WAVES Vibrations that carry energy from one place to another

Types of Wave Mechanical. Examples: slinky, rope, water, sound, & earthquake Mechanical. Examples: slinky, rope, water, sound, & earthquake Electromagnetic. Examples: light, radar, microwaves, radio, & x-rays Electromagnetic. Examples: light, radar, microwaves, radio, & x-rays

What Moves in a Wave? Energy can be transported over long distances Energy can be transported over long distances The medium in which the wave exists has only limited movement The medium in which the wave exists has only limited movement Example: Ocean swells from distant storms Example: Ocean swells from distant storms Path of each bit of water is ellipse

Periodic Wave Source is a continuous vibration Source is a continuous vibration The vibration moves outward The vibration moves outward

Wave Basics - Vocabulary Wavelength is distance from crest to crest or trough to trough Wavelength is distance from crest to crest or trough to trough Amplitude is maximum height of a crest or depth of a trough relative to equilibrium level Amplitude is maximum height of a crest or depth of a trough relative to equilibrium level

Frequency and Period Frequency, f, is number of crests (waves) that pass a given point per second Frequency, f, is number of crests (waves) that pass a given point per second Period, T, is time for one full wave cycle to pass Period, T, is time for one full wave cycle to pass T = 1/f f = 1/T (inverses or reciprocals) T = 1/f f = 1/T (inverses or reciprocals) Waves /second = seconds/wave = Waves /second = seconds/wave = fT

Unit of Frequency Hertz (Hz) Hertz (Hz) Second -1 same as 1/second or per second Second -1 same as 1/second or per second Used to be “cycles per second” Used to be “cycles per second”

Wave Velocity Wave velocity,v, is the velocity at which any part of the wave moves Wave velocity,v, is the velocity at which any part of the wave moves If wavelength =  v = f If wavelength =  v = f Example: a wave has a wavelength of 10m and a frequency of 3Hz (three crests pass per second.) What is the velocity of the wave? Hint: Think of each full wave as a boxcar. What is the speed of the train? Example: a wave has a wavelength of 10m and a frequency of 3Hz (three crests pass per second.) What is the velocity of the wave? Hint: Think of each full wave as a boxcar. What is the speed of the train?

v = f  v/f f = v/ v = f  v/f f = v/   lambda  wavelength  f frequency  v is sometimes called velocity of propagation (speed wave moves in medium)

Example A ocean wave travels from Hawaii at 10 meters/sec. Its frequency is 0.2 Hz. What is the wavelength? A ocean wave travels from Hawaii at 10 meters/sec. Its frequency is 0.2 Hz. What is the wavelength? = v/f = 10/0.2 = 50 m

Second example What is the wavelength of 100 MHz FM radio waves? Use v = c = 3 x 10 8 m/s What is the wavelength of 100 MHz FM radio waves? Use v = c = 3 x 10 8 m/s = v/f = 3 x 10 8 m/s ÷ 100 x 10 6 s -1 = v/f = 3 x 10 8 m/s ÷ 100 x 10 6 s -1 = (300 x 10 6 ) ÷ (100 x 10 6 ) m = (300 x 10 6 ) ÷ (100 x 10 6 ) m = 3.0 m = 3.0 m

Another example Waves travel 75 m/s on a certain stretched rope. The distance between adjacent crests is 5.0 m. Find the frequency and the period. Waves travel 75 m/s on a certain stretched rope. The distance between adjacent crests is 5.0 m. Find the frequency and the period. f = v/ f = v/ f = 75 m/s ÷ 5.0 m = 15 Hz = 15 s -1 f = 75 m/s ÷ 5.0 m = 15 Hz = 15 s -1 T = 1/15 = s T = 1/15 = s

Longitudinal vs. Transverse Waves Transverse: particles of the medium move perpendicular to the motion of the wave Transverse: particles of the medium move perpendicular to the motion of the wave Longitudinal: vibrations in same direction as wave Longitudinal: vibrations in same direction as wave

Longitudinal Wave Can be thought of as alternating compressions (squeezing) and expansions or rarefactions (unsqueezing) Can be thought of as alternating compressions (squeezing) and expansions or rarefactions (unsqueezing)

Longitudinal Wave

Sound Wave in Air Compressions and rarefactions of air produced by a vibrating object Compressions and rarefactions of air produced by a vibrating object

Waves and Energy Waves with large amplitude carry more energy than waves with small amplitude Waves with large amplitude carry more energy than waves with small amplitude

Resonance Occurs when driving frequency is close to natural frequency (all objects have natural frequencies at which they vibrate) Occurs when driving frequency is close to natural frequency (all objects have natural frequencies at which they vibrate) Tacoma Narrows bridge on the way to destruction– large amplitude oscillations in a windstorm

Interference Amplitudes of waves in the same place at the same time add algebraically (principle of superposition) Amplitudes of waves in the same place at the same time add algebraically (principle of superposition) Constructive interference: Constructive interference:

Destructive Interference Equal amplitudes(complete): Equal amplitudes(complete): Unequal Amplitudes(partial): Unequal Amplitudes(partial):

Reflection Law of reflection: Law of reflection: Angle of Incidence equals angle of Reflection Angle of Incidence equals angle of Reflection

Hard Reflection of a Pulse Reflected pulse is inverted Reflected pulse is inverted

Soft Reflection of a Pulse Reflected pulse not inverted Reflected pulse not inverted

Soft (free-end) Reflection

Standing Waves Result from interference and reflection for the “right” frequency Result from interference and reflection for the “right” frequency Points of zero displacement - “nodes” (B) Points of zero displacement - “nodes” (B) Maximum displacement – antinodes (A) Maximum displacement – antinodes (A)

Formation of Standing Waves Two waves moving in opposite directions Two waves moving in opposite directions

Examples of Standing Waves Transverse waves on a slinky Transverse waves on a slinky Strings of musical instrument Strings of musical instrument Organ pipes and wind instruments Organ pipes and wind instruments Water waves due to tidal action Water waves due to tidal action

Standing Wave Patterns on a String “Fundamental” = “Fundamental” =

First Harmonic or Fundamental

Second Harmonic

Third Harmonic

Wavelength vs. String length

String length = How many waves? L =

String length = How many waves? L = 3/2 

Wavelength vs. String Length Wavelengths of first 4 harmonics Wavelengths of first 4 harmonics L f =v

Frequencies are related by whole numbers Example Example f 1 = 100 Hz fundamental f 1 = 100 Hz fundamental f 2 = 200 Hz 2 nd harmonic f 2 = 200 Hz 2 nd harmonic f 3 = 300 Hz 3 rd harmonic f 3 = 300 Hz 3 rd harmonic f 4 = 400 Hz 4 th harmonic f 4 = 400 Hz 4 th harmonic etc etc Other frequencies exist but their amplitudes diminish quickly by destructive interference Other frequencies exist but their amplitudes diminish quickly by destructive interference

Wave velocity on a string Related only to properties of medium Related only to properties of medium Does not depend on frequency of wave Does not depend on frequency of wave v 2 = T/m/l Tension divided by mass per unit length of string v 2 = T/m/l Tension divided by mass per unit length of string

Standing Waves in Open Tubes

First Three Harmonics in Open Tube Amplitudes are largest at the open ends Amplitudes zero at the nodes

Tube Closed at One End L  /4 L =  /4 L =  /4 No even harmonics present f = v air /

Beats Two waves of similar frequency interfere Two waves of similar frequency interfere Beat frequency equals the difference of the two interfering frequencies

Acknowledgements Diagrams and animations courtesy of Tom Henderson, Glenbrook South High School, Illinois Diagrams and animations courtesy of Tom Henderson, Glenbrook South High School, Illinois