Homework #4

Due at 5pm on Friday, Feb 15, 2013.

Q1: Decompose a waveform into normal modes

We started this problem in class on Tuesday

A guitar string (mass per unit length μ, under tension τ) is anchored at x = 0 and x = L. It is then displaced by a musician's finger so that it has the following profile at t = 0 (there is a discontinuity in the first derivative at x = L/3 and the scale of the y-axis is exaggerated).

The transverse velocity at all points is zero before the string is released at t = 0. Give a complete mathematical description of the resulting motion of the string, including a Mathematica animation.

Here is a guide to help you:

( a ) Write this wave form very generally as a superposition of the allowed standing wave modes of vibration of this string (Remember the group activity in class? Use that and generalize to all the possible wave vectors).

( b ) Make sure that the time dependence satisfies what you know about the velocity at t = 0, and that the space dependence satisfies what you know about where the string is anchored.

( c ) Turning to the space dependence only, find out what contribution each “normal mode” makes at t = 0. (Fourier series will be useful here even though this is not a periodic function – why?)

( d ) Now return to the full wave function which include the time dependence of each normal mode. Plot the function (you need to include enough terms to form a reasonable approximation to the function), and animate it. Describe what you see.

Q2: Energy in waves

Main 9.11

Note: All problems in chapter 9 of Main are useful practice for the final.

Q3. Quantum waves

  • McIntyre 5.1
  • McIntyre 5.2 ( a ), ( b ) & ( c ).

Challenge problem

Reflection and transmission of waves from a point mass on a rope: Consider a rope that has a tension τ, uniform mass density µ and a mass m attached to it at position x = 0. A harmonic, single frequency, traveling wave is incident from the left. You are to discuss, qualitatively and quantitatively by following (a) – (f), the reflection and transmission of the wave. Understand that this mass is a point mass – that is, its mass is concentrated at a true mathematical point. Experimentally, this might be approximated by a small, dense ball bearing crimped onto a very much lighter string.

( a ) First, think about the limits your answer should yield. What would happen if the mass were not there (i.e. m = 0)? What if the mass were very heavy?

( b ) Now we set up the problem. On the left, there is an incident wave (with unit amplitude) and a reflected one

ψ_left(x,t) = ψ_inc(x,t) + ψ_refl(x,t).

On the right there is a transmitted wave ψ_right(x,t). Write down an expression for ψ_left(x,t) and ψ_right that uses complex exponentials. (It is understood that when we actually plot the displacement, the real part of these expressions will be plotted.)

( c ) What can you say about the frequencies and k vectors of the incident, transmitted and reflected waves, and why is this so?

( d ) Consider what conditions must be placed on the displacement ψ(0, t) at x = 0, and on the slope ∂ψ/∂x at x = 0. (This second part that is different from what we did in class … you have to apply Newton's law to the mass).

( e ) Solve your equations to show that

Show that the phase angle of the reflected and transmitted waves changes in general. Was your intuition in (a) justified?

( f ) Plot and animate the string profile for various values of ε (small, intermediate and large) and describe what you see. In particular, for large values of ε, can you see when the discontinuity in the slope at x = 0 is greatest?