Sound: An Interactive eBook consists of 33 interactive simulations which require the reader to click buttons, move sliders, etc. in order to answer questions about the behavior of waves and sound in particular.  The goal of this book is to create an engaging text that integrates the strengths of printed, static textbooks and the interactive dynamics possible with simulations to engage the student in actively learning the physics of sound.

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Sinusoidal waves have the property, called superposition, that their amplitudes add linearly if they arrive at the same point at the same time. This gives rise to several interesting phenomena in nature which we will investigate in this and the next few simulations.

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In this chapter various physics concepts and definitions needed for the study of sound, acoustics and musical instruments are presented.

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This simulation shows particles interacting with a slight attraction which will cause them to stay connected with each other to form a a solid at low temperature. But if they have enough thermal energy they will begin to move around each other to act like liquid. Additional thermal energy causes them to act like a gas.

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All sound starts with something that vibrates. The reed in a clarinet vibrates, the vocal cords in a singer's throat vibrate, the air flowing over the mouthpiece of a flute oscillates, and the speaker cone on your stereo or in an ear-bud vibrates. In this chapter we investigate a particular kind of vibration called simple harmonic motion.

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Resonance occurs in an oscillating system when the driving frequency happens to equal the natural frequency. For this special case the amplitude of the motion becomes a maximum. An example is trying to push someone on a swing so that the swing gets higher and higher. If the frequency of the push equals the natural frequency of the swing, the motion gets bigger and bigger. Resonance is a key concept in the production of sound in instruments and in acoustics and we will come across it many more times in this book.

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This simulation shows five different masses, each attached to a spring of the same stiffness. The springs are mounted on a mechanical device that shakes the springs and attached masses.

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Many systems, including musical instruments, have a wide range of frequencies at which the system will resonate.  How does this range depend on the damping coefficient?

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Transverse waves are the type of wave you usually think of when you imagine a wave. The motion of the material constituting the wave is up and down so that as the wave moves forward the material moves perpendicular (or transverse) to the direction the wave moves.

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Longitudinal waves are waves where the motion of the material in the wave is back and forth in the same direction that the wave moves. Longitudinal waves are sometimes called compressional waves. Sound waves (in air and in solids) are examples of longitudinal waves.

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If you stretch a slinky out between two points and gently twist it at one end, the twist will travel down the slinky as a wave pulse. This is an example of a torsional wave.

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This simulation shows an oscillating electron in a sending antenna on the left. Because electrons have an electric field, an accelerating electron will create a wave in the electric field around it. Magnetic fields are created by moving charges so a magnetic wave is also formed by an accelerating charge. Only the y-component of the change in the electric field is shown (so an oscillation frequency of zero will show nothing, because there is only a constant electric field).

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There are two different speeds involved with describing a wave. In previous chapters we saw that the individual points on a wave oscillate (up and down for transverse waves, back and forth for longitudinal waves) with simple harmonic motion, just like masses on springs. But the up and down speed of a point on a transverse wave doesn't tell us how fast the wave moves from one place to the next. The wave speed, v, is how fast the wave travels and is determined by the properties of the medium in which the wave is moving.

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In many cases waves of all types will travel in a straight line, reflecting off of objects and surfaces at the same angle that they strike the surface. This is called the law of reflection and is true for sound waves as well as light as long as the surface is smooth relative to the wavelength.

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A wave that changes speed as it crosses the boundary of between two materials will also change direction if it crosses the boundary at an angle other than perpendicular. This is because the part of the wavefront that gets to the boundary first, slows down first. The bending of a wave due to changes in speed as it crosses a boundary is called refraction.

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A wave that passes all the way through a piece of material with parallel sides leaves the material at the same angle that it entered. The wave un-bends when it exits the material by the same amount that it bent when entering but this is only true if the sides of the material are parallel.  Convex and concave lenses have sides that are not parallel (except near the center). In this case parallel rays of light end up exiting in different directions.

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The speed of a wave can depend on the frequency of the wave, a phenomenon known as dispersion.  Although this effect is often small. it is easy to observe with a prism.

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When two waves of the same type come together it is usually the case that their amplitudes add. So two overlapping water waves have an amplitude that is twice as high as the amplitude of the individual waves. This is called constructive interference and it can occur for sinusoidal waves as well as pulses.

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Sinusoidal waves have the property, called superposition, that their amplitudes add linearly if they arrive at the same point at the same time. This gives rise to several interesting phenomena in nature which we will investigate in this and the next few simulations.

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If two sources of waves are in phase to start with, when they reach a distant location they may be in-phase (leading to constructive interference) or out-of-phase (leading to destructive interference) depending on slight differences in the distance traveled. This path difference gives rise to many interesting phenomena such as interference patterns (in the case of light) and dead spots in auditoriums (in the case of sound).

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This simulation shows a top view of waves interfering on the surface of a tank of water (imagine tapping the surface of a pond with the end of a stick at regular intervals). The white circles coming from the spot represents the wave crests with troughs in between. Two sources can be seen at the same time and the separation between them and the wavelength of both can be adjusted.

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Sometimes waves don't travel in a straight line, even if their speed does not change (as in the case of refraction). For example, you can hear the conversation in the next room even though you cannot see the source. This is because sound waves undergo diffraction, bending as they go through the doorway between the two rooms.

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If the wave source or receiver is moving, the waves will appear to have a different frequency. For example if you are moving towards a sound source you catch up with the next peak in the wave sooner than you would expect because you are moving towards it. This effect is called the Doppler Shift and occurs for both light and sound.

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The mechanism of human hearing does not operate as a perfect scientific instrument. In this chapter we relate a few subjective measurements of sound (things people report after hearing a sound) to objective, scientific measurements (measurements made in a laboratory using scientific instruments). The three subjective quantities of pitch, loudness and timbre are related to laboratory measurements of a sound wave's fundamental frequency, amplitude and waveform, respectively.

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This simulation explores the aural texture of four basic periodic waveforms: sine, triangle, square, and sawtooth. The sine waveform has a single frequency and is the building block of other periodic waves by summing harmonics in a Fourier Series as we will see in the next section. The richness of the sound is called the timbre (defined in the previous chapter) and is determined by the amplitude of the harmonics in the Fourier sum.

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The French Mathematician Jean Baptiste Joseph Fourier showed any periodic function can be formed from an infinite sum of sines and cosines. This is very convenient because it means that everything we know about sines and cosines applies to a periodic function of any shape. Although the sum is infinite in theory, in many cases using just a few terms may be close enough to provide a good approximation.

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This is an exercise using a Fast Fourier Transform to analyze sound from your computer or mobile devices microphone.

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We look at how the ear turns vibrations into the perception of sound. Some of the exact details of this process are still not completely understood but the general picture of how we hear is fairly well established. Once again we will see that the human hearing mechanism gives us experiences that do not correspond exactly to laboratory measurements.

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The phenomena of beats occurs when two notes are close together in frequencies and we perceive one note which varies in loudness. A guitar string can be tuned by comparing a note with a known pitch and tuning the string until the beats disappear. What happens if the two frequencies get further and further apart?

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In this simulation a string is driven at one end by an oscillating driver. The result is that a wave will eventually form on the string.  At certain frequencies the wave will become large and we refer to this resonance phenomena as a standing wave.

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In this section a set of initial conditions for a vibrating string is shown. The first is the fundamental frequency of the string, the second is the second harmonic. The third and fourth initial conditions simulate plucking in the center and at a location one fourth of the way along the string.

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The Vibrating Plates simulation allows you to examine vibrational modes on a rectangular surface. The surface is fixed at the edges so the nodal lines occur in different places compared to a rectangle with free edges. The model assumes that the surface is very thin and very flexible; real surfaces which are stiffer will have slightly different nodal lines and anti-nodes. The effect is very similar, however. Free edged surfaces, thin flexible surfaces and thick stiff surfaces all have nodal lines and anti-node areas.

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This section shows a simulation that compares the fundamental, second, third and forth harmonics of standing waves on a string with standing waves in a tube. Notice that for a tube open on both ends the displacement nodes occur where the string has nodes and the displacement anti-nodes in the tube occur where the string has displacement nodes. The pressure nodes in a tube open at both ends occur in the same place as the string nodes.

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Waves on a string form standing waves because the wave reflects from each end of the string where there is a fixed node. How do standing waves form in a tube full of air?  This section shows that waves reflect from the end of a tube and that this reflection can be of two types, depending on whether the boundary is 'soft' or 'rigid'.

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But why do sound waves reflect from the open end of a tube or when the size of tube changes abruptly? The resistance to the movement of a wave crossing a boundary from one medium into another is called impedance and occurs for waves on a string, sound waves in air and electronic signals in a circuit. When a wave tries to travel from a medium with one impedance to a region where the impedance is different, there will be a partial reflection. The reflection means that not all the energy of the wave is transmitted to the new medium.

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This section allows you to see and manipulate the modes for a square drum head. You can change the modes using the sliders to change the mode numbers n and m. For a membrane there are nodal lines which do not vibrate similar to the nodes we saw on the string but now in two dimensions. You can rotate and enlarge the surface by dragging the mouse over the image. Just like the case for a vibrating string, more than one mode can be present on the two dimensional surface at the same time.

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As we know, musical instruments consist of a vibration which is amplified by resonance. The human singing voice is no different. The vocal chords are the vibrating part and the throat, mouth, nasal cavities and bronchial tubes constitute the resonance cavities that amplify these vibrations into sound. Because every person's combination of throat, mouth, nasal cavities and bronchial tubes is slightly different, we all sound slightly different. The fact that we can change the shape of some of these cavities at will enables us to produce a wide range of pitches, depending on the initial structure and training.

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The notes on a musical instruments are organized into scales and we would like to have a scale where we get the greatest number of combinations that sound good together. We also would like to standardize the scale in such a way that if we build other instruments, two instruments playing together can play the same pitch. This turns out to be more difficult than it would seem. The choice of scale is arbitrary; we can choose any combination of notes that we like and in fact some cultures have chosen scales very different from the ones used in western music. However there are advantages and disadvantages for any of these various choices, as explained in this chapter.

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The study of what happens to sound in an enclosed space or as the result of interactions with large objects such as buildings is called acoustics. Humans have been trying to improve the acoustics of auditoriums and other public spaces since the time of the ancient Romans.  Reflection, refraction, path difference, diffraction and interference will govern how sound behaves inside rooms, auditoriums and concert pavilions.

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The electrical resistance of an electrical conductor is a measure of the difficulty to pass an electric current through that conductor. It is measured in Ohms and the relation between resistance (R), current (I) and electrical potential (V) is Ohm's law: V = IR. Ohm's law says that a larger voltage makes more current flow if resistance is fixed. Or if resistance is lower at the same voltage, more current will flow.

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In this section we study the magnetic field of either a permanent magnet or the field produced by a flow of current in a coil. Field is measured in Gauss. The compass, magnet and coil are all draggable. The earth's magnetic field can also be demonstrated.

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In this section we study electric and magnetic fields with different orientations to see their effects on neutral, positive and negative charges. For the electric field case the particles have zero initial velocity. In second case with a magnetic field in the x-direction the initial velocity is zero but there is a check-box so that you can give the particles an initial velocity in the +x direction. In the third case the magnetic field is rotated so that it points into the screen (which is now the x direction) and the particles have an initial velocity in the +z direction. For this case a black arrow shows the direction of the force on the particle.

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If a changing magnetic field is present near a wire that is part of a circuit it will cause current to flow in the circuit. This is known as Faraday's law and is the basis for a lot of modern technology. Electric generators, traffic detectors embedded in the road, metal detectors, the read head on a computer hard drive, credit card readers, cassette tape readers, and transformers (both the ones on the utility pole outside your house and the little boxes that plug into the wall to run electronic gear) all use Faraday's law to operate. We will see several applications for sound reproduction.

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It is not possible to record, transmit and replay sounds perfectly so that they sound exactly as they were heard originally. This chapter explains several electronic devices used in sound recording and reproduction using concepts that were introduced in previous chapters.

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