Vibrations and Waves Simple Harmonic Motion, Waves, and the Science of Sound

Simple Harmonic Motion, Waves, and the Science of Sound n What types of motion have we studied thus far? n What kinds of events in your life occur over and over again? n How can you create a vibration or wave? n Why study waves?

December 26, 2004 Indian Ocean Tsunami n 200,000 Casualties n Millions Displaced from Homes n $Billions in Property Loss

With no warning the people were caught unaware.

The Destruction and Devastation

December 26, 2004 Indian Ocean Tsunami

Periodic Motion Repeats itself over a fixed and reproducible period of time. Mechanical devices that do this are known as oscillators.

Examples of periodic motion… A weight attached to a spring which has been stretched and released. A pendulum “bobbing” back and forth. A record spinning on a turntable The surf lapping up on the shore. The tapping of a pencil on your desk. A bouncing ball. Can you think of any?

Simple Harmonic Motion (SHM) Periodic motion where there is a equilibrium position that is centrally located. This can be described by a sine or cosine function. Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs).

Equilibrium The midpoint of the oscillation of a simple harmonic oscillator. Position of minimum potential energy and maximum kinetic energy.

Amplitude (A) How far the wave is from equilibrium at it’s maximum displacement. Waves with high amplitude have more energy than waves with low amplitude.

Period (T) The length of time it takes for one cycle of periodic motion to complete itself.

Frequency (f): How fast the oscillation is occurring. Frequency is inversely related to period. The unit of frequency is the Hertz (Hz) where 1 Hz = 1 s -1.

Springs n A very common type of Simple Harmonic Oscillator. n Our springs will be ideal springs. – They are massless. – They are compressible and extensible. n They will follow a law known as Hooke’s Law.

Restoring force n The restoring force is the secret behind simple harmonic motion. n The force is always directed so as to push or pull the system back to its equilibrium (normal rest) position.

Hooke’s Law mmm x F x F F = -kx Spring compressed, restoring force out Spring at equilibrium, restoring force zero Spring stretched, restoring force in Equilibrium position

Pendulums n Springs are not the only type of oscillators. n The pendulum can be thought of as an oscillator. n The displacement needs to be small for it to work properly.

Mechanical Wave A disturbance which propagates through a medium with little or no net displacement of the particles of the medium. Examples would be water waves, sound, seismic waves, waves on a spring.

Diagram of a Wave x(m) y(m) A: amplitude : wavelength crest trough equilibrium

Speed of a wave n The speed of a wave is the distance traveled by a given point on the wave (such as a crest) in a given interval of time. n v = d/t – d: distance – t: time

Derivation of Wave Speed

Speed of a wave n v = ƒ – v : speed (m/s) – : wavelength (m) – ƒ : frequency (s –1, Hz)

8.0 m 3.5 m 20.0m/s Determine amplitude, wavelength, frequency and period

Wave types: transverse A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves. Example: waves on a string

Wave types: transverse

Transverse Wave Notice the motion of each dot relative to the propagation of the wave.

Wave types: longitudinal A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave moves. Longitudinal waves are also called compression waves. Example: sound

Wave types: longitudinal

Longitudinal Wave

Longitudinal vs Transverse

Longitudinal described as a Transverse Wave

Other Wave Types n Earthquakes: combination n Ocean waves: circular n Light: electromagnetic* *Not Mechanical

Reflection of waves Occurs when a wave strikes a medium boundary and “bounces back” into original medium. Completely reflected waves have the same energy and speed as original wave.

Reflection of waves n Fixed-end reflection: wave reflects with inverted phase. This occurs when reflecting medium has greater density. n This can also be called reflection off of a rigid medium.

Reflection of waves n Open-end reflection: wave reflects with same phase. This occurs when reflecting medium has lesser density. n This can also be called reflection off of a non-rigid medium.

Open End reflection Fixed End Reflection

Refraction of waves Transmission of wave from one medium into another. Refracted waves may change speed and wavelength. Refracted waves do not change frequency.

Refraction of Waves n Waves can be bent when they pass from one medium into another. n An example is when light refracts from the water to the air as it is reflected off of a spoon. The effect you see is a bent or broken spoon.

Refraction of Waves

Diffraction n The bending of a wave around a barrier. n Diffraction of waves combined with interference of diffracted waves causes “diffraction patterns”.

Diffraction

Principle of Superposition n When two or more waves pass a particular point in a medium simultaneously, the resulting displacement at that point in the medium is the sum of the displacements due to each individual wave. n The waves interfere with each other.

Types of interference. n If the waves are “in phase”, that is crests and troughs are aligned, the amplitude is increased. This is called constructive interference. n If the waves are “out of phase”, that is crests and troughs are completely misaligned, the amplitude is decreased and can even be zero. This is called destructive interference.

Constructive Interference crests aligned with crest waves are “in phase”

Destructive Interference crests aligned with troughs waves are “out of phase”

Green + Blue = Black

Standing Wave n A standing wave is a pattern of wave crests and troughs that remains stationary in a medium when a wave reflects and interferes with itself. Nodes Antinodes

Standing Wave Reflection and Interference n A standing wave is a wave which is reflected back and forth between fixed ends (of a string for example). n Reflection may be fixed or open- ended. n Superposition of the wave upon itself results in constructive and destructive interference and an enhanced wave.

Standing Wave on a Rope

Nodes and Antinodes Nodes are positions of minimum amplitude Antinodes are positions of maximum amp. Red + Blue = Black

Sound is a longitudinal wave

Sounds in the Real World Because of superposition and interference, real world waveforms may not appear to be pure sine or cosine functions. That is because most real world sounds are composed of multiple frequencies.

Musical Instruments

Fixed-end standing waves (violin string) Fundamental First harmonic = 2L First Overtone Second harmonic = L Second Overtone Third harmonic = 2L/3 L

Open-end standing waves (organ pipes) Fundamental First harmonic = 2L First Overtone Second harmonic = L Second Overtone Third harmonic = 2L/3 L

Closed End standing waves (some organ pipes) First harmonic = 4L Second harmonic = (4/3)L Third harmonic = (4/5)L L

Beats n The characteristic loud-soft pattern that characterizes two nearly (but not exactly) matched frequencies. n Musicians call this “being out of tune”

What word best describes this to physicists? Amplitude Answer: beats

Resonance The vibration of a body at its natural frequency due to the action of a vibrating source of the same frequency.

The Music of Resonance by Robert Tiso

Tacoma Narrows Bridge Collapse

Doppler Effect n The Doppler Effect is the raising or lowering of the perceived pitch of a sound based on the relative motion of the observer and the source of the sound.

Doppler Effect n When an ambulance is racing toward you, the sound of its siren appears to be higher in pitch. n When the ambulance is racing away from you, the sound of its siren appears to be lower in pitch.

Boat Wake

Breaking the Sound Barrier

The Human Ear

Decibel Scale

A final word about “pitch” n Pitch rises when frequency rises and the sound is “higher”. n Pitch is lowered when the frequency is lowered and the sound is “lower”.

Question 1 Explain why the restoring force is always opposite the displacement from equilibrium for a spring.

Question 2 Why do we consider the pendulum a simple harmonic oscillator?

Question 3 Draw 2 periods of a wave with wavelength 6 cm and amplitude 3 cm. Try to be accurate with scale. Label wavelength and amplitude

A B C Question 4 Explain the energy transfers of a pendulum. Indicate the points A, B, and C.

Question 5 Explain how the stadium wave is like a transverse wave, and the domino fall is like a longitudinal wave.

Question 6 Does the following poem really explain waves? Why or why not?

Ocean’s Waves Calm emotions cause small waves to flow, Calm emotions cause small waves to flow, Anger causes tsunamis to blow. Waves from each emotion, all controlled by me, Ocean. Wondered how waves are formed, now you know. by Ocean Myranda