An Introduction to Waves and Wave Properties Ch. 14 Waves and Sound An Introduction to Waves and Wave Properties

Animation courtesy of Dr. Dan Russell, Kettering University 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. Water Waves Wave “Pulse” People Wave Animation courtesy of Dr. Dan Russell, Kettering University

Wavelength ( λ ): distance over which wave repeats Period (T): time for one wavelength to pass a given point Frequency f: How often the wave repeats itself. Note: T = 1/ f or f = 1/T

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

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 = ƒ v : speed (m /s)  : wavelength (m) ƒ : frequency (s–1, Hz)

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

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?

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?

Problem: The frequency of an oboe’s A is 440 Hz 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.

Problem: The frequency of an oboe’s A is 440 Hz 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.

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

Wave types: transverse

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: longitudinal

Longitudinal vs Transverse

Other Wave Types Earthquakes: combination Ocean waves: surface Light: electromagnetic

Water waves are a combination of transverse and longitudinal waves.

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 Animation courtesy of Dr. Dan Russell, Kettering University

Animation courtesy of Dr. Dan Russell, Kettering University 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. Animation courtesy of Dr. Dan Russell, Kettering University

Wave transfer from a low density to a high density material.

Transmitted Wave Wave speed decreases. Amplitude decreases. Wavelength decreases. Polarity is the same.

Reflected Wave Wave speed & length stays the same. Amplitude decreases Polarity is reversed.

Wave transfer from a high density to a low density material.

Transmitted Wave Amplitude increases. Wave speed increases Wave length increases Polarity remains the same.

Reflected Wave Wave length and speed stays the same. Polarity remains the same. Amplitude decreases.

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 vibrations of some other material, which starts the air moving. Animation courtesy of Dr. Dan Russell, Kettering University

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. © Tom Henderson, 1996-2004

Sound Waves Here, the wave is a series of compressions and stretches. Sound waves are longitudinal waves, similar to the waves on a Slinky: Here, the wave is a series of compressions and stretches.

Sound Waves In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

Sound Waves The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.

Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic. Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.

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

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.

The Oscilloscope With the Oscilloscope we can view waveforms in the “time domain”. Pure tones will resemble sine or cosine functions, and complex tones will show other repeating patterns that are formed from multiple sine and cosine functions added together.

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.

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 car blowing its horn races toward you, the sound of its horn appears higher in pitch, since the wavelength has been effectively shortened by the motion of the car relative to you. The opposite happens when the car races away.

The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other. When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.

The new frequency is: If the observer were moving away from the source, only the sign of the observer’s speed would change:

To summarize:

The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:

We find:

Here is a comparison of the Doppler shifts for a moving source and a moving observer. The two are similar for low speeds but then diverge. If the source moves faster then the speed of sound, a sonic boom is created.

Combining results gives us the case where both observer and source are moving:

Sample Problem: A bus approaching a bus stop at 24 m/s blows its horn Sample Problem: A bus approaching a bus stop at 24 m/s blows its horn. What the perceived frequency that you hear, if the horn’s true frequency is 150 Hz?

Doppler Effect Stationary source Moving source http://www.kettering.edu/~drussell/Demos/doppler/mach1.mpg Animations courtesy of Dr. Dan Russell, Kettering University Supersonic source http://www.lon-capa.org/~mmp/applist/doppler/d.htm

The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements. At left, a Doppler radar shows the hook echo characteristic of tornado formation. At right, a medical technician is using a Doppler blood flow meter.

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

Sample Problem: Draw the waveform from its two components.

Sample Problem: Draw the waveform from its two components.

Two-dimensional waves exhibit interference as well Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.b

Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

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

A standing wave is fixed in location, but oscillates with time A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible. Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.

In a piano, the strings vary in both length and density In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape. Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension. Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.

Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes. As indicated in the drawing, one end is a node (N), and the other is an antinode (A).

Fixed-end standing waves (violin string) 1st harmonic 2nd harmonic Animation available at: 3rd harmonic http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

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

If the tube is open at both ends, both ends are antinodes, and the sequence of harmonics is the same as that on a string.

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

In this case, the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

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

Sample Problem 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. C) What is the wavelength and frequency of the 2nd harmonic? Draw the standing wave

Sample Problem How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s. A) Draw the situation. B) Calculate the pipe length. C) What is the wavelength and frequency of the 2nd harmonic?

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. Demonstration.

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”. Let’s hear (and see) a demo of this phenomenon.

Beats are an interference pattern in time, rather than in space Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency.

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

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

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”. http://www.falstad.com/ripple/

Double-slit or multi-slit diffraction 

Double slit diffraction nl = d sinq n: bright band number (n = 0 for central) l: wavelength (m) d: space between slits (m) q: 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.

Single slit diffraction nl = s sinq n: dark band number l: wavelength (m) s: slit width (m) q: angle defined by central band, slit, and dark band

Sample Problem Light 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?

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 x = 0, and another is formed at x = 0.03 m? What is the spacing between the slits?

Sample Problem Light 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?