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.