1. http://paws.kettering.edu/ ~drussell/Demos/supe rposition/superpositio n.html We can invent a solution to the boundary problem by superposing a reflected wave An inverted reflection ensures that y=0 at the boundary

2. Boundary condition for a fixed end at x=0: y(0,t) = 0 This is satisfied by adding an inverted reflection. You could have a sine wave in, and an inverted sine wave out (on chalkboard…) Any specified wave in (initial condition), just add its inverted reflection. Pulse, sinusoid, anything! What is the Boundary condition for a free end?

3. Reflection off a free end - A Non-inverted reflection keeps the slope=0 at the boundary

4. Demonstration with Wave Table - Free and Fixed Ends What about junctions between two media / ropes? Consider the “fixed end” to be an infinitely massive rope, v=0 Consider the “free end” to be a massless rope, v=  Recall: wave speed  √(restoring force/inertia) We might guess that a reflection off a medium with a slower wave speed is inverted; off a medium with a higher wave speed is not. What do you predict will happen to a pulse initiated in the long rods, that reflects off the short rods? A] it will be inverted B] it will be non-inverted

5. Standing Waves The reflection off a perfect boundary gives a wave that does not appear to travel. It is a sum of a left-going and a right-going wave. http://paws.kettering.edu/~drussell/Demos/superposition/superposition.html Note that there are other places, besides x=0, where y(x,t)=0 always. If our string is finite in length and pinned at the other end, y(x,t) =0 at both ends. Only certain wavelengths will fit! (Chalkboard…)

6. Standing Waves on a String

7. Are “normal modes” the only allowed vibrations of a string? “Nothing is as it appears to be, nor is it otherwise.” YES! And, NO! A mathematical theorem (truth) from Fourier states that any continuous function on the interval [0,x] (zero at the ends) can be written as a sum of sinusoids. We know, from the linearity of the wave eqn, that a sum of solutions is also a solution.

8. So: YES, only (sums of) normal modes are allowed But: ANY initial wave shape can be written as a sum of normal modes! The amplitude of each normal mode (or harmonic) depends on the initial shape of the string Guitar Demo http://mysite.verizon.net/vzeoacw1/harmonics.html