WAVES I

  1. Look at waves in extended elastic systems
    1. Forces on particles determined by relative displacements of neighbors
    2. Particles oscillate about equilibrium positions and do not travel with wave
  1. Wave equation gives particle displacement as a function of position on x-axis and time
    1. If the disturbance is simple harmonic in time then for a wave moving in the +x-direction the particle displacement at x is

y = Asin[2p f(x/v -t)]

y = Asin(kx-w t)

 

where A = amplitude, f = frequency, v = speed of wave, k = 2p f/v, and w = 2p f

 

  1. Definitions
    1. Amplitude - maximum displacement of particle
    2. Wavelength (l ) - distance traveled in one complete cycle
    3. Frequency (f) - number of cycles per second
    4. These are related: v = l f
    5. We also find : k = 2p /l and w = 2p f
    6. Transverse wave - particle motions are at right angles to direction of wave travel
    7. Longitudinal wave - particle motions are along the direction of travel
  1. Principle of superposition - two or more waves traverse the same space independently of each other and the displacements caused by each wave simply add.
  2. Interference

y1 = Asin(kx-w t)

y2 = Asin(kx-w t + j )

y = y1 + y2

where j is the phase difference

    1. What can cause a phase difference?
    2. Waves are added using trigonometric identities or the method of phasors
    3. If j is 0, 2p , 4p , ., then we have constructive interference
    4. If j is p , 3p , ., then we have destructive interference
  1. Standing transverse waves can occur when a wave is reflected back on itself and the original wave and the reflected wave interfere with each other.

y1 = Asin(kx-w t)

y2 = Asin(kx+w t)

y = [2Asinkx]cosw t

Each point moves with SHM of amplitude [2Asinkx]

Nodes occur every l /2