Vibrations (Oscillations in Time)

• Properties of SHMF =-ksxPeriod (time for a full cycle): T=2π√(m/ks)Frequency (number of cycles per second): f=1/TAmplitude: A = Maximum displacement from

equilibriumVelocity: v = speed of massAngle swept: θ(t) = ωt and: ω = 2πf = 2π/Ty-motion: y(t) = A sin(ωt) = A sin(2πft)= A sin(2πt/T)Review Circular Analog from SHM ppt slides.

• (next slide should look familiar)

Circle Analog to SHM

•Suppose we have an object of mass (m) in uniform circular motion with angular velocity (ω) and radius (A).

•Angle swept: θ(t) = ωt and: ω = 2πf = 2π/T

•y-motion: y(t) = A sin(ωt) = A sin(2πft)= A sin(2πt/T)

•Projection or “shadow” of vertical axis traces out an SHM with period (T).•Speed of circular motion: v= Aω = A(2π/T) _____ •Max speed of SHM: vmax=√(ks/m) A _____•Equate to give: T=2π√(m/ks)

Wave Motion• Harmonic Motion in Time and Space.

Freezing time: See oscillation over spaceLook at one point: See oscillations over time. y(x,t) = A sin(2πx/λ -2πt/T)

Waves transport Energy

Characteristics of Waves• Amplitude (A)

Maximum displacement from equilibriumSound (density or pressure)E&M (Electric field) (E)

• Intensity (I ~ A2 ~ Power)Sound Wave: LoudnessE&M Wave: Brightness

• Wavelength (λ) (m)Distance between successive crests (time frozen)

• Period (T) (s)Time between successive crests (distance frozen)

• Frequency (f) (Hz or 1/s)Number of cycles per second (pitch or color)

• Speed (v = f λ) (m/s)Speed at which energy is transported.

Types of Waves

• Transverse WavesAmplitude is

perpendicular to propagation direction

Ocean Waves, Waves in strings, (E&M) Waves

• Longitudinal WavesAmplitude is parallel to

propagation directionSound Waves, parallel

compression of springs, p-waves of earthquakes.

Follow any one particle in the animation; it oscillates in time.

Period vs. Wavelength(watch what’s plotted on horizontal axis)

Behavior of a Wave at an Interface

• Interface: boundary between two “media” (materials)

• A wave incident on an interface can beReflected: Surface appears mirror-likeAbsorbed: Surface can appear black (heats

up; if hot enough can emit visible radiation)Transmitted: Surface is invisibleCombination of the above: Real world.

Superposition and Interference

• When two (or more) waves pass through the same region at the same time:they interfere with each other to create a new

waveformCombined waveform is the sum of the

displacements at each point.Constructive Interference: combined waveform

larger than originals.Destructive Interference: combined waveform

smaller than originals.

Superposition and Interference (cont’d)

• Total Constructive InterferenceWaves add “in phase” Phase shift: 0º, 360 º or 0, 2π, …Maximum combined waveform

• Total Destructive InterferenceWaves add “out of phase”Phase shift: 180 º, 540 º or π/2, 3π/2, …Minimum combined waveformCan result in extinction.

• Intermediate waveforms dependent on wave properties.

Phase Shift: Relative Difference in angle between two waves. (See Circular Analog)

Mechanical Wave at a Reflective Boundary

• When a wave reflects off a fixed boundary, it sustains a π/2 wave shift or λ/2 phase shift.Pulse gets inverted by reaction

force.

• When a wave reflects off a loose boundary, there is no wave shift.

Self-Interference and Standing Waves

• A wave incident on a reflective boundary Can interfere with its reflectionAmplitudes vibrate up and down (antinodes)Stationary points of zero amplitude (nodes)Net energy flow is zero (flow to right = flow to left)

• ResonanceAll vibrations and waves have natural or resonant

frequencies (ex: pendulum of length L)Leads to highest possible amplitudes when

• Forces are applied at natural frequencies. – Pushing a kid on a swing (or pushing a pendulum)

• Frequencies match geometry (next slide)

Standing Waves (cont’d)

• For waves traveling along length (L) between two fixed ends, to form standing waves:Displacement = 0 at endsIntegral number of half wave lengths must fit on L.Leads to: (n = 1, 2, 3, ….)

• Allowed wavelengths: L = n (λn/2) or λn = 2L/n• Natural Frequencies: fn = v/λn = nv/2L• Fundamental Frequency or first harmonic: n=1: f1

• Second Harmonic or First Overtone: n=2: f2 = 2f1

• And so on: nth Harmonic or (n-1)th Overtone: fn = n f1

Application to stringed instrumentsApplet and Illustrations for Standing Waves on Strings

Sound Waves

• Longitudinal Waves

• Caused by vibration of materials (e.g. tuning fork)

• Cannot travel through a vacuum.

• Travel faster and louder through solids and liquids than in gases (340 m/s in dry air at 20°C).

• Pitch: describes how we hear frequency.Hearing range: 20Hz to 20,000HzLower (infrasonic), Higher (ultrasonic)

• Loudness: describes how we hear intensity (square of amplitude)

Decibel Scale (dB)• Measures Loudness• An increase of 10dB = 10 times increase in intensity.

The Doppler Effect

How does our perception of a wave change when its source is moving towards us or away from us?

Sound Wave AppletIf the relative motion of the source is towards us we

perceive higher frequency (higher pitch or bluer).If the relative motion of the source is away from us

perceive lower frequency (lower pitch or redder).The observed shift in frequency is called the “Doppler

Shift”.

What if a source of sound moves as fast as the sound wave itself?

• The source will move with the wave itself.• Both source and wave arrive at our ear at the

same time. Sound Wave Applet• We hear all of the amplitudes compressed

together (LOUD) at one instant (SUDDEN)It’s a “SONIC BOOM”

• What if the source exceeds speed of sound?Source arrives before the sound it makes. We don’t

hear it until it’s past us.A “CONICAL SHOCK WAVE”.

A Plane in Supersonic Flight

The Doppler Shift also tells us….

• Whether a storm system is coming to us or not.By bouncing radio waves off rain droplets and

analyzing the shift between sent and received waves.

• Whether we’re exceeding the speed limit.By bouncing radio waves off your car (RADAR)

• Whether we have blood clots in our legs or the rate of blood flow through our hearts.By bouncing sound waves off red blood cells.

• Whether or not the Universe is expanding.What does that mean?

Resonance and Music

• Stringed Instruments (Already discussed)Integral number of half wave lengths must fit on

L. Use finger pressure to vary L.Leads to: (n = 1, 2, 3, ….)

• Allowed wavelengths: L = n (λn/2) or λn = 2L/n

• Natural Frequencies: fn = v/λn = nv/2L

• Fundamental Frequency or first harmonic: n=1: f1

• Second Harmonic or First Overtone: n=2: f2 = 2f1

• And so on: nth Harmonic or (n-1)th Overtone: fn = n f1

Organ Pipes and Wind Instruments

• Closed Pipe: Open mouth (A): antinodeClosed end (D): nodeBoundary Condition: L = n λn/4 or

λn = 4L/n; n=1,3,5, ...Natural frequencies: f1 = v/4L;

fn=n f1; n=1,3,5, ...All even harmonics are missing.

• Open Pipe: Open mouth (A): antinodeOpen end (D): antinodeBoundary Condition : L = nλn/2 or λn =

2L/n; n=1,2,3,...Natural Frequencies: f1 = v/2L; fn= f1;

n=1,2,3,...

Antinodes exist at open ends; freer movement

Harmonics of Open and Closed Pipes• Pressure and Displacements in open and closed pipes

Closed pipe Open and closed pipes