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Wave solutions of short duration, or transients, are shown to be equivalent to the sum of a large number of sinusoidal solutions over a range of frequencies. The range of frequencies is inversely proportional to the time duration. This result can be expressed mathematically as an uncertainty principle and explains why waves of very short duration do not have an identifiable pitch. A similar phenomenon occurs during a rapid rise or fall of a signal. Spectrographs can be used to represent a changing spectrum as a function of time, where data are collected and analyzed with a moving time window, similar to what appears to happen for human perception. In contrast to periodic signals, for a short transient signal, the phase factors in the Fourier series are most important for perception.
The basics of sound propagation in three dimensions are considered. Simple models for sources are presented, including isotropic and nonisotropic sources, such as the dipole, quadruple, line sources, and oscillating baffles. Sound propagation from those sources is treated in the ray approximation, where effects due to the wave nature of sound are suppressed. For finite sources, it is shown that in the absence of reflections, sound intensity falls off following an inverse square law. Reflections in a room are considered and lead to reverberation. The time sound remains in a room is characterized using a reverberation time, and a simple equation is used to relate that time to room dimensions, materials used, and possible contents of the room. Sound in a room from multiple sources is considered using sound-energy density rather than intensity. One example of the treatment of multiple sources is known as the cocktail party effect.
The results for the vibrating string are generalized to discuss vibrating systems that are not necessarily harmonic. Solutions are characterized by counting the nodes for solutions—that is, locations where the vibrational amplitude is always zero. One-dimensional solutions for uniform bars and rods, relevant for various chimes and bars used for music, are presented. The nonuniform bars used for xylophones and marimbas, and how they are used for tuning, are presented. The procedure is generalized for two-dimensional objects, where node lines are considered, and applied to membranes, such as drum heads, and plates. The complications that arise with degeneracies, or multiple solutions with the same frequency, are presented. A general, symbolic way of representing vibrational modes is shown, as well as how that can, very symbolically, represent any possible solution for a vibrating object as a sum of those with sinusoidal time dependence.
Frequency, as events per second measured in hertz (Hz), is introduced as one type of rate that is important for music. Tones associated with music are typically found to correspond to a few hundred events per second. Other rates are considered as examples. Two types of rates are distinguished: additive and multiplicative. For an additive rate, a quantity is added for each interval. A multiplicative rate involves a multiplicative change, such as a percentage change, for each interval. Multiplicative rates lead to exponentially increasing and decreasing behavior. Exponential behavior is often described using logarithms. It is found that the frequency of tones is an additive rate, but the change in the frequency going up and down the keyboard is multiplicative. In particular, octaves are a factor of 2 in frequency.
The speed of sound in gases is derived in a manner similar to what was done for the speed of waves on a string. The mass of the gas molecules provides the inertial contribution. The gas laws, along with approximation techniques, are used to estimate the return force. The appropriate process is adiabatic, and so the speed of sound depends on the adiabatic constant. The adiabatic constant, in turn, depends largely on the shape of the molecules. Excellent agreement is found between the rather simple theory and measured results. The speed of sound in a gas is found to depend on the (average) mass of the molecules, temperature, and the adiabatic constant (the shape of the molecules); however, there is very little pressure dependence when the pressure is near 1 atmosphere.
Newton’s laws and consideration of units are used to present and discuss the mass on a spring as an example of a harmonic oscillator, a mechanical oscillator with a sinusoidal time dependence. Both the transient solution, where the oscillator is started away from equilibrium and left on its own, and the driven solutions, where a sinusoidal driving force is applied, are presented. The quality factor, Q, is introduced, which characterizes the relative amount of damping forces present, such as those of friction or air drag. The quality factor is related to the number of oscillations that are made when left on its own and to the excitation bandwidth, the range of frequencies over which resonance is observed. Musically relevant examples include the ocarina, tuning fork, some speaker enclosures, and the phenomenon of sympathetic resonance.
How electronics can be used to alter and/or create musical signals is considered. Emphasis is on those techniques where physics is evident, recognizing that much is now done using programming languages. Basic effects done using simple physics include distortion, reverberation, and tremolo. Various signal-processing and playback techniques are also used, along with psychoacoustics, to provide surround sound and other effects. Electronic music can be created using the Hammond organ, a theremin, and electronic synthesizers. A basic synthesizer will combine a carrier tone and a time-dependent amplitude for that tone in order to create different sounds. Although synthesizers are often controlled by keyboards, other user interfaces have also been developed that, for example, resemble a drum kit or a woodwind instrument.
The harmonic vibrational frequencies of an ideal string under tension, fixed at the ends, are developed, starting with the results from the harmonic oscillator. The corresponding resonances result in sinusoidal standing waves. The end conditions restrict the possible solutions to be those with an integer number of half wavelengths between the ends of the string. A general vibration of the string, such as from a pluck, can be described as the sum of sinusoidal solutions. More general end conditions include a free end, in which case there is an odd number of quarter wavelengths between the ends.
This chapter presents a discussion and derivation of the ideal gas law, starting with atoms and molecules. Collisions are characterized by the impulse, the change in momentum due to the collision, determined by the product of the average force during and the time duration of a collision. A free atom or molecule in a box is used to develop the concept of pressure on a wall and, ultimately, the ideal gas law for many noninteracting molecules. The distinction between gauge pressure and absolute pressure is necessary to understand before applying the gas laws. The root mean square (rms) speed for a typical molecule is estimated to be a bit faster than the speed of sound. That is, the molecules can be expected to be supersonic. Included is a discussion of isotopes and carbon dating and their connection to the musical scale.
The basics of magnets, magnetism, induced magnetism, magnetic fields, and electromagnets are introduced. The force between a magnet and an electromagnet is the basis for speakers, devices that turn electrical signals into sounds, as well as other electromechanical devices. The process works in reverse, in that a sound incident on a speaker can produce an electrical signal. The latter principle is used for some microphones. In general, a time-dependent magnetic field, for example, due to a moving magnet, will tend to induce electrical currents that oppose the change. This is known as Faraday’s law of induction. Several of the principles of magnetism are used together to create an electric guitar pickup. A scheme to use a pair of pickups to cancel out environmental signals, known as a humbucker, is shown. Electrical transformers work based on a time-changing magnetic field from one electromagnet experienced by another, and they are useful for generating electrical signals that better match the destination for the signals—typically an amplifier.
Combinations of tones that are consonant (“nice”) are those that exhibit no slower beats. Beats occur when two periodic signals are close to the same frequency. For sinusoidal signals, the beat frequency is simply the difference between the two signals. For complex signals, one must also consider differences between multiples of the signal frequency. Thus, consonant combinations are those where the ratio of the frequencies is equal to a rational number. Of particular importance are rational numbers involving the ratio of small integers. The musical fifth corresponds to a frequency ratio of 3 to 2 and is an important part of music. A set of note frequencies used for a musical scale can be justified based on consonant combinations, and variations of the details of those choices, known as temperaments, are useful in music for practical reasons. In particular, the equal-tempered scale used for keyboards, based on multiples of the 12th root of 2, is very common.
Various sound-recording technologies are presented that use different mediums and different encoding methods for the recorded signal. Analog and digital are two broad categories used for encoding. Sound signals can be recorded using a physical change to a material, as is used for vinyl records and some CDs and DVDs. Signals can also be recorded using magnetic materials and their interaction with electromagnets. Optically encoded signals involve a change in the reflection or transmission of light from a material, including the analog signals used in older movie films and digitally recorded signals for newer recordings. Two examples of digital recording encoding are 1-bit and base-2 binary. Nyquist folding, used when sampling is too slow, is introduced.
The behavior of fluids is complicated. One approximate solution is that of sinusoidal traveling waves. Those waves traveling in air are what we hear. The amplitude of sound waves is very small compared to 1 atmosphere, and the range of amplitudes we experience is very large. A logarithmic scale—in particular, decibels—is used to compress that range. The contrast is made between loudness, a perceived quantity, and intensity, a measurable quantity. For human hearing, a change of intensity of approximately 10 dB corresponds to a perceived change in loudness of approximately a factor of 2. Several examples are presented that show how multiple sound levels, expressed using decibels, can be compared and combined.
Electrical forces, electric fields, and electric field lines are considered and can be understood as arising from positive and negative electrical charges. Of most importance on an everyday scale will be the charge imbalance between positive and negative, known as the net charge. A net charge can be stored using two electrical conductors that are near to each other but separated by an insulating region. A device designed to do this is a capacitor. Resistors are devices that act like friction in an electric circuit and are one type of impedance found in electronics. Capacitors and resistors are characterized by capacitance, C, and resistance, R, respectively. A resistor connected to a capacitor will have behavior characterized by a time constant, RC—the product of the resistance and capacitance. It is one example of many systems that have a characteristic time scale. Some basic electronic schematic symbols are presented and used to demonstrate the exponential behavior of RC circuits. Similar circuits can be used to filter electrical signals—for example, to block low- or high-frequency signals and thus change the overtone content.
The basics of wave propagation in three dimensions, including the wave nature of sound, are considered. When waves from a common source reach a common end point, but by different paths, wave interference can result. Wave amplitudes that add or subtract result in constructive and destructive interference, respectively. Diffraction results when the components of a continuum of sources, initially from the same source, arrive at an end point following different paths. Diffraction becomes very significant when the wavelength becomes comparable to, or larger than, the objects in its path. Diffusion of sound energy in a room will occur if the reflections look as if they are randomized. Increasing the amount of sound diffusion in a room is often desirable, so special acoustic treatments have been devised for this purpose that are known as diffusers. Some of those treatments are based on variable-depth walls, where the depths are determined by a numerical series, including pseudo-random numbers and some results from numbers theory.