Standing WAVES

Standing WAVES

1. modes

   • normal mode
     • A motion in which all particles of the string move sinusiodally w the same frequency
       • Fundamental frequency n=1, L=lambda/2
     • n= # of anti nodes

2. modes

   • linear resonator
     • n=#of antinodes
   • musical instruments may play the same fundamental frequency but have different undertones.

3. mathematical annalysis

   • 1D
     • y(x,t) f(x) cos (wt+phi)
       • we can choose to say that the max displacement occurs at t=0 then phi=0
         • the 1D wave eqn can then be subbed in
           • thus f(x)=Asin(w/v x)+Bcos (w/v x)
             • Apply boundary conditions
             • A cannot =0
   • any function can be approximated by sim

4. General soln for taught string y(x,t)=Asin (npix/L)cos (wt)

5. Energy in a wave

   • K=u
     • small angle apptoximation sin theta = theta
   • ∆S>∆x

6. super position of normal modes

   • when you activate one wave you are activating a multitude of standing waves, which is why we consider the summation see eqn 17
     • timber results from different wave shapes
       • by adding more modes we achieve better agreement esp w respect to the sharp corner
     • the first harmonic is most important
     • gob only interested in sin expansion

7. Energy of Vibration of a string

   • 0