AstroPhysics 5
Homework 9
Due: Tuesday, November 16 in class
1) A harmonic, travelling wave representing a quantum bunny is
given by f(x,t) = 2 sin(2π x / 5 + 2π (0.3) t)
where f and x are in cm, and t is in seconds.
In what direction is the bunny travelling, and how fast?
2) We have seen that we can add, or superpose, two oppositelytravelling, harmonic waves to create a “standing wave”.
(a) Visit the website,
http://www.smaphysics.ca/phys30s/waves30s/waveadd1.html
Enter the correct numbers in the text boxes underneath f(x,t)
and g(x,t) in order to add these two waves:
f(x,t) = 2 sin(2π x / 5 + 2π (0.3) t) AND
g(x,t) = 2 sin(2π x / 5 - 2π (0.3) t)
Hit “Forward” to see the resulting superposed wave go forward
in time. Include a snapshot (printout) of this website’s window
with your homework so we can see the values you entered, and
the waves you created.
Note: You can also input these equations for f and g by typing them
in where it says “Alter the equations of the waves directly in the boxes
below” . But your grader would like to see the correct values for
“Amplitude”, “Wavelength”, ... etc typed into the text boxes.
(b) Using the same website, add the two waves below, which are
oppositely-travelling, but of unequal amplitudes:
f(x,t) =2 sin(2π x / 5 + 2π (0.3) t) AND
g(x,t) = 0.9 sin(2π x / 5 - 2π (0.3) t)
Describe what the sum of these waves is doing. Travelling at a
constant speed? Standing still? Other?
(c) Standing waves arise very naturally in various physical
situations. For example, visit the website
http://www.walter-fendt.de/ph14e/stwaverefl.htm
It shows an elastic string of beads that supports waves.
Describe in a couple of sentences what happens in this applet,
and what circumstance causes these standing waves to become
established. Include a couple of snapshots of the simulation if
you wish, to illustrate your point.
3) Prove that the formula for a standing wave formed of two
oppositely-travelling waves is:
A sin(kx + ωt) + A sin(kx - ω t) = 2A sin(kx)cos(ω t)
HINT: Use these two trig identities:
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
4) A standing wave in one-dimension has “nodes” which are
points where it is always zero, and “antinodes” where it has its
maximum amplitude. In two-dimensions, waves will have nodes
that are are curves, rather than points. In three-dimensions
the nodes of standing waves will be two-dimensional surfaces.
(a) Below is a double slit diffraction pattern for quantum
squirrels that can run through two narrow holes in a hedge.
Please treat the green curves as if they were the “troughs”, or
maximally negative places on the squirrel waves, and red curves
like the “crests” or maximally positive places. Reproduce the
picture and on top of it sketch the nodal curves - the set of
locations at which there will be no squirrels.
(b) At the website
http://www.youtube.com/watch?v=GtiSCBXbHAg&feature=related
is a 2D “Cladne plate” a metal plate which can support waves.
Sand is sprinkled on it. The sand is undisturbed along the
nodes, but vibrates off of the plate everywhere else. The video
shows several patterns as frequency is increased. Sketch one
of them.
(c) Go to
http://www.falstad.com/circosc/index.html
where you will see a simulation of a 2D vibrating circular
membrane. You can also see a real life example of this at
http://www.youtube.com/watch?v=v4ELxKKT5Rw
These websites shows several patterns as frequency is
increased. Sketch two different nodal patterns. Please also tell
how they are ordered in terms of their oscillation frequency.
(That is, say which one has the longer and which the shorter
period of motion.)
(d) At the website
http://www.falstad.com/qmatom/
are simulations of the 3D wave functions of hydrogen. Sketch
the nodal surfaces for these three cases:
n=2, l = 0
n=2, l =1
n=3, l = 0
5) Below is a diffraction pattern for light that passes through
two narrow slits and falls on a screen (the black line). Please
treat the green curves as if they were the “troughs”, or
maximally negative values on the electric field wave and red
curves like the “crests” or maximally positive places.
Consider points A, B, C on the screen and say whether each is:
a very bright spot OR
a perfectly dark spot OR
a spot of an intermediate brightness
Please explain your reasoning!
6) Slide 10a from Week 10 of our class is appended to this
homework. Your professor did her best to position the two
cursors the right distance apart so that they measured the
distance between adjacent bright fringes in the double slit
pattern:
for blue light
for red light
How did your professor do? In other words, use the
information given on Slide 10a for λ, d and L and also the
formula that predicts the separation of adjacent bright or dark
regions in the limit that the screen is far from the slits. In this
way, confirm what those two x values should be.