Waves and Wave Motion in elastic media

Simple Harmonic Motion Any object moving under the influence of Hooke’s Law type forces exhibits a particular type of motion about its equilibrium position. Any object moving under the influence of Hooke’s Law type forces exhibits a particular type of motion about its equilibrium position. This motion is called “sinusoidal” and has certain specific characteristics. This motion is called “sinusoidal” and has certain specific characteristics.

Nomenclature A - Amplitude - the maximum displacement from equilibrium A - Amplitude - the maximum displacement from equilibrium - Wavelength - Distance between any two successive identical points. - Wavelength - Distance between any two successive identical points. f - Frequency - Number of cycles or oscillations per second. ( = 1/T) f - Frequency - Number of cycles or oscillations per second. ( = 1/T) T - Period - Time to complete one full oscillation or cycle. (=1/f) T - Period - Time to complete one full oscillation or cycle. (=1/f)

Plot of Amplitude vs. Position

Mechanical Waves Waves in an elastic (or deformable) media Waves in an elastic (or deformable) media Originate as a displacement of some portion of the media from its equilibrium condition Originate as a displacement of some portion of the media from its equilibrium condition This portion then begins to oscillate about its equilibrium position. This portion then begins to oscillate about its equilibrium position. The elastic properties of the media transmit this disturbance from one layer to the next. The elastic properties of the media transmit this disturbance from one layer to the next.

Types of mechanical waves Transverse - oscillation  to direction of propagation Transverse - oscillation  to direction of propagation Longitudinal - oscillation  to direction of propagation Longitudinal - oscillation  to direction of propagation One, two, or three dimensional One, two, or three dimensional Pulse Pulse Periodic Periodic Harmonic Harmonic Any combination of these! Any combination of these!

One, Two, & Three Dimensions One Dimension One Dimension Spring / Slinky Spring / Slinky Two Dimension Two Dimension Ripples on water Ripples on water Three Dimension Three Dimension Sound / Earth Quake Sound / Earth Quake

Pulse wave A single oscillation sent through a media due to a non-repeating event. A single oscillation sent through a media due to a non-repeating event. e.g. Sound from an explosion or crash. Splash from a stone dropped in water.

Periodic Wave Wave produced by an oscillation of the media which repeats with a regular period. e.g. Sound from a steady drum beat

Harmonic Wave Created by a simple harmonic oscillations of the media. e.g. Music

v = f

Different types of waves

Reflection of Waves When a wave hits a “hard” surface it reflects back “inverted”. ( Newton’s 3rd law) When a wave hits a “hard” surface it reflects back “inverted”. ( Newton’s 3rd law) When a wave hits a “soft” surface it reflects back in a “non-inverted” manner. When a wave hits a “soft” surface it reflects back in a “non-inverted” manner. “Hard” surfaces have lower velocity for that type of wave.

Wave reflection

Reflection of a wave from a free end Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Reflection of a wave from a fixed end Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Transmission When a wave hits a surface or interface not all of the energy is reflected. Some is transmitted into the next medium. When a wave hits a surface or interface not all of the energy is reflected. Some is transmitted into the next medium. The transmitted wave is “non-inverted” and the “harder” the surface the less is transmitted. The transmitted wave is “non-inverted” and the “harder” the surface the less is transmitted.

Travel between media of different density Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Longitudinal Wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Sound Pressure wave - longitudinal Pressure wave - longitudinal Frequency = pitch Frequency = pitch v = 334 m/s in air at room temperature v = 334 m/s in air at room temperature Velocity is dependent upon the material Velocity is dependent upon the material

Sound is a pressure wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Tuning fork creating a sound wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Guitar String creating a sound wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Sound wave and Eardrum Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Ear

Reverberation vs Echo Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Standing Waves Created by the interference of two waves of the same frequency traveling in opposite directions. Created by the interference of two waves of the same frequency traveling in opposite directions. In a string or tube of limited length the reflections off the ends will create two traveling waves moving in opposite directions. In a string or tube of limited length the reflections off the ends will create two traveling waves moving in opposite directions. Only certain wavelengths ( ) will create standing waves in a string or tube of limited length. Only certain wavelengths ( ) will create standing waves in a string or tube of limited length. These wavelengths ( ) correspond to frequencies (f = v/ ) called natural or resonant frequencies These wavelengths ( ) correspond to frequencies (f = v/ ) called natural or resonant frequencies

Traveling Wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Standing Wave Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fundamental Frequency Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Second Harmonic Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Third Harmonic Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fourth Harmonic Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Fifth Harmonic Animations courtesy of Paul Hewitt and borrowed from physicsclassroom.com

Harmonics

Speeds of Sound at T = 20 C Air Air Helium Helium Hydrogen Hydrogen Water Water Sea water Sea water Iron/Steel Iron/Steel Glass Glass Aluminum Aluminum 343 m/s 343 m/s 1005 m/s 1005 m/s 1300 m/s 1300 m/s 1440 m/s 1440 m/s 1560 m/s 1560 m/s ≈5000 m/s ≈5000 m/s ≈ 4500 m/s ≈ 4500 m/s ≈ 5100 m/s ≈ 5100 m/s

Intensity of Sound Unit is the “Bel”. Named after Alexander Graham Bell Unit is the “Bel”. Named after Alexander Graham Bell More commonly used is the decibel (dB) More commonly used is the decibel (dB) = 0.1 Bel = 0.1 Bel  (in dB) = 10 log(I/I 0 )  (in dB) = 10 log(I/I 0 ) I is the intensity (  A 2 ) I is the intensity (  A 2 ) I 0 = 1.0 x W/m 2 the “threshold of hearing” I 0 = 1.0 x W/m 2 the “threshold of hearing”

Some Intensities (in dB) Jet plane at 30 m Jet plane at 30 m Threshold of pain Threshold of pain Indoor rock concert Indoor rock concert Auto interior (50mph) Auto interior (50mph) Street traffic Street traffic Conversation (50cm) Conversation (50cm) Whisper Whisper Rustle of leaves Rustle of leaves x x x x10 -11

Interference When two (or more) waves pass through each other they “interfere”. When two (or more) waves pass through each other they “interfere”. Simultaneous displacements of the medium will add together. -Principle of Superposition- Simultaneous displacements of the medium will add together. -Principle of Superposition- Constructive and/or Destructive interference. Constructive and/or Destructive interference.

Interference - two waves 1 = 2 m, 2 = 1.5 m

Beats and Beat Frequency Caused by two traveling wave of different frequencies, f 1 and f 2. Caused by two traveling wave of different frequencies, f 1 and f 2. Carrier frequency = Average (f 1 and f 2 ) Carrier frequency = Average (f 1 and f 2 ) Beat frequency = | f 1 - f 2 | Beat frequency = | f 1 - f 2 | Demo Demo

Doppler Effect Apparent change in frequency (pitch) of a sound from a moving source. Apparent change in frequency (pitch) of a sound from a moving source. Source moving toward observer: Source moving toward observer: f’ = f / (1-v s /v) f’ = f / (1-v s /v) Source moving away from observer: Source moving away from observer: f’ = f / (1+v s /v) f’ = f / (1+v s /v) f = frequency of source, f’=frequency heard by observer, v s = velocity of source, v = velocity of sound