Chapter 11: Vibrations and Waves

Periodic Motion When a vibration (aka oscillation) repeats itself back and forth over the same path the motion is called periodic. One of the most common examples of this is an object bouncing at the end of a spring, so we will look at this example closely.

Spring – Mass Systems Assume a massless spring attached sideways with an object of mass m attached to one end. All springs will have a relaxed length where the mass will just sit still, this is called the equilibrium position.

Choices Choices There can be 3 possible scenarios. The mass could be at the equilibrium position. The mass could be between the equilibrium position and the other end of the spring (the spring is squished). The mass could be beyond the equilibrium position (the spring is stretched).

You’re making me uncomfortable A spring hates to be away from its equilibrium position. If moved away from its EP, a spring will try to move back to it. Remember chapter 6, springs create a force equal to spring constant times displacement. F = -kx

A sad truth So let’s say we have a mass on a spring and we squish it in a little bit then let go. The spring puts a force on the mass in the opposite direction we pushed it, trying to get back to EP. The spring over shoots the EP and exerts a force pulling the mass back to EP, but it over shoots it again!

Warning: Vocab The distance we move the mass away from EP is called the displacement. The maximum displacement is called the amplitude. One complete back and forth motion is called a cycle. Period is the time per cycle and frequency is the number of cycles per time.

Simple Harmonic Motion Any vibrating system for which the restoring force is directly proportional to the negative of the displacement is said to exhibit simple harmonic motion. Simple: force is caused only by displacement, it’s not rocket propelled. Harmonic: continuous back and forth movement. Motion: duh

Simple Harmonic Oscillator A system that creates simple harmonic motion is called a simple harmonic oscillator. Today we examine the energies involved in such a system. Relax, it’s just a review of chapter 6.

Getting things started The simplest simple harmonic oscillator is our mass and spring again. To get the system bouncing we first need to squish or stretch the spring. When we do that we give the spring potential energy, remember PE = 1/2kx2 (chapter 6)

The return of conservation of energy In chapter 6 we learned that the total energy of a system, E, is always equal to the kinetic energy plus the potential energy. E = KE + PE E = 1/2mv2 + 1/2kx2

Extreme Measures It is simplest to look at the extreme points of the system to find its total E. At the maximum displacement (called amplitude, A) all of E is PE E = 1/2m(0)2 + 1/2kA2 = 1/2kA2 At the equilibrium position, EP, all of E is KE E = 1/2mv02 + 1/2k(0)2 = 1/2mv02 Note: in this chapter v0 is the MAX velocity.

The Period of Simple Harmonic Motion V0 = Solving for T gives us T = Remember 1/2kA2 = 1/2mv2, so A/v0 = √(m/k) So T =

Frequency of SHM Because f = 1/T, f =

Position as a function of time How can we figure out the distance our object is from the equilibrium point at any given time? x = A cos ωt x = A cos 2πft x = A cos (2πt / T)

Velocity and acceleration as a function of time v = -v0 sin(2πt / T) a = -(kA/m) cos(2πft) = -a0 cos(2πt / T)

Wave Motion In Chapters 11 and 12 we are only concerned with one family of waves, mechanical waves. Mechanical waves are waves created by mechanical forces Shaking a slinky An earthquake A car going over a bump

A common misconception Many people think that waves carry matter like surfing. This is NOT true A wave is energy, a chain reaction.

Types of waves Transverse wave – a wave that travels perpendicular to the vibrations Longitudinal wave – a wave that travels parallel to the vibrations Aka density wave or pressure wave

Vocab Pulse – a single wave bump. (What you made with the slinkies) Continuous/Periodic wave – when the force making the wave is a vibration. (What you did when you shook the slinky constantly) Amplitude – the max height of the wave

Even more vocab Wavelength (λ) the distance from peak to peak. Frequency (f) – the number of peaks per unit time. Wave Velocity (v) – the velocity that wave peaks move.

The speed of waves What is speed? For a wave, what is distance? Time? So v = λ / T Because T = 1/f, we can write v = λf speed of a wave = frequency x wavelength

Speed Limit Remember the slinky lab? What happened as frequency increased? The speed of a mechanical wave for any given medium is fixed.

Superposition of Waves Last time we learned different sound waves in the same medium travel at the same speed. So two different sounds played at the same time and distance reaches your ear at the same time. How can 2 waves be in the same place at the same time?

Superposition of Waves 2 Waves are not matter, they are displacements of matter. They can occupy the same space. The combination of two overlapping waves is called superposition.

Interference When two waves overlap they have an effect on each other. This effect can be observed by studying the interference pattern of the waves. We will first look at the two extreme cases of interference.

Constructive Interference Superposition principle – the amplitude of the resulting wave can be found by adding the amplitudes of each wave. When the displacement is on the same side for each, the sign is the same and the wave is bigger. This is constructive interference

Destructive Interference When the displacement of two overlapping waves are in opposite directions their signs are different. So, when added they produce a smaller wave. This is destructive interference. When 2 equal and opposite waves overlap, their sum is zero. This is called complete destructive interference.

Reflection of waves If you shook your slinky hard enough you saw it start to come back. If waves are not matter then how do they bounce off things?

Newton’s Third strikes again Picture a string tied to a pole so that the knot can move freely As the pulse travels down the string the knot moves up. Tension pulls the knot back down creating a disturbance in the rope. This creates a new wave back in the same direction.

If the end was fixed If the string fixed to the pole instead how would it change things? When the pulse hits the pole it wants to move up, but the pole exerts a downward force. This force creates a wave that is in the opposite displacement of the original wave.

Standing waves In the last slide, what would happen if the string was moved up and down not just once but constantly? This This is what is known as a standing wave

Nodes and Antinodes Nodes (N)- places where complete destructive interference occurs. Antinodes (A) – the midway point between two nodes. The amplitude is highest at this point.

More on standing waves Only particular frequencies, and therefore, particular wavelengths can produce standing waves. The wavelength of a standing wave depends on the length of the string.

Standing Waves Yesterday we ended class by observing the standing wave. The points of destructive interference, where the string stays still, are called nodes. The points of constructive interference, where the string reaches its max height, are called antinodes.

All Natural A standing wave can occur at more than one frequency. However, they can only occur at specific frequencies. The frequencies that do create standing waves are called the natural frequencies or resonant frequencies of the string.

Harmonics The wavelengths of the natural frequencies depend on the length of the string itself. The lowest frequency, called the fundamental frequency, has a wavelength given by the following: L = ½λ1

Overtones The other natural frequencies are called overtones. The wavelengths of each overtone can be found using the following: L = nλn / 2 Or λn = 2L / n

Frequency From v = λf we know f = v/ λ So the frequency of each harmonic can be found using: fn = v / λn = nv / 2L Finally, v =√(FT / (m/L)) With these tools we can know exactly how long to make certain strings so that they make specific noises. This is what makes music possible!