Vibrations and Waves Vibrations & Waves.

Vibrations and Waves Vibrations & Waves

Periodic Motion Motion that repeats in a regular cycle is called periodic motion. The revolution of a planet about its sun is an example of periodic motion. The highly reproducible period (T) of a planet is also called its year. Mechanical devices on earth can be designed to have periodic motion. These devices are useful timers. They are called oscillators.

Periodic Motion Motion that repeats in a regular cycle is called periodic motion or simple harmonic motion. Pendulum - Mass on a spring

Simple Pendulum Simple harmonic motion can be demonstrated by the swing of a pendulum. A simple pendulum consists of a massive object, called the bob, suspended by a string or light rod of length L.

Forces on Pendulum At the left and right positions, the net force and acceleration are maximum, and the velocity is zero. At the middle position in the figure, the net force and acceleration are zero, and the velocity is maximum q L x

SHM - Pendulum q L x You can see that the net force is a restoring force; that is, it is opposite the direction of the displacement of the bob and is trying to restore the bob to its equilibrium position.

Pendulum GPE max GPE max Fnet and a max Fnet and a max KE 0 KE 0
v zero v zero GPE zero Fnet and a zero KE max v max 7

Simple Harmonic Motion
Requres a RESTORING FORCE - force that restores object to its equilibrium position that is directly proportional to the displacement of the object Period (T): time it takes the object to complete one cycle of motion Units - seconds Frequency (f): number of cycles in one second. Units - seconds-1 or Hertz Amplitude (A) : maximum distance that the object moves from the equilibrium position Units - meters

Experimentally determine what T depends on before derive an expression

Experimental Design Purpose? Controlled Experiments
Determine relationship between two different variables Controlled Experiments Manipulate only one variable in an experiment Observe its effect on a second variable Hold ALL other variables in the experiment CONSTANT

Variables Any factor that might affect the behavior of an experiment.
Independent Variables Factor that is changed or manipulated during the experiments Always plotted on the x-axis Time is usually the independent variable Dependent Variables Factor that depends on the independent variable Always plotted on the y-axis

Collecting and Recording Data
At least 6 data points are necessary for a good graph. Independent variable should cover a range of at least 10 fold if possible (eg to 2.0 m) Raw data is recorded in a data table immediately as it is collected in the lab. Data Table Construct data table before collecting the data Independent variable in leftmost column of data table Every column is labeled with the variable name being measured AND the units in parentheses Values in table do not have units. Same number of decimal places in each column

Graphing Data Purpose Determine relationship between two variables
Plot data as scatter graphs (do not connect the data points) Graphs Always include Title (in WORDS) DEPENDENT vs. INDEPENDENT variable Label each axis with the variable and the UNITS Recognize common relationships in graphs Connect the data points with a line or curve of best fit to show the relationship between variables

Axes labeled with variable symbols (not words) and units
Graphing Data Title (words) Force Applied vs. Mass F=2m Direct Relationship Dependent variable Axes labeled with variable symbols (not words) and units Independent variable

Simple Harmonic Motion for a Pendulum
independent of mass independent of amplitude Dependent on g (gravitational strength)

Example Problem On a planet with an unknown value of g, the period of a 0.75 m long pendulum is 1.8 sec. What is g for this planet?

Resonance London Millenium bridge Resonance is a special form of simple harmonic motion in which the additions of small amounts of force at specific times in the motion of an object cause a larger and larger displacement. Resonance from wind, combined with the design of the bridge supports, may have caused the original Tacoma Narrows Bridge to collapse. Tacoma Narrows Bridge Tacoma Narrows Bridge2 glass shattering montage

Waves Disturbance that travels through a medium from one location to another location.

Waves Disturbance that carries energy through matter and space. A wave transports energy NOT matter Waves travel through matter or space Newton’s laws of motion & conservation of energy govern the motion of waves A wave transports energy NOT matter. When a wave is present in a medium (that is, when there is a disturbance moving through a medium), the individual particles of the medium are only temporarily displaced from their rest position. There is always a force acting upon the particles that restores them to their original position. In a slinky wave, each coil of the slinky ultimately returns to its original position. In a water wave, each molecule of the water ultimately returns to its original position. And in a stadium wave, each fan in the bleacher ultimately returns to its original position. It is for this reason, that a wave is said to involve the movement of a disturbance without the movement of matter. The particles of the medium (water molecules, slinky coils, stadium fans) simply vibrate about a fixed position as the pattern of the disturbance moves from one location to another location.

Electromagnetic Waves
Mechanical Waves Mechanical waves require a medium to travel through Water Air Ropes Travel through the medium, but do not carry the medium away Electromagnetic Waves Electromagnetic waves do NOT require a medium to travel through

What type of wave??? ME or EM
X-rays Sound waves Light waves ripples Earthquake or seismic waves Microwaves Radio waves Surfing wave Stadium wave Ultrasound waves EM ME EM ME ME EM EM ME ME ME

Transverse Waves Wave that vibrates perpendicular to the direction of the wave’s motion.
Trough – lowest point on the wave Crest – highest point on the wave Wavelength – shortest distance between two identical points on a wave Amplitude – maximum distance from equilibrium (related to energy of the wave

Longitudinal Waves Wave that vibrates parallel to the direction of the wave’s motion. Example: Vibrate a slinky back and forth Sound travels as longitudinal waves

Longitudinal Waves Transverse Waves
Direction of travel Disturbance Longitudinal Waves Transverse Waves Direction of travel Disturbance

Measurements of a Wave Amplitude – depends on source, not on speed or medium Period/Frequency - depend on source, not on speed or medium Speed – depends only on medium (not on amp or frequency) Wavelength – depends only on medium Phase Measuring/describing a wave (wave properties) A – max disp fro, equilib A depends on how was generated but not how fast. More work to generate wave with larger A (strong winds produce larger water waves). Waves with >A transfer more E. Waves carry energy that can do work - TSUNAMI - speed – same as car find delta d of wave peak/dt While AMP of mechanical wave determines amount of energy it carries, only the medium determines the speed - wavelength – crest DEMO online – with tdflashzone use online stopwatch to measure wave period (and frequency) takes that many sec to go one wavelenght ….

Measuring a wave – Trough crest

Measuring a wave – AMPLITUDE
2xs amp 4xs energy

Period & Frequency Frequency – number of waves per second
Measured in Hertz (Hz) Period – time it takes to complete one cycle Measured in seconds (s)

Period & Frequency The frequency of a wave is equal to the reciprocal of the period. Both the period and the frequency of a wave depend only on its source. They do not depend on the wave’s speed or the medium.

Measuring a wave – Wavelength, l
large Wavelength l medium Wavelength l small Wavelength

Measuring a wave -speed
Speed of wave depends on the properties of the medium it travels in eg. Wave speed in a string depends on tension and strings mass/length eg. Wave speed in water depends on depth and g

Transverse Wave Longitudinal Wave A transverse wave is one that vibrates perpendicular to the direction of the wave’s motion. 2) A quick shake of a rope sends transverse waves in both directions. 3) Waves obtained in threads and ropes are transverse waves. A longitudinal wave is one in which the particle displacement is in the same direction as, or parallel to, the direction of the wave’s motion. 2) The squeeze and release of a coiled-spring toy sends out longitudinal wave pulses in both directions. 3) Waves obtained in springs and sounds are longitudinal waves.

DO NOW A sound wave has a frequency of 192 Hz and travels the length of a football field, m, in s. 337 m/s a. What is the speed of the wave? b. What is the wavelength of the wave? c. What is the period of the wave? d. If the frequency was changed to 442 Hz, what would be the new wavelength and period? 1.76 m s Same medium so same v (337m/s) New T=0.0023s, new l=0.76m

The time required for the sound waves (v = 340 m/s) to travel from the tuning fork to point A is ____ . The wavelength of the sound is ______ 0.059 s 0.664 m

Two waves are traveling through the same
container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The speed of wave B must be ________ the speed of wave A. a. one-ninth b. one-third c. the same as d. three times larger than Same medium so same v

The water waves below are traveling along the surface of the ocean at a speed of 2.5 m/s and splashing periodically against Wilbert's perch. Each adjacent crest is 5 meters apart. The crests splash Wilbert's feet upon reaching his perch. How much time passes between each successive drenching? Answer and explain using complete sentences.

f = 6 waves/2 sec = 3 waves/sec = 3 Hz
Suppose I wiggle a slinky back and forth, and count that 6 waves pass a point in seconds. What would the frequency be? f = 6 waves/2 sec = 3 waves/sec = 3 Hz

Sound Waves Sound is a type of wave.
Longitudinal As the bell shown in the figure moves back and forth, the edge of the bell strikes particles in the air.

When the edge moves forward, air particles are driven forward
Air particles bounce with greater velocity Greater pressure When the edge moves backward, air particles are no longer driven forward Air particles bounce with lower velocity Lower pressure

This results in alternating regions of slightly high and slightly low pressure.
The collisions among air particles cause the pressure variations to move away in all directions. These pressure variations are transmitted through matter as sound waves.

All Sound is Caused By Vibration of Something-
Example - Sound Field radiated by a Tuning Fork 43

Properties of Sound Speed Pitch – frequency of sound Loudness – amplitude of sound Quality or timbre

Pitch A measure of how high or low a sound is
Pitch depends on the frequency of a sound wave Louder (larger Amp) Softer (Smaller Amp) Low pitch Low frequency Longer wavelength High pitch High frequency Shorter wavelength Phet sound and speaker sim

Measurements of a Wave Amplitude – depends on source, not on speed or medium Period/Frequency - depend on source, not on speed or medium Speed – depends only on medium (not on amp or frequency) Wavelength – depends only on medium Phase

Wave Behavior (all waves)
When the wave encounters the boundary of the medium in which it is traveling, it often reflects back into the medium. In other instances, some or all of the wave passes through the boundary into another medium often changing direction - refraction. Many properties of wave behavior result from the fact that two or more waves can exist at the same time in the same medium (unlike particles).

Waves at Boundaries – wave speed depends on the medium
Incident Wave - wave that strikes the boundary Transmitted or Refracted Wave – wave that transmits to the new medium Reflected Wave – returning wave on the original medium

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 from fixed boundary
Reflects back - same speed Inverted same amp

Reflection from free boundary
Reflects back - same speed - upright Reflection from free boundary

Refraction of Waves Transmission of wave from one medium to another.
Refracted waves may change speed and wavelength. Refraction is almost always accompanied by some reflection. Refracted waves do not change frequency.

No boundary Rigid boundary Free Boundary When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary. Low to high density boundary High to Low density boundary

Reflection and Transmission of Waves
same slower Same speed slower

faster Same speed High to Low density boundary

MORE dense LESS dense Reflected wave Same speed Refracted wave slower High to Low density boundary 57

MORE dense LESS dense Reflected wave Same speed Refracted wave faster High to Low density boundary 58

MORE dense LESS dense LESS dense MORE dense Reflected Transmitted Speed (l)* same faster waveform upright amplitude smaller larger Reflected Transmitted Speed (l)* same slower waveform inverted upright amplitude larger smaller *Transmitted waves DO NOT change frequency

DO NOW The speed of sound in water is 1498 m/s. A sonar signal is sent straight down from a ship at a point just below the water’s surface, and 1.80 s later, the reflected signal is detected. How deep is the water? 1348.2m = 0.84 mile 60

DO NOW 1m/s 2cm vtransmitted vreflected
TOP: An incident pulse is traveling at a speed of 1 m/s in a string (blue) to which a 2nd string of a different density (red) is attached. BOTTOM: Part of the wave is reflected at the boundary and part is transmitted. What is the amplitude of the incident pulse? What are the wavelengths of the incident, reflected and transmitted pulses? What are the frequencies of the incident, reflected and transmitted pulses? What are the speeds of the reflected and transmitted pulses? Which string is denser, the blue or the red one? 4 cm li=0.8m, lr=0.8m, lt=0.4m 1.25hz vr=1m/s, vt=0.5m/s Red

Superposition of Waves
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. The waves interfere with each other. 62

Wave Interference Constructive Interference – wave displacements in same direction Antinode Destructive Interference – wave displacements in opposite direction Node

Principle of Superposition
The displacement of a medium caused by two or more waves is the algebraic sum of the displacements caused by the individual waves. In other words, two or more waves can combine to form a new wave - interference. Constructive interference – result in a new wave with greater amplitude. Destructive interference – result in a new wave with lesser amplitude.

Wave Interference

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

Standing Waves Wave that appears to be standing still.
Standing wave is the interference of two traveling waves (with equal f and l), moving in opposite directions. Nodes are at the ends of the rope. Antinodes are in the middle.

Standing Waves If you double the frequency of the vibration, you can produce one more node and one more antinode in the rope. Further increases in frequency produce even more nodes and antinodes.

Resonance London Millenium bridge Resonance is a special form of simple harmonic motion in which the additions of small amounts of force at specific times in the motion of an object cause a larger and larger displacement. Resonance from wind, combined with the design of the bridge supports, may have caused the original Tacoma Narrows Bridge to collapse. Tacoma Narrows Bridge Tacoma Narrows Bridge2 glass shattering montage

Standing Waves Nodes Antinodes
Incident wave Reflected wave Nodes Poinst of complete destructive interference Do not move Antinodes Poinst of complete constructive interference Largest amplitude points of the standing wave

Fixed end Standing Waves (violin string)
Third Harmonic Standing Wave Pattern First Harmonic Standing Wave Pattern Fixed end Standing Waves (violin string) 1st harmonic L= ½l= ½ v/f1 2nd harmonic (one octave higher) L= l= v/f2 3rd harmonic L= 3/2 l = 3/2 v/f3 If a guitar string is simply plucked, the fundamental frequency dominates. The first harmonic can be produced by touching the string lightly in the middle when plucking it. Touching the string lightly one-third the length of the string from one end will produce the second harmonic

Second Harmonic Standing Wave Pattern
Third Harmonic Standing Wave Pattern First Harmonic Standing Wave Pattern Second Harmonic Standing Wave Pattern Standing Waves Harmonic # of Nodes Antinodes Pattern Resonant Frequency 1st 2 1 L = l1/2 = v/2f1 2nd 3 L = l2= v/f f2 =2f1 3rd 4 L = 3l3/2 = 3v/2f3 f3 = 3f1 4th 5 L = 2l4= 2v/f f4 = 4f1 5th 6 L = 5l5/2 = 5v/2f5 f5 = 5f1 6th 7 L = 6l6 /2= 3v/f6 f6 = 6f1 nth n + 1 n -- L = nln/2= nv/2fn fn = nf1 guitar strings

Example: If a violin string vibrates at 440 Hz as its fundamental frequency, what are the frequencies of the first four harmonics 75

Example: Violin A 0.32 m long violin string is tuned to play A above middle C at 440 Hz What is the wavelength of the fundamental string vibration? 1st harmonic L= ½l1

Wind Instruments Sound is generated by vibrations, so when air is blown into one end of a pipe or tube and then bounces off of the sides, the air vibrates. When the air inside the tube vibrates at the same frequency, or in resonance, with the vibration of your lips, a sound is produced.

Wind Instruments The vibrating reed or lip produces sound waves with many frequencies. This sound wave of alternate high- and low-pressure variations moves down the air column. When the wave reaches the end of the column, it is reflected back up the column and can set up standing waves.

Resonance in an Open Pipe
Ends must be same - both ends are pressure nodes (displacement antinodes) Harmonics increase by 1: 1st, 2nd, 3rd, 4th, 5th, etc. 1st Harmonic L = l1/2 2nd Harmonic L = l2 3rd Harmonic L = 3l3/2 … Pressure nodes Pressure nodes nth Harmonic L = nln/2 sound wave in pipes 79

Resonance in a Closed Pipe
Ends must be opposite: open–nodes, closed-antinodes Harmonics increase by 2 (only odd harmonics). Pressure node 1st Harmonic L = l1/4 = v/4f1 3nd Harmonic L = 3l3/4 = 3v/4f3 f3 = 3f1 5th Harmonic L = 5l5/4 = 5v/4f5 f5 =5f1 … Pressure antinode nth Harmonic (odd only) L = nln/4 = nv/4fn fn = nf1 80

Open Pipes Closed Pipes Ends same Ends are opposite Every Harmonic
Odd Harmonics Pan Pipes Sax Clarinet Trumpet Flute Shakuhachi

Do Now OPEN PIPE For an open tube with a length of 0.3 m,
a) What is the fundamental resonant frequency? b) What is the frequency of the 2nd harmonic? The speed of sound waves in the tube is 343 m/s OPEN PIPE 1st Harmonic f1 = 572 Hz 2nd Harmonic 1st 2nd 3rd f2 = 1143 Hz (= 2f1)

Do Now Closed PIPE For closed tube with a length of 2 m,
a) What is the fundamental resonant frequency? b) What is the frequency of the 3rd harmonic? The speed of sound waves in the tube is 343 m/s Closed PIPE 1st Harmonic 1st f1 = Hz 3rd 2nd Harmonic f3 = Hz (= 3f1)

Determine the length of a closed-end air column that produces a fundamental frequency (1st harmonic) of 480 Hz. The speed of waves in air is known to be 340 m/s. Draw a diagram to help you solve. L 1st Harmonic v = 340 m/s f = 480 Hz

The lead instrumentalist of a band uses a test tube (closed-end air column) with a 17.2 cm air column. The speed of sound in the test tube is 340 m/sec. Find the frequency of the first harmonic played by the instrument. L=0.172m 1st Harmonic f1=494 Hz

Doppler Effect

Stationary Sound source emitting sound with frequency fs
I hear fs I hear fs

Doppler Effect Sound source moving with vs emitting sound with frequency fs I detect higher pitch fO>fs I detect lower pitch fO< fs

Breaking the Sound Barrier
Sound source moving at the speed of sound (Mach 1) emitting sound with frequency fs I detect lower pitch fo< fs OW, I hear sonic BOOM

I detect lower pitch fo< fs
Supersonic Sound source moving faster than the speed of sound (Mach 1.4) emitting sound with frequency fs I detect lower pitch fo< fs OW, I hear sonic BOOM

Doppler Effect - Observer moving away - Source approaching
+ Observer moving towards - Observer moving away + Source receding - Source approaching

Doppler Effect The Doppler effect occurs in all wave motion, both mechanical and electromagnetic. Astronomers observe light from distant galaxies and use the Doppler effect to measure their speeds and infer their distances. Radar detectors use the Doppler effect to measure the speed of baseballs and automobiles. Physicians can detect the speed of a moving heart wall in a fetus by means of Doppler effect in an ultrasound.

Doppler Effect A trumpet player sounds C above middle C (524 Hz) while traveling in a convertible at 24.6 m/s. If the car is coming toward you, what frequency would you hear? Assume that the temperature is 20°C. vs = 24.6 m/s Fs = 524 Hz

Doppler Effect A trumpet player sounds C above middle C (524 Hz) while traveling in a convertible at 24.6 m/s. Once the car passes and is going away from you, what frequency would you hear? The speed of sound is 343 m/s. vs = 24.6 m/s Fs = 524 Hz What is freq when car passes? (vs is neg)

One foggy morning, Benny is driving his speed boat toward a lighthouse as the fog horn blows with a frequency of Hz. As he approaches, he hears a frequency of 188 Hz. What speed is Kenny traveling to hear this change in frequency? The speed of sound in air is 343 m/s. Givens: fs = 180Hz fO = 188 Hz v = 343 m/s vs = 0

Echolocation tutorial

Vibrations and Waves Vibrations & Waves.

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