the speed of the wave.
