# An Introduction to Waves and Wave Properties

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An Introduction to Waves and Wave Properties
SPH3UW Waves and Sound An Introduction to Waves and Wave Properties

Mechanical Wave A mechanical wave is a disturbance which propagates through a medium with little or no net displacement of the particles of the medium. A Pulse is a single disturbance which carries energy through a medium or space Water Waves Wave “Pulse” People Wave

Parts of a Wave crest : wavelength 3 equilibrium A: amplitude 2 4 6
x(m) -3 trough y(m)

In the drawing, one cycle is shaded in color.
The amplitude , A , is the maximum excursion of a particle of the medium from the particles undisturbed position. The wavelength is the horizontal length of one cycle of the wave. The period is the time required for one complete cycle. The frequency is related to the period and has units of Hz, or cycles/second.

Velocity of a Wave The velocity of a wave is the distance traveled by a given point on the wave (such as a crest) in a given interval of time.

Example 1 The Wavelengths of Radio Waves
AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3.00x108m/s. A station broadcasts AM radio waves whose frequency is 1230x103Hz and an FM radio wave whose frequency is 91.9x106Hz. Find the distance between adjacent crests in each wave.

AM FM

Problem: Sound travels at approximately 340 m/s, and light travels at 3.0 x 108 m/s. How far away is a lightning strike if the sound of the thunder arrives at a location 2.0 seconds after the lightning is seen? The speed of light can almost seem instantaneous at these distances, so we need only concern ourselves with the sound component of lightening.

Types of Waves Refraction and Reflection

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

Wave types: transverse
Energy transport A transverse wave is a moving wave that consists of oscillations occurring perpendicular (or right angled) to the direction of energy transfer. If a transverse wave is moving in the positive x-direction, its oscillations are in up and down directions

Wave types: longitudinal
Longitudinal waves, also known as "l-waves", are waves whose direction of vibration is the same as their direction of travel, meaning that the movement of the medium is in the same direction

Other Wave Types Earthquakes: combination (p and s waves)
Ocean waves: surface Light: electromagnetic

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 Types Fixed-end reflection: The wave reflects with inverted phase. Open-end reflection: The wave reflects with the same phase

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.

Sound is a longitudinal wave
Sound travels through the air at approximately 340 m/s. It travels through other media as well, often much faster than that! Sound waves are started by vibration of some other material, which starts the air moving.

Hearing Sounds We hear a sound as “high” or “low” depending on its frequency or wavelength. Sounds with short wavelengths and high frequencies sound high-pitched to our ears, and sounds with long wavelengths and low frequencies sound low-pitched. The range of human hearing is from about 20 Hz to about 20,000 Hz. The amplitude of a sound’s vibration is interpreted as its loudness. We measure the loudness (also called sound intensity) on the decibel scale, which is logarithmic.

Doppler Effect The Doppler Effect is the raising or lowering of the perceived pitch of a sound based on the relative motion of observer and source of the sound. When a ambulance siren is sounding when it races toward you, the sound of its siren appears higher in pitch, since the wavelength has been effectively shortened by the motion of the ambulance relative to you. The opposite happens when the ambulance moves away.

The Doppler Effect When a moving object emits a sound, the wave crests appear bunched up in front of the object and appear to be more spread out behind the object. This change in wave crest spacing is heard as a change in frequency. Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 12, Questions 9, 10, and 20. A nice Doppler effect applet can be found here: It will be found toward the bottom of this page. The results will be similar when the observer is in motion and the sound source is stationary and also when both the sound source and observer are in motion.

Doppler Effect Stationary source Moving source Supersonic source

The Doppler Effect formula
fo is the observed frequency. fs is the frequency emitted by the source. vo is the observer’s velocity. vs is the source’s velocity. v is the speed of sound. Note: take vs and vo to be positive when they move in the direction of wave propagation and negative when they are opposite to the direction of wave propagation.

Example: A source of sound waves of frequency 1. 0 kHz is stationary
Example: A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.5 times the speed of sound. (a) What is the observed frequency if the observer moves toward the source? (b) Repeat, but with the observer moving in the other direction.

Example Solution: A source of sound waves of frequency 1
Example Solution: A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.5 times the speed of sound. (a) What is the observed frequency if the observer moves toward the source? fo is unknown; fs= 1.0 kHz; vo = 0.5v; vs = 0; and v is the speed of sound.

Example Solution: A source of sound waves of frequency 1
Example Solution: A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.5 times the speed of sound. (b) Repeat, but with the observer moving in the other direction. fo is unknown; fs = 1.0 kHz; vo = +0.5v; vs = 0; and v is the speed of sound.

Pure Sounds Sounds are longitudinal waves, but if we graph them right, we can make them look like transverse waves. When we graph the air motion involved in a pure sound tone versus position, we get what looks like a sine or cosine function. A tuning fork produces a relatively pure tone. So does a human whistle. Later in the period, we will sample various pure sounds and see what they “look” like.

Graphing a Sound Wave

Sensitivity of the Human Ear
Sound Sensitivity of the Human Ear 05/02/2006 We can hear sounds with frequencies ranging from 20 Hz to 20,000 Hz an impressive range of three decades (logarithmically) about 10 octaves (factors of two) compare this to vision, with less than one octave! An Octave is a series of eight notes occupying the interval between (and including) two notes, one having twice or half the frequency of vibration of the other.. Lecture 10

Complex Sounds Because of the phenomena 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. The human voice and most musical instruments produce complex sounds. Later in the period, we will sample complex sounds.

Speakers: Inverse Eardrums
Sound 05/02/2006 Speakers: Inverse Eardrums Speakers vibrate and push on the air pushing out creates compression pulling back creates rarefaction Speaker must execute complex motion according to desired waveform Speaker is driven via “solenoid” idea: electrical signal (AC) is sent into coil that surrounds a permanent magnet attached to speaker cone depending on direction of current, the induced magnetic field either lines up with magnet or is opposite results in pushing or pulling (attracting/repelling) magnet in coil, and thus pushing/pulling on center of cone Lecture 10

Speaker Geometry Sound 05/02/2006 Lecture 10

Superposition of Waves

Principle of Superposition
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.

Types of interference. If the waves are “in phase”, that is crests and troughs are aligned, the amplitude is increased. This is called constructive interference. 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”

Constructive Interference

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

Destructive Interference

Resonance Certain devices create sound waves at a natural frequency. If another object, having the same natural frequency is impacted by these sound waves, it may begin to vibrate at this frequency, producing more sound waves. This phenomenon is know as resonance. i.e. opera singer shattering glass with voice

Standing Waves

Standing Wave A standing wave is a wave which is reflected back and forth between fixed ends (of 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. Let’s see a simulation.

Open and Closed Tubes Many musical instruments depend on the musician moving air through the instrument. Musical instruments like this can be divided into two categories, open ended or closed ended. A Tube or Pipe can be a musical instrument, it is often bent into different shapes or has holes cut into it. An open ended instrument has both ends open to the air. An example would be an instrument like a trumpet. You blow in through one end and the sound comes out the other end of the pipe. A closed ended instrument has one end closed off, and the other end open. An example would be an instrument like some organ pipes, or a flute. Although you blow in through the mouth piece of a flute, the opening you’re blowing into isn’t at the end of the pipe, it’s along the side of the flute. The end of the pipe is closed off near the mouth piece.

Closed Ended Pipes L First harmonic  = 4L
Remember that it is actually air that is doing the vibrating as a wave inside the Pipe. The air at the closed end of the pipe must be a node (not moving), since the air is not free to move past the sealed end it must be reflected back. There must also be an antinode (maximum movement) where the opening is, since that is where there is maximum movement of the air. The simplest, smallest wave that I can possibly fit in a closed end pipe is shown Below. This is ¼ of a wavelength. Since this is the smallest stable piece of a wave we can fit in this pipe, this is the Fundamental, or 1st Harmonic this is the lowest possible frequency that any instrument can play. L First harmonic  = 4L

Closed Ended Pipes Since the length of the tube is the same as the length of the ¼ wavelength We know that the length of this tube is ¼ of a wavelength so this leads to our first formula: “L” is the length of the tube in metres. On it’s own this formula really doesn’t help us much. Instead, we have to solve this formula for λ and then combine it with the formula v=fλ to get a more useful formula: When the wave reaches the closed end it’s going to be reflected as an inverted wave . This does not change the length of the wave in our formula, since we are only seeing the reflection of the wave that already exists in the pipe. L

Closed Ended Pipes So what does the Next Harmonic look like? I know this name might seem a little confusing (I’m the first to agree with you!) but because of the actual notes produced and the way the waves fit in, musicians refer to the next step up in a closed ended pipe instrument as the 3rdharmonic… there is no such thing as a 2nd harmonic for closed ended pipes. In fact all of the Harmonics in closed ended Pipes are going to be odd numbers.

Closed Ended Pipes Harm. # # of Waves in Column # of Nodes Antinodes
Length- Wavelength Relationship 1 1/4  λ= (4/1)*L 3 3/4 2  λ= (4/3)*L 5 5/4 λ= (4/5)*L 7 7/4 4  λ= (4/7)*L 9 9/4  λ= (4/9)*L

The speed of sound waves in air is 340 m/s. Determine the fundamental frequency (1st harmonic) of a closed-end air column that has a length of 76.5 cm. Since this is the First Harmonic: Now for the Frequency: You could also have just used:

Open Ended Pipes I know you’re probably thinking that there couldn’t possibly be any more stuff to learn about Resonance, but we still have to do Open Ended Pipes. Thankfully, they’re not that hard, and since you already have the basics for closed pipes it should go pretty easy for you. The fundamental (first harmonic) for an open ended pipe needs to be an antinode at both ends, since the air can move at both ends. So this is the smallest wave we can fit into a open ended pipe: ½ of a wavelength L Fundamental First harmonic  = 2L

Open Ended Pipes So the basis for drawing the standing wave patterns for air columns is that vibrational antinodes will be present at any open end and vibrational nodes will be present at any closed end. If this principle is applied to open-end air columns, then the pattern for the fundamental frequency (the lowest frequency and longest wavelength pattern) will have antinodes at the two open ends and a single node in between. For this reason, the standing wave pattern for the fundamental frequencies) for an open-end air column looks like the diagrams below.

Open Ended Pipes The relationships between the standing wave pattern for a given harmonic and the length-wavelength relationships for open ended tubes are summarized in the table below. Harmonic # # of Waves in Column # of Nodes Antinodes Length- Wavelength Relationship 1 1/2 2 Wavelength = (2/1)*L 1 or 2/2 3 Wavelength = (2/2)*L 3/2 4 Wavelength = (2/3)*L 2 or 4/2 5 Wavelength = (2/4)*L 5/2 6 Wavelength = (2/5)*L

A 4 m long organ pipe (open at both ends) produces a musical note at its fundamental frequency. Determine the wavelength of the note produced. What is the frequency of the pipe given the speed of sound is 346 m/s? Since this is the First Harmonic: Now for the Frequency: Did you notice, that if we made the pipe longer, the wavelength would be bigger, and since wavelength and frequency are inversely related, that means the frequency would be smaller.

Example: An organ pipe that is open at both ends has a fundamental frequency of 382 Hz at 0.0 °C. What is the fundamental frequency for this pipe at 20.0 °C? How long is this organ pipe? At Tc = 0.0 °C, the speed of sound is 331 m/s. At Tc = 20.0 °C, the speed of sound is 343 m/s. The fundamental frequency is: Speed of sound:

Symbol definitions: f1,20 is the fundamental frequency at 20C and f1,0 is the fundamental frequency at 0C. The speeds of sound are labeled as v20 and v0 for the speed of sound at a temperature of 20C and at 0C respectively.

How long is this organ pipe?

Fixed-end standing waves (violin string)
Fundamental First harmonic  = 2L If the string vibrates in more than one segment, the resulting modes of vibration are called overtones First Overtone Second harmonic  = L Second Overtone Third harmonic  = 2L/3

12.7 Beats When two waves with nearly the same frequency are superimposed, the result is a pulsation called beats. Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 12, Question 7.

Beats “Beats is the word physicists use to describe the characteristic loud-soft pattern that characterizes two nearly (but not exactly) matched frequencies. Musicians call this “being out of tune”.

Two waves of different frequency
Superposition of the above waves The beat frequency is

Beats If the beat frequency exceeds about 15 Hz, the ear will perceive two different tones instead of beats.

What word best describes this to physicists?

What word best describes this to musicians?
Amplitude Answer: bad intonation (being out of tune)

A tuning fork with a frequency of 256 Hz is sounded together with a note played on a piano. 12 beats are heard in 4 seconds. What is the frequency of the piano note? Beat Frequency: Since Beat Frequency also equals: Then: Without further information, there is no way to know which answer is correct...

Echolocation Sound waves can be sent out from a transmitter of some sort; they will reflect off any objects they encounter and can be received back at their source. The time interval between emission and reception can be used to build up a picture of the scene.

Example A boat is using sonar to detect the bottom of a freshwater lake. If the echo from a sonar signal is heard s after it is emitted, how deep is the lake? Assume the lake’s temperature is uniform and at 25 C.

Example 1: A boat is using sonar to detect the bottom of a freshwater lake. If the echo from a sonar signal is heard s after it is emitted, how deep is the lake? Assume the lake’s temperature is uniform and at 25 C. The signal travels two times the depth of the lake so the one-way travel time is s. From table 12.1, the speed of sound in freshwater is 1493 m/s.

The Speed of Sound Waves
The speed of sound in different materials can be determined as follows: B is the bulk modulus of the fluid and  its density. In fluids Y is the Young’s modulus of the solid and  its density. In thin solid rods Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 12, Questions 8, 13, 14, and 15. The bulk modulus (B) of a substance measures the substance's resistance to uniform compression. While Young’s modulus (K) measures the Stiffness or elasticity of a solid

The Speed of Sound Waves
In ideal gases Here v0 is the speed at a temperature T0 (in kelvin) and v is the speed at some other temperature T (also in kelvin). For air, a useful approximation to the above expression is where Tc is the air temperature in C.

Materials that have a high restoring force (stiffer) will have a higher sound speed.
Materials that are denser (more inertia) will have a lower sound speed.

Example 1: A copper alloy has a Young’s Modulus of 1.11011 Pa and a density of 8.92103 kg/m3. What is the speed of sound in a thin rod made of this alloy? The speed of sound in this alloy is slightly less than the value quoted for copper (3560 m/s) in table 12.1.

Example 2 Bats emit ultrasonic sound waves with a frequency as high as 1.0105 Hz. What is the wavelength of such a wave in air of temperature 15.0 C? The speed of sound in air of this temperature is 340 m/s.

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