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**An Introduction to Waves and**

Waves and Sound An Introduction to Waves and Wave Properties

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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.

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**Wave is moving in this direction Matter**

Transverse Wave Anatomy: 2 wavelengths shown Crest Amplitude Rest Line λ Trough Wave is moving in this direction Matter

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Speed of a wave 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. v = d/t d: distance (m) t: time (s) v : speed (m /s) v = λƒ λ: wavelength (m) ƒ : frequency (s–1, Hz)

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Period of a wave T = 1/ƒ T : period (s) ƒ : frequency (s-1, Hz)

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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? d = vst d = (340m/s)(2.0s) = 680m Problem: The frequency of an oboe’s A is 440 Hz.. What is the period of this note? What is the wavelength? Assume a speed of sound in air of 340 m/s.. T = 1/f = 1/440s-1 = s λ = v/f = 340m/s / 440s = 0.77m

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**Refraction and Reflection**

Types of Waves Refraction and Reflection

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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

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Examples Transverse

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**Longitudinal (also called compressional)**

And… Longitudinal (also called compressional)

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**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.

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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 off human hearing is from about 20 Hz to about 20,,000 Hz. The amplitude off a sound’s vibration is interpreted as its loudness.. We measure the loudness (also called sound intensity) on the decibel scale, which is logarithmic.

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Otther Wave Types Eartthquakes:: combiinattiion Ocean waves:: surfface Liightt:: ellecttromagnettiic

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Other Wave Types Earthquakes: combination Ocean waves: surface Ocean waves have transverse and longitudinal properties Light: electromagnetic (Transverse waves)

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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..

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**Fixed-end reflection: the wave reflects with inverted phase**

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

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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.

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Doppler Effect Causes waves to change frequency because either the source or the observer is moving. Greater speed, greater Doppler Effect Source and observer moving closer = higher f. Source and observer moving apart = lower f. Light: Stars moving closer appear more blue, stars moving away appear more red. Works for all waves, but most commonly discussed with sound

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**Stationary Sound Source**

Moving at Speed of Sound Moving Sound Source Moving Faster than Speed of Sound

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Equation Toward Away

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**What if the train is moving away from the platform?**

A train approaches a platform at a speed of 10 m/s while the whistle blows. The whistle sounds with a frequency of 261 Hz. What is the frequency the observer hears? What if the train is moving away from the platform? vsound = 340 m/s vsource = 10 m/s f = 261 Hz = Hz = Hz

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**Superposition of Waves**

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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.

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Graphing a Sound Wave

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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.

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The Oscilloscope Shows complex wave patterns

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The Fourier Transform We will also view waveforms in the “frequency domain”. A mathematical technique called the Fourier Transform will separate a complex waveform into its component frequencies.

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**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.

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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..

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**Constructive example:**

Destructive example out of phase speakers can produce dead spots

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**Constructive Interference**

crests aligned with crest waves are “in phase”

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**Constructive Interference**

Waves are “in phase” (add together) Crests align with crests

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**Destructive Interference**

crests aligned with troughs waves are “out of phase”

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**Destructive Interference**

Waves are “out of phase” Crests align with troughs

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**Examples Constructive and destructive applet Sample waveform exercise**

Superposition of Waves Example of interference example 2 Sample waveform exercise

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Standing Wave 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.

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**Fixed-end standing waves (violin string)**

1st Harmonic Fundamental λ = 2L 2nd Harmonic 1st Overtone λ = L 3rd Harmonic 2nd Overtone λ = 2L/3 Violin string

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= 1 ‘bouncy’ f = v/2L = 2 ‘bouncies’ f = 2v/2L = v/L = 3 ‘bouncies’ f = 3v/2L

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**Open-end standing waves (organ pipes)**

1st Harmonic Fundamental λ = 2L 2nd Harmonic 1st Overtone λ = L 3rd Harmonic 2nd Overtone λ = 2L/3

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**Mixed standing waves (some organ pipes)**

1st Harmonic Fundamental λ = 4L 3nd Harmonic λ = 4L/3 5th Harmonic λ = 4L/5

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**Sample Problem λ = 2L L = v/2f = (340 m/s) / [(2)(256 s-1)]**

How long do you need to make an organ pipe that produces a fundamental frequency of middle C (256 Hz)? The speed of the sound in air is 340 m/s. A) Draw the standing wave for the first harmonic B) Calculate the pipe length. λ = 2L L = v/2f = (340 m/s) / [(2)(256 s-1)] v = λf L = 0.66 m C) What is the wavelength and frequency of the 2nd harmonic? Draw the standing wave λ = L = 0.66 m f = 2 x fundamental = 512 Hz

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Resonance Resonance occurs when a vibration from one oscillator occurs at a natural frequency for another oscillator. The first oscillator will cause the second to vibrate. Beats “Beats is the word physicists use to describe the characteristic loud-soft pattern that characterizes two nearly (butt not exactly) matched frequencies. Musicians call this “being out of tune”.

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**What word best describes this to physicists?**

Answer: beats

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**What word best describes this to musicians?**

Answer: bad intonation (being out of tune)

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Diffraction

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Diffraction Diffraction is defined as the bending of a wave around a barrier.. Diffraction of waves combined with interference of the diffracted waves causes “diffraction patterns”.. Let's look at the diffraction phenomenon using a “ripple tank

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n

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Wave behavior of Light Diffraction: the spreading out of a wave as it passes an edge or opening Interference: when waves overlap and produce new larger waves

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**Double slit diffraction**

nλ = d sinθ n: bright band number (n = 0 for central) λ: wavelength (m) d: space between slits (m) θ: angle defined by central band, slit, and band n This also works for diffraction gratings consisting of many, many slits that allow the light to pass through. Each slit acts as a separate light source..

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**Single slit diffraction**

nλ = s sinθ n: dark band number λ: wavelength (m) s: slit width (m) θ: angle defined by central band, slit, and dark band

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Sample Problem Lights of wavelength 360 nm is passed through a diffraction grating that has 10,000 slits per cm. If the screen is 2.0 m from the grating, how far from the central bright band is the first order bright band? nλ = d sinθ d = (1 cm/10000 slits)(1 m/100 cm) = 10-6 m λ = (360 nm)(10-9 m/nm) = 3.6(10-7) m n = 1 (1)(3.6)(10-7 m) = (10-6 m) sinθ => θ = sin-1 (0.360) = 21o Δy = x tan θ = (2.0 m) tan21o = 0.77 m θ x Δy

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Sample Problem Light of wavelength 560 nm is passed through two slits. It is found that, on a screen 1.0 m from the slits, a bright spot is formed at y = 0, and another is formed at y = 0.03 m. What is the spacing between the slits? l = √x2+Δy2 = 1.0 m nλ = d sinθ d = nλ/sinθ d = (1)(560)(10-9)/(0.03/1.0) d = 1.9(10-5)m l Δy=0.03m θ X=1.0m

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**Sample Problem 2nd order (n=2)**

Light of wavelength 360 nm is passed through a single slit of width 2.1 x 10-6 m. How far from the central bright band do the first and second order dark bands appear if the screen is 3.0 meters away from the slit? 1st order (n=1) nλ = s sinθ => sinθ = nλ/s θ = sin-1(1)(360)(10-9)/2.1(10-6) θ = 9.9o Δy = 3.0 tan(9.9o) = 0.52 m 2nd order (n=2) θ = sin-1(2)(360)(10-9)/2.1(10-6) θ = 20.1o Δy = 3.0 tan(20.1o) = 1.09 m θ 3.0 m Δy

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