## Wave, Sound, and Music

### Contents

Wave
Polarization
Sound
Music
Musical Instruments
Music and the Brain

### Wave

One of the most characteristic features of the quantum theory is the wave-particle duality, i.e., the ability of matter or light quanta to demonstrate the wave-like property of interference (such as standing wave), and yet to appear subsequently in the form of localizable particles, even after such interference has taken place. Atomic and molecular theory depends on the computation of probability wave. Elementary particle theory starts from a field equation. The concepts of standing wave and fourier superposition are fundamental to quantum theory. Therefore in addition to its application to natural phenomena, an understanding of wave is one of the pre-requisites for the studying of modern physics.  Wave motion is one of the most familiar of natural phenomena. When a medium, whether gas, liquid, or solid, is disturbed, the disturbance moves out in all directions until it encounters a boundary at which point it will either be absorbed, reflected, or refracted depending on the nature of the discontinuity. In reality, the wave would fade away gradually by damping in the medium. The physics of wave motion can be illustrated best in one dimension such as in a string. Figure 01 shows a pulse generated by a single up and down motion of the string. The pulse moves out as shown in successive time frames. Now if the up and down motion of

#### Figure 02 Traveling Wave[view large image]

the string is driven by a motor, it would generate a traveling wave in the form of the sine function as shown in Figure 02.

It also shows that the shape of the string repeats itself in every distance interval , which is called the wavelength. The frequency of a wave is how frequently the wave crests pass a given point. If 100 wave crests pass a point in 1 second, the frequency is 100 cycle per second (it is sometimes expressed as 100 c/s, or 100 cps, or 100 Hz). The frequency and the period is related by a simple formula: = 1/ ----------- (1)

It is related to the wavelength and wave velocity by another formula:  = / = v, ---------- (2)

where v is the velocity of the traveling wave.

The velocity v is related to the tension T and mass density (per unit length) of the medium by yet another formula:

v = (T/ )1/2, ---------- (3)

which implies that the wave moves faster when the tension of the medium is high and the density is low.

Mathemetically, the displacement u of the wave in three dimensional space is expressed by the differential equation: ---------- (4a)
where x, y, z are the spatial coordinates, t is the time, and v is the propagation velocity of the wave (referring to a certain phase) in the medium; it is also known as the phase velocity. For wave motion in one dimension, e.g., along the x axis, the 2nd and 3rd terms in Eq.(4a) vanish; the solution for u at any point x and any time t is expressed by the sine function:

u = A sin(kx - t) ---------- (4b)

where = 2  , k = 2 / , and A is the amplitude as shown in Figure 02. It can be shown that the cosine function in similar form as Eq.(4b) is also a solution for Eq.(4a) with the initial and boundary conditions of u = A at t =0 and x =0.  If the vibrating string is attached to a rigid support at the other end, the traveling wave will be reflected and will begin to travel back toward the driven end. If the frequency of vibration is not properly chosen, the direct wave and the reflected wave will combine to produce a jumbled wave pattern. It is found that only a number of particular frequencies can produce regular patterns of motion along the string. At these frequencies, certain positions along the string remain stationary (the nodes) while the rest of the string vibrates with a constant

#### Figure 04 Standing Wave [view large image]

amplitude at any one point (see Figure 03). These regular wave patterns are called standing waves as shown in Figure 04. This condition is sometimes called resonance. In order to satisfy the requirement that nodes exits at both ends of the string (because the ends are fixed), the condition for setting up these standing waves is:

L = n ( /2), ---------- (5)

where L is the distance between the two end points, and n = 1, 2, 3, 4, ...

The lowest frequency at which a standing wave can be set up is called the fundamental frequency o for the particular string. The higher frequencies with integer multiple 2 o, 3 o, and 4 o, ... are called harmonics or overtones. Usually, the dominant standing wave is the fundamental as shown in Figure 11a, which displays the proportion of the fundamental to the various harmonics for different kinds of musical instruments. Since Eq.(4a) is a linear differential equation, the sum of the separate solutions is also a solution. Thus, superposition of the the fundamental and harmonics can generate different kind of waveform as shown in Figure 05. Mathematically, it is expressed by the Fourier series f(x) with u = f(x) sin(n vt/L):   ----- (6)

#### Figure 06 Fourier Series and Waveforms[view large image]

where f(x) is the maximum displacement of the wave at x, L= /2 and n = 1, 2, 3, ... Figure 06 depicts the various waveforms produced by the respective Fourier series.
The summation sign represents sum over all the variables with an index n. For example, n2 = 12 + 22 + 32 + ... The integral sign represents sum over a continuous variable x from x = -L to x = +L. A trivial example is dx = 2L

The wave moves along a string is called transverse wave since the vibration is perpendicular to the direction of propagation. The electromagnetic waves and the ripples in a pond are another examples. Electromagnetic waves are generated by acceleration of electric charges such as in lightning, hot filament, electrical circuit, etc. An idealized source that can emit infinitely long sinusoidal waves at one fixed frequency - such as the wave trains in Figure 02 - is said to emit a monochromatic wave at that frequency. Such is not usually the case with sources of electromagnetic radiation. According to classical electrodynamics the frequency of the wave from an oscillating charge is broadened and shifted as shown in Figure 07a due to the loss of energy in the process of emitting the  wave. The electromagnetic waves usually do not propagate in unidirection either, Figure 07b shows the radiation pattern for charge accelerated in its direction of motion. The "8" shape pattern (a no hole doughnut in 3 dimension) is emitted at low velocity, while the lobes (a thick cone in 3 dimension) are generated at speed close to the velocity of light.

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