Environmental and Architectural Acoustics

Z. Maekawa, Jens Rindel, P. Lord

• Publisher: CRC Press
• Year: 2010
• ISBN:
  9781134060023
• Edition: 2
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

Adopting a multi-disciplinary approach to the practice of achieving a more acceptable acoustic environment, this book draws on the same basic principles to cover both the outdoors and indoor space. It starts with the fundamentals of sound waves and hearing and goes on to the measurement of noise and vibration, room acoustics, sound absorption, airb

1
Fundamentals of sound waves and hearing

Sound can be visualised physically as a wave motion, which is transmitted through a whole range of elastic media. It is called a sound wave. On the other hand, it is also a sensation subjectively perceived by the ear, which is stimulated by the sound wave. This is referred to as auditory sensation, a phenomenon which is the subject of advanced research and comes under the general heading of psychophysiology.

1.1 Sound waves

A sound wave is transmitted through a medium that has both inertia and elasticity. The space in which sound waves travel is called the sound field. In a sound field the medium particles exhibit a repetitive movement backwards and forwards about their original position. Since a particle in the medium causes a neighbouring particle to be displaced by ξ, the repetitive movement produces a wave motion, i.e. vibration that is transmitted from particle to particle successively in the medium. The direction of the particle’s movement is the same as that of the transmission path of the sound wave. Therefore, it is called a longitudinal wave. As shown in Figure 1.1, the medium particles are crowded together at a certain point, producing a high pressure while at a neighbouring point they are dispersed resulting in a reduced pressure. Two such points of condensation and rarefaction exist alternately in the wave motion. Thus, at a fixed point, the dense and rare parts of the wave arrive alternately and the pressure consequently repeatedly rises and falls. This pressure fluctuation is called sound pressure p and the velocity of motion of the particles of the medium is called the particle velocity v.

The number of fluctuations in 1 s is called the frequency generally expressed by f, the unit of which is the hertz (Hz). The distance that a sound travels in 1 s is called the sound speed and generally denoted by c (m s⁻¹). If we let λ represent the wavelength, then

λ = c/f(m)

(1.1)

Figure 1.1 Particle movement and wave propagation; ξ: particle displacement, v: particle velocity; p: sound pressure; λ: wavelength; and T: period.

In order to express the wave motion in the form of a mathematical equation, the displacement ξ of a medium particle in the x direction is expressed as follows,

ξ = AF(t − x/c)

(1.2)

where F is a function which has two independent variables, time t and distance x; c is a constant and A is the amplitude, which is constant in this case. When a sound source produces simple harmonic motion the equation becomes

ξ = A cos ω(t − x/c) = A cos (ωt − kx)

(1.3)

where ω is the angular frequency; ω = 2πf and k = ω/c. The sound pressure and the particle velocity can also be expressed in the same form. In Equation (1.3) when t becomes (t + 1 s) and x becomes (x + c × 1 s),

ξ = A cos ω[(t + 1) − (x + c)/c] = A cos ω(t − x/c)

Therefore, the expression does not change. This means the phenomenon shifts through a distance c in 1 s, therefore, c is the sound speed. If we consider a certain position, for instance, where x = 0 in Equation (1.3), then ξ = A cos ωt, which means that it is a sine wave motion. Also at a certain time, for example when t = 0, ξ = A cos (−kx), which shows that the positions of moving particles also follow a sinusoidal pattern. In this case kx = 2π(x/λ), therefore, when (x/λ) is an integer the values of ξ are the same, and it is found that particles separated by λ are in the same phase. Also, k = ω/c...