Properties of Waves A wave is a Traveling disturbance Carries energy from place to place without the transfer of matter

Common Properties of Mechanical Waves Require a physical medium Air, a string, body of water, etc. Require a driving excitation to get the wave started Vibrations then propagate, via interactions between particles, through the medium

Types of Waves Transverse Wave A wave whose particles vibrate perpendicularly to the direction the wave is traveling

Types of Waves Longitudinal Wave A wave whose particles vibrate parallel to the direction the wave is traveling Examples: Sound waves in the air Slinky Production of compressed and stretched regions that travel along the spring

Water waves are partially transverse and partially longitudinal.

Periodic Wave Continuous wave motion that requires a disturbance from an oscillating source Wave pulse Caused by a single disturbance

Wave Characteristics AMPLITUDE Maximum displacement of a particle in a wave from its equilibrium position Can also be defined as the height of the peak Always a positive value

Wave Measures WAVELENGTH (λ) Distance between two adjacent similar points on a wave Crest to crest Trough to trough CREST Highest point above the equilibrium position TROUGH Lowest point below the equilibrium position

Wave Characteristics PERIOD Time it takes a complete cycle to occur SI Unit = Seconds (s) FREQUENCY Number of cycles or vibrations per second Reciprocal of the period SI Unit = Hertz, Hz (Hz = s -1 )

Wave Speed Derivation of the expression for the speed of a wave in terms of its period or frequency v = Δx/Δt  Speed is = to displacement ÷ the time it takes to undergo that displacement For waves, a displacement of 1λ occurs in a time interval equal to 1 period of vibration (T) v = λ/T Frequency and period are inversely related f = 1/T Substituting this frequency relationship into the previous equation for speed gives a new equation for the speed of a wave v = λ/T  or λf

Wave Speed How fast a wave travels… v = λ/T OR v = λf v = Speed of wave (m/s) T = Period (s) f = Frequency (Hz) λ = Wavelength (m)

Problem A piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string. v = 343 m/sf = 262 Hzλ = ? v = fλ  λ = v/f  λ = 343 m/s/262 Hz = 1.31m

AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3 x 10 8 m/s. A station broadcasts AM radio waves whose frequency is 1230 x 10 3 Hz and a FM radio wave whose frequency is 91.9 x 10 6 Hz. Find the distance between adjacent crests in each wave. Wavelengths of Radio Waves

AM FM

Period Period of a Simple Pendulum T = 2π √(L/a g ) T = Period (s) L = Length (m) a g = 9.8 m/s 2 (free fall acceleration) Period of a Mass-Spring System T = 2π √(m/k) T = Period (s) m = Mass (kg) k = Spring constant (N/m)

Problem You need to know the height of a tower, but darkness obscures the ceiling. You note that the pendulum extending from the ceiling almost touches the floor and its period is 12 s. How tall is the tower? T = 12 sa g = 9.8 m/s 2 L = ? T = 2π√(L/a g )  L = 36 m

KEEPING TIME Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s. T = 2π√(L/g) 

Problem The body of a 1275 kg car is supported on a frame by 4 springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of s. For the first few seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

Answer m = (1275 kg kg)/4 = 357 kg T = s k = ? T = 2π√(m/k)  T 2 = 4π 2 (m/k)  k = 4π 2 m/T 2 k = 4π 2( 357kg)/(0.840s) 2 = 20,000 N/m

A Body Mass Measurement Device The device consists of a spring-mounted chair in which an astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of the astronaut.

T = 2π √(m/k)

Boundary Behavior of Waves As a wave travels through a medium, it will often reach the end of the medium and encounter an obstacle or perhaps another medium through which it could travel Example: Canyon Echoes….. A sound wave reflects off canyon walls to produce an echo Boundary Behavior Behavior of a wave upon reaching the end of a medium Boundary: Where 1 medium ends and another begins

Fixed vs. Free End Reflections FIXED End Reflection: Returning disturbance off of fixed boundary is referred to as an inverted reflected pulse Reflected & incident pulses have the same speed & wavelength Amplitude of the reflected pulse is less than the amplitude of the incident wave FREE End Reflection Reflected pulse is NOT inverted

Boundary: Less to More Dense When a wave travels from a less dense to a more dense medium, the incident pulse will be… Inversely reflected off of the boundary Transmitted as a pulse through the new, denser medium (never inverted) Transmitted pulse (in the more dense medium) travels slower & has a smaller wavelength compared to the reflected pulse (in the less dense medium) Incident & reflected pulses have the same speed & wavelength

Boundary: More to Less Dense When a wave travels from a more dense to a less dense medium, the incident pulse will be… Reflected off of the boundary (No inversion) Transmitted as a pulse through the new, less dense medium (never inverted) Transmitted pulse (in the less dense medium) travels faster & has a larger wavelength compared to the reflected pulse (in the more dense medium) Incident & reflected pulses have the same speed & wavelength

Summary of Boundary Behavior Waves in Ropes Wave speed is always greatest in least dense rope Wavelength is always greatest in least dense rope Frequency of a wave is not altered by crossing a boundary Reflected pulse becomes inverted when a wave in a less dense rope is heading towards a boundary with a more dense rope Amplitude of the incident pulse is always greater than the amplitude of the reflected pulse

Wave Interactions A material object, such as a rock, will NOT share its space with another rock Mechanical Waves Are not matter but rather displacements of matter, therefore, 2 waves can occupy the same space at the same time If you listen carefully at a concert, you can distinguish the sounds of different instruments Trumpet sounds are different from flute sounds Sound waves of each instrument are unaffected by the other waves passing through the same space at the same moment If you drop 2 rocks in water, the waves produced by each can overlap and form an interference pattern

Wave Interference Occurs when 2 waves meet when traveling along the same medium Two types: Constructive Interference Destructive Interference

Constructive Interference Occurs at any location along the medium where the two interfering waves have a displacement in the same direction

Destructive Interference Occurs along any location of the medium where the two interfering waves have a displacement in the opposite direction

Wave Behavior upon Interference Meeting of 2 waves along a medium does not alter the individual waves or even deviate them from their path All in all, The 2 waves meet Produce a net resulting shape of the medium And then continue on doing what they were doing before the interference

Principle of Superposition When 2 waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location

Did not do Doppler -2012

Doppler Effect Imagine a bug jiggling its legs and bobbing up and down in the middle of a quiet puddle – merely treading water in a fixed position Crests of the wave it makes are concentric circles because the wave speed is the same in all directions Bug bobs in the water at a constant frequency Distance between wave crests (the wavelength) will be the same for all successive waves Waves encounter point A as frequently as they encounter point B

Doppler Effect Top view of circular water wave made by a stationary bug jiggling in still water A B

Doppler Effect Now, the jiggling bug moves across the water at a speed less than the wave speed Centers of the circular crests move in the direction of the swimming bug A B

Doppler Effect Observer B – Encounters a higher frequency, because each successive crest has a shorter distance to travel so they arrive at B more frequently A B

Doppler Effect Observer A – Encounters a lower frequency because of the longer time between wave-crest arrivals A B

Doppler Effect This change in frequency due to the motion of the source (or receiver)  DOPPLER EFFECT A B

Doppler Effect Evident when you hear the changing pitch of a car horn as the car passes you When the car approaches Pitch is higher than normal because the sound wave crests are encountering you more frequently When the car passes & moves away You hear a drop in pitch because the wave crests are encountering you less frequently

Doppler effect: The change in frequency or pitch of the sound detected by an observer because the sound source and the observer have different velocities with respect to the medium of sound propagation

Doppler Effect A change in the frequency of a wave due to motion of the source and/or the listener * When the sound source is stationary Sound has the same wavelength and frequency in all directions * When the sound source moves toward the listener Sound wave-fronts arrive closer together Listener hears a higher frequency

Doppler Effect * When the sound source moves away from the listener Sound wave-fronts arrive farther apart Listener hears a lower frequency * Preview Kinetic Books

Completed Standing waves 2012

Standing Waves Interference Phenomenon Requires a perfectly timed interference of 2 waves passing through the same medium, in opposite directions, at the same frequencies  Interference of the incident wave & reflected wave occur in such a manner, specific points along the medium appear to be standing still  STANDING WAVE PATTERN

Standing Waves Nodes Locations in a standing wave that have NO Displacement Antinodes Locations where particles vibrate between a maximum upward displacement to a maximum downward displacement ** Nodes and antinodes stay at constant positions in a standing wave

Harmonics Standing wave patterns that correspond to a particular frequency Harmonic Frequencies 1 st Harmonic: Lowest possible frequency (fundamental) 2 nd Harmonic 3 rd Harmonic 4 th Harmonic n th Harmonic Frequency associated with each harmonic is dependent upon the speed at which waves move through the medium & the wavelength of the medium ** Handout: Standing Waves- Harmonics & Patterns **

Mathematics of Standing Waves 1 st Harmonic: Length of the string is one-half the length of a complete wave L = 1/2λ 2 nd Harmonic L = 2/2λ  L = λ 3 rd Harmonic L = 3/2λ n th Harmonic L = n/2λ