WAVES I

- Look at waves in extended elastic systems

- Forces on particles determined by relative displacements of neighbors
- Particles oscillate about equilibrium positions and do not travel with wave

- Wave equation gives particle displacement as a function of position on x-axis and time

- 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

- Definitions

- Amplitude - maximum displacement of particle
- Wavelength (l ) - distance traveled in one complete cycle
- Frequency (f) - number of cycles per second
- These are related: v = l f
- We also find : k = 2p /l and w = 2p f
- Transverse wave - particle motions are at right angles to direction of wave travel
- Longitudinal wave - particle motions are along the direction of travel

- Principle of superposition - two or more waves traverse the same space independently of each other and the displacements caused by each wave simply add.
- Interference

y_{1} = Asin(kx-w
t)

y_{2} = Asin(kx-w
t + j
)

y = y_{1 }+ y_{2}

where j is the phase difference

- What can cause a phase difference?
- Waves are added using trigonometric identities or the method of phasors
- If j is 0, 2p , 4p , …., then we have constructive interference
- If j is p , 3p , …., then we have destructive interference

- 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.

y_{1} = Asin(kx-w
t)

y_{2} = Asin(kx+w
t)

y = [2Asinkx]cosw t

Each point moves with SHM of amplitude [2Asinkx]

Nodes occur every l /2