1. Classical (I.e. not Quantum) Waves

2. Find speed of wave on a string, which is flexible and not displaced much
method of Tait:
the force downward, radial with respect to the curve at the top of the bump, is equal to the component of the tension on each side:

3. The mass of a segment of a string with mass per unit length  is
The idea of this calculation is to equate the radial force with the centripetal force:
solving for v

4. Another calculation, with the kink, is instructive:

5. There is an unbalanced force on the kink
There is an unbalanced force on the kink. The right hand end has no y component force, the left end has a force
the impulse equals change in momentum:
and now we use the small angle approximation that sin x = tan x, so

6. Reflections at the end:
open end: pulse reflected with no change in sign
fixed end: pulse reflected inverted
momentum conservation can be used to conclude this behavior

7. Reflections at the end:
open end: pulse reflected with no change in sign
fixed end: pulse reflected inverted
momentum conservation can be used to conclude this behavior

8. Traveling Waves-start with Harmonic Waves, exciting with a simple harmonic motion of frequency f, wavelength  = v/f
We have seen that the travelling wave travels with speed v, so in full generality the wave function is (NON-DISPERSIVE):
and the harmonic wave excited by SHM is
Note k is called “wave number” and has nothing to do the the k in Hooke’s law used in the previous slide on SHM!!
It is useful to write the wave function in some other ways:

9. Sometimes we want the choice of the origin of x and t to be arbitrary, so we have to throw in an “initial phase” 
We like to define another “2 killer” =2f
The key here is the traveling wave form, which comes from the wave equation, which we will soon derive for the string

10. Review of Simple Harmonic Motion

11. SHM formula application to wave:
Energy transported by wave--any bit of the string is performing SHM, with amplitude A
and in a time dt the oscillation moves along by a distance dx=vdt, to the power passing by in one second is

12. String fixed at 0 and L: Standing Waves
the end points have to stay at zero y=0
at the point x=0, this follows from the form sin kx. At the other end, it implies a relation between k and L, sin kL = 0

13. Some trig: what if we have two traveling waves going in opposite directions:
and with identity
sin(a±b)=sin a cos b ± cos a sin b
and we see that this is the wave function for the standing waves

14. What if we add (traveling) waves with different wave lengths?
So we get a “difference” frequency or “beat frequency” -this can be written as a oscillation with frequency f1+f2 and variable amplitude:
amplitude =

15. Wave Equation on String, generalize our analysis of kink:
the mass of this bit of string is  ds= dx, and F=ma 

16. Or, canceling the dx
We have noted that this has any solution of the form
that is, traveling, non-dispersive waves, which lead to standing wave on the finite string.

17. Solutions of the Wave Equation 1. Traveling Wave solutions

18. 2. The “Normal Modes” of the string with fixed ends:
A successful separation of variables! The equation has to be true for all x and t, can only be so if each is equal to the same constant, call it -2

19. This is the familiar SHO equation, with solutions
but the cos kx does not fit the fixed ends, y(0)=y(L)=0, so kn=n/L, son=n v/L
The wave equation is linear, so sums of solutions are also solutions. The shape at time zero can be what we please, so we need both sin and cos in t, and then the general solutions are

20. Power Spectrum of String-
Special Problem, preparation for Test, Due Monday 18 October, questions answered 13,15 Oct.
Power Spectrum of String-
A string is 4 meters long, fixed at both ends. It weighs 80 grams in total, and carries a tension of 100 Newtons.
The middle half is lifted from the equilibrium position by 5mm, like this:
What is the power in the vibrations in the first ten active modes, or harmonics?