1. Waves A pulse on a string (demos) speed of pulse = wave speed = v
depends upon tension T and inertia (mass per length )
y = f(xvt) (animation)
actual motion of string
motion of “pulse”

2. Periodic Waves: coupled harmonic motion (animations)
aka sinusoidal (sine) waves
wave speed v: the speed of the wave, which depends upon the medium only.
wavelength : the distance over which the wave repeats,
frequency f : the number of oscillations at a given point per unit time. T = 1/f.
distance between crests = wave speed  time for one cycle
 = vT
 Wavelength, speed and frequency are related by:
v =  f

3. Mathematical Description of Periodic Waves

4. The Wave Equation

5. Transverse Wave Velocity: lifting the end of a string Tension F
Linear Mass Density (m/L) 
Transverse Force Fy
Fnet
vyt Fy vt F F
l = vt

6. Reflections at a boundary: fixed end = “hard” boundary
Pulse is inverted
Reflections at a boundary: free end = “soft” boundary
Pulse is not inverted

7. Reflections at an interface
light string to heavy string = “hard” boundary
faster medium to slower medium
heavy string to light string = “soft” boundary
slower medium to faster medium

8. Principle of Superposition: When Waves Collide!
When pulses pass the same point, add the two displacements
(animation)

9. vibrations in fixed patterns
Standing Waves
vibrations in fixed patterns
effectively produced by the superposition of two traveling waves y(x,t) = (ASW sin kx) cost
constructive interference: waves add
destructive interference: waves cancel
 = 2L
 = 2L
 = 2L
 = 2L
node
antinode
antinode

10. Example: The A string on a violin has a linear density of 0
Example: The A string on a violin has a linear density of 0.60 g/m and an effective length of 330 mm. (a) Find the Tension in the string if its fundamental frequency is to be 440 Hz. (b) where would the string be pressed for a fundamental frequency of 495 Hz?

11. Standing Waves II pipe open at one end
node
antinode
 = 4L
 = 4L
 = 4L
 = 4L