Chapter 15 Waves

1. Chapter 15 Waves • Wave Basics • Water Waves • We can watch a single wave move to shore • Motion is very regular • Water moves toward shore but none accumulates on the beach: Complex Motion • Standing on beach, you can feel the KE of the waves • Waves in water are a familiar example • Light, radio, microwaves, infrared, sound, etc… are other examples • Wave motion explains may physics phenomena • Wave Pulses • Simple example = slinky (spring) • Single, fast, compression at one end causes motion along the spring • Follow movement to opposite end; may even reflect back • Pulse = single wave

2. What is really happening to cause and carry the Pulse? • Spring is in same position before and after the pulse • Parts of the spring move, taking turns as the pulse propagates • A local compression of the spring is what is moving • Compressed area switches places with relaxed areas • Compressed area ends up at the opposite end • Each individual loop moves forward and then back • Features of Wave Motion • Medium = object or material through which a wave moves • A wave must use some kind of matter to move • Water waves use the water itself as the medium • Disturbance within the material • Spring—compression of the loops • Water—up and down “displacement” of the water

3. Longitudinal Wave = displacement is parallel to the direction of wave travel (spring compression, sound) Transverse Wave = displacement is perpendicular to direction of wave travel (wave on a string, light) • Velocity = how fast the disturbance is traveling through the medium • Determined by the properties of the medium • Tension—tighter the spring, the faster the pulse travels • Density—denser the spring, the slower the pulse travels • Transmission of Energy • Moving compression of a spring = KE + PE • Energy moves with the pulse • Water waves—noise, movement of the sand on the beach, etc…

4. Periodic Waves • Wave = continuous set of pulses moving in the same way on the medium • Periodic Wave = equal times and distances between pulses • Period = time between to pulses of a periodic wave = T • Frequency = number of pulses per unit time = f = 1/T • Units: 27 per second = 27/s = 27 s-1 = 27 Hz • 1 Hertz = 1 Hz = 1 per second = 1/s-1 • Wavelength = distance between pulses of a periodic wave = l (lambda) • Velocity of a periodic wave = frequency times the wavelength • Waves on a Rope • Rope similar to a very stiff spring • Hard to compress, so you don’t get longitudinal waves • Easy to move up and down, so you can make transverse waves Small f = large l Large f = small l

5. Graphing Waves • At any given time, a single transverse pulse is easy to graph • Periodic waves can be more complicated • Complex periodic wave can be generated on a string by complex (but still regular) motion of the hand up and down • Simple Harmonic Motion gives a Harmonic Wave • Each point on the rope is moving in harmonic motion • Simple sin graph, like in harmonic motion • All complex waves can be broken down into the sum of several harmonic waves = Fourier or Harmonic Analysis

6. Velocity of a wave on a rope • Velocity does not depend on frequency (f) or the shape of the wave • Why do pulses move? • Initial Force, called Tension, starts the motion (acceleration) • Tension then acts on a neighboring area of the string, which gets accelerated, and so on down the string • Velocity depends on rate of acceleration at succeeding points on rope • F = ma or a = F/m • Large force causes a large acceleration to a fast velocity • Large mass makes acceleration and velocity slower • m = mass per unit length of the rope (similar to density) • Equation for velocity only includes F and m, not f or l

7. Sample problem: A rope with a periodic wave on it has the following properties L = 10 m, m = 2 kg, F = 50 N, f = 4 Hz • v = ? • l = ? • Interference and Standing Waves • Interference • Waves on a rope reflect back and “interfere” with the input wave • Water waves reflect off of the beach and interfere with incoming waves • Interference = combination of two or more waves • Waves on a spliced rope • Principle of Superposition = when combining waves, the total displacement is the sum of the individual wave displacements • Constructive Interference = combine waves of same f, l to give larger total displacement (add together) • Destructive Interference = combine waves of same f, l to give smaller total displacement (cancel out)

8. Constructive Interference Destructive Interference • In Phase = two waves move the same way at the same time (constructive) • Out of Phase = two waves don’t move the same way at the same time (destructive) • Amplitude = height of the displacement • Completely constructive—amplitude = 2 x individual amplitude • Completely destructive—amplitude = 0 • Not quite in phase—amplitude somewhere between 0-2 x amplitude • Standing Waves • Above, we have multiple waves traveling in the same direction • Interaction of waves traveling in opposite directions • A periodic wave and its reflection are the simplest example (same f, l) • Apply the Principle of Superposition to see what happens to Amplitude

9. At certain times, the two amplitudes cancel out (+ and – added) for each point • Let the two waves move l/4 each, and look again. i. Points A still add to 0 amplitude ii. Points B now have 2 x amplitude c) Standing Wave = oscillating pattern in a rope when a periodic wave and its reflection interfere to give a stable pattern of zero and 2 x amplitudes • Node = point of no motion in a standing wave (points A) • Antinode = point of greatest motion in a standing wave (points B) • Node—Antinode distance = l/4 • Node—Node distance or Antinode—Antinode distance = l/2