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”

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

Mathematical Description of Periodic Waves

The Wave Equation

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

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

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

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

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

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?

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