Homework 1

due at 5pm Wednesday, Feb 6, 2013

For #1 and #2, print out the Mathematica code and output and turn it in as part of your writeup.

Q1. Mathematical description of a 1d wave

( a ) for A = 1 unit; k = 2π per meter; ω = π rad/s. What are the wavelength, period and amplitude of the disturbance? Discuss the dimensions of A.

( b ) Plot in Mathematica two spatial cycles of the waveform and animate for two time periods.

( c ) Which direction does the wave travel and with what speed? Which direction does it travel if you change the sign of the position term? Of the time term? Of both? Why?

( d ) Focus on the position x = 0 m. At what rate is the quantity represented by ψ changing at t = 0, ¼, ½, and 1 s? Describe the variation over one cycle.

Q2. Mathematical description of a 1d wave

Write down & plot in Mathematica a sinusoidal waveform ψ(x,t) that has the following properties:

  • Amplitude 2 m, wavelength 10 m, travels to the right at 1 m/s, ψ = 2 m at x = 5 m and t = 0 s.
  • Standing wave, amplitude 5 m, period 1 s, wavelength 1 m that is momentarily flat at t = 0 s.

Q3. Wave Equation

Describe the following waveforms in words (waveform, period, phase angle, direction & speed of travel … etc.). Demonstrate whether they are, or are not, solutions to the non-dispersive wave equation (∂²/∂t²)ψ(x,t)=v²(∂²/∂x²)ψ(x,t).

( a ) ψ(x,t) = 4cos(4πx + 3πt) − 4sin(4πx + 3πt)

( b ) ψ(x,t) = 3cos(2πx)sin(πt)

( c ) ψ(x,t) = 3exp(−αx)cos((2π/3)x − πt) , α is a constant

Hints

Animating functions of two variables: You will find it very useful to be able to view functions of 2 variables as an animated sequence. Mathematica and other programs are very good at this. You must become familiar with this animation capability.

Mathematica has an excellent interface command for animations, with slider bars. Look up the Animate and Manipulate commands in the help files. In the Manipulate mode, you can vary many parameters at once. For example if the function is A*sin(kx - ωt), you could plot the function as a function of x, with t varying, and set a particular value of k with a slider bar. Then you can move the slider to change ω, and run the t animation again.