Vibrations and Waves Periodic Motion Periodic motion is

Vibrations and Waves

Periodic Motion Periodic motion is that motion in which a body moves back and forth (oscillates) over a fixed path An oscillating system exhibits SHM when the restoring force is proportional to the displacement (from equilibrium)

Simple Harmonic Motion, SHM Simple harmonic motion is periodic (repeating) motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. x F A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx

Hooke’s Law (for a mass/spring system) – the amount of stretching or compressing of a spring (displacement) is proportional to the force applied. F = -kx F = force (Newton) k = spring constant (N/m) x = displacement (m) from equilibrium The negative sign means that the direction of the force is opposite the direction of motion.

SHM Simple Harmonic Motion

Energy in a system Work is done in stretching or compressing a spring (or displacing a pendulum). So potential energy is stored in the system. The total energy is the sum of the potential and kinetic energy. Mass/spring Pendulum Etotal = ½ kx 2 + ½ mv 2 Etotal = mgh + ½ mv 2

Energy changes: Etotal = PEel + KE Note: The first, third and last diagrams indicate max PE. (v = 0) The second and fourth indicate max KE. (a = 0) Etotal = ½ kx 2 + ½ mv 2

Properties of Waves

What are waves? A wave is a transfer of energy from one place to another. Wave-types take many forms. Wave Characteristics include: • • • Period Frequency Wave speed Amplitude Wavelength

Period (T) – the time needed for one full cycle (back and forth) Measured in units of time – seconds for SI Frequency (f ) - the number of complete cycles (waves) that pass a given point in the medium in 1 second. Measured in Hertz (Hz) 1 Hz (s-1) = 1 cycle/second NOTE: Frequency is the inverse of the period It is set by the vibration source so it stays the same as the wave moves from one medium to another

Pendulum Note: the restoring force is proportional to the displacement. It is equal to mgsinθ FT FT Restoring force mg mg T = period (sec) l = length of pendulum (m) g = 9. 81 m/s 2

Example. What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)? L T 2 = 4π2 L/g L = (T 2 g)/(4π2) L = 0. 993 m

Mass and spring system T = period (sec) m = mass (kg) k = spring constant A) F and Accel = 0 velocity at max B) F and Accel at max in the + direction velocity = 0 C) F and Accel at max in the neg. direction velocity = 0 (N/m)

Example: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? x F Or 2. 0 s-1

Amplitude - the maximum displacement of the medium measured from the rest (equilibrium) position. • Proportional to the amount of energy carried by the wave • Emax = 1/2 kx 2 so the energy ~ amplitude 2 or x 2 Crest - highest point on waveform • maximum positive displacement of medium Trough - lowest point on waveform • maximum negative displacement of medium

Energy is proportional to x 2 -where x is the displacement from equilibrium (amplitude)

Mechanical Waves that require a medium to transfer energy (a physical disturbance in a medium) May be either transverse or compressional (longitudinal). Examples: • Sound waves, water waves, earthquake waves, waves in a string

Wave Types Two types of periodic waves – • Transverse – the vibrations are perpendicular to the direction of motion • Longitudinal (compression) – the vibrations are parallel (in the same direction) as the direction of motion

Transverse Waves particles in the medium vibrate perpendicular to the direction of wave motion or speed Motion of particles Motion of wave

Longitudinal Waves (Compression Waves) particles in the medium vibrate parallel to the wave velocity • one wave is made up of alternating areas of high pressure (compressions) and low pressure (rarefactions) v Motion of particles Motion of wave

Comparison of transverse and longitudinal waves: The “crest” on a transverse wave (max amplitude) corresponds to the compression point on a longitudinal wave. The “trough” on transverse corresponds to the rarefaction point on a longitudinal wave.

Wavelength The distance between correspondi ng points on consecutive waves. The symbol “λ” for wavelength is “lambda” This is a distance measurement (meters).

Wave Speed velocity of a wave as it travels through a medium • Mechanical wave: medium determines the speed • EM wave: velocity constant (“c”) in a vacuum v = lf

Example 2: A vibration sends waves down a string. The vibrator makes 600 complete cycles in 5 s. For one complete vibration, the wave moves a distance of 20 cm. What are the frequency, wavelength, and velocity of the wave? f = 120 Hz The distance moved during a time of one cycle is the wavelength; therefore: l = 0. 20 m v = fl v = (120 Hz)(0. 20 m) v = 24. 0 m/s

Wave Speed in a String The wave speed v in a vibrating string is determined by the tension F and the linear density μ, or mass per unit length v = speed of the transverse wave (m/s) F = tension on the string (N) m or m/L = mass per unit length (kg/m)

Example 1: A 5 g section of string has a length of 2 m from the wall to the top of a pulley. A 200 g mass hangs at the end. What is the speed of a wave in this string? F = (0. 20 kg)(9. 8 m/s 2) = 1. 96 N (weight of mass) 200 g

Wave Interactions

Superposition and Interference patterns When two or more waves occupy the same space in a medium at the same time they interfere with each other. • The medium’s displacement will be the vector sum of the displacements caused by the individual waves. • After the waves pass each other they return to original form There are two types of interference patterns.

Constructive Interference When two waves interfere in a manner such that the amplitude of the resultant wave is greater than the amplitude of the individual waves.

Destructive Interference When two waves interfere in a manner such that the amplitude of the resultant wave is smaller than the amplitude of the individual waves. Note: Complete cancellation does not always occur in destructive interference.

Reflection/Refraction and Boundaries When a wave encounters the boundary between two media some will be reflected and some will be transmitted. How much goes each way is determined by the relative densities of the media. The transmitted pulses will always be upright (right side up) NOTE: A BOUNDARY is a change between media. A BARRIER is a fixed point that causes a reflection – no transmission. (See Standing Waves)

Crossing Boundaries More Dense to Less Dense When a wave is moving from a more dense medium to a less dense medium, the reflected pulse is upright. v Before After More dense medium v Less dense medium v

Crossing Boundaries Less Dense to More Dense When a wave is moving from a less dense medium to a more dense medium, the reflected pulse is inverted. v Before Less dense medium More dense medium v After v

Cool link for boundaries Scroll down to the last two animations http: //paws. kettering. edu/~drussell/Dem os/reflect. html

Diffraction The bending of a wave around a barrier, such as an obstacle or opening. Occurs when a wave passes an edge, passes through a narrow gap or goes past an object. None of the properties of a wave are changed by diffraction. The wavelength, frequency, period and speed are same before and after diffraction. The only change is the direction in which the wave is traveling.

Resonance A condition that exists when the frequency of a force applied to a system matches the natural frequency of vibration of a second system that causes a dramatic increase in amplitude. Example: a guitar body. The string vibrates and produces a sound wave. This energy causes a “sympathetic” reaction in the sounding board of the guitar and amplifies the sound.

Doppler Effect See “The Physics Classroom” – • Wave Interactions

Standing Waves

Standing Waves Caused by a wave and its reflected wave interfering in the medium Reflection caused when waved encounters a fixed barrier. Wave pattern of alternating nodes and antinodes. Nodes - areas of no displacement of the medium caused by destructive interference Antinodes - areas of maximum displacement of the medium caused by constructive interference. Note: The wavelength of a standing wave consists of 2 antinodes. 1 wavelength is shown. Antinode

Only particular frequencies will produce a standing wave on a string of a certain length. The ends produce nodes. Lowest frequency: 1 antinode = ½ λ Note: 1 λ = 2 L Next possible frequency: 2 antinodes = 1 λ Note: 1 λ = L Next frequency: 3 antinodes = 1. 5 λ Note: 1 λ = 2/3 L

Possible Wavelengths for Standing Waves General equation to find the wavelength on a given length of string. n = 1, 2, 3, 4 etc (or the number of antinodes)

Possible Frequencies f = v/l : f = 1/2 L f = 2/2 L f = 3/2 L General equation to find the specific frequency that will produce a standing wave on a given length. f = 4/2 L f = n/2 L n = 1, 2, 3, 4 etc (number of antinodes) v = speed of a wave on the string

Electromagnetic Waves that consist of an electric field and a magnetic field working to propagate it through free space (a vacuum) No medium required Best modeled using transverse waves All EM waves travel at the speed of light, “c” c = 3 x 108 m/s Examples: • light waves, microwaves, ultraviolet, infrared, radar, radio waves, gamma rays, x-rays, etc.