A Pocket-Sized Introduction to Acoustics Keith Attenborough, Michiel Postema

A Pocket-Sized Introduction to Acoustics Keith Attenborough, Michiel Postema

A pocket-sized introduction to acoustics Keith Attenborough, Michiel Postema To cite this version: Keith Attenborough, Michiel Postema. A pocket-sized introduction to acoustics. The Univerisity of Hull, 80 p., 2008, 978-90-812588-2-1. hal-03188302 HAL Id: hal-03188302 https://hal.archives-ouvertes.fr/hal-03188302 Submitted on 6 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A pocket-sized introduction to acoustics Prof. Dr. Keith Attenborough Dr. Michiel Postema Department of Engineering The University of Hull 2 ISBN 978-90-812588-2-1 °c 2008 K. Attenborough, M. Postema. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechan- ical, photocopying, recording or otherwise, without the prior written permission of the authors. Publisher: Michiel Postema, Bergschenhoek Printed in England by The University of Hull Typesetting system: LATEX 2" Contents 1 Acoustics and ultrasonics 5 2 Mass on a spring 7 3 Wave equation in fluid 9 4 Sound speed in air 11 5 Solutions of the 1-D wave equation 13 6 Energy 15 7 Point and line sources 17 8 Root-mean-square pressure 19 9 Superposition of waves 21 10 Complex representation of a plane, harmonic wave 23 11 Decibel scale 25 12 Sound perception 29 13 Vectorial notation of the wave equation 33 14 Plane waves in isotropic media 35 15 Mechanisms of wave attenuation 39 16 Sound propagation and attenuation outdoors 43 3 4 17 Derivation of Snell's law 47 18 Reflection and transmission of waves on a plane fluid–fluid inter- face 51 19 Aspects of sound insulation 57 20 Room acoustics 61 A Fourier transform 65 B Equal loudness contours 67 C List of symbols 69 D Bibliography 75 E Biographies 77 1 Acoustics and ultrasonics Sound waves are a form of mechanical vibration. They correspond to particle displacements in the matter. Unlike electromagnetic waves, which can propagate in vacuum, sound waves need matter to support their propagation: a solid, a liquid, or a gas. The ear is an excellent acoustic detector in air but its sensitivity is limited to an interval between 20 Hz and 20 kHz. Audio frequency sound is essential in communication and entertainment. The acoustics of buildings, particularly concert halls, has been the subject of considerable study. Unwanted audio-frequency sound is called noise. The study of noise and noise control is an important part of engineering. Ultrasound refers to sounds and vibrations at frequencies above this upper limit of 20 kHz to values that can reach 1 GHz. Consequently, ultrasonics involve higher frequencies and smaller wavelengths than audio acoustics. Conversely, infrasound involves sounds and vibrations at low frequencies (below 20 Hz) and long wavelengths. Because the physiological sensation of sound has disappeared at these frequencies, our perceptions of infrasound and ultrasound are di®erent. Ultrasonic waves in fluids and solids are used for non-destructive evaluation. The general principle is to excite and detect a wave at ultrasonic frequencies and to deduce information from the signals detected. Example applications include the detection of flaws and inhomogeneities in solids, SONAR, SODAR, medical imaging, and acoustic microscopy. 5 6 For applications of acoustics, ultrasonics, and noise control, it is important to have a good understanding of wave propagation in in¯nite fluids and solids. The mathematical formulation of wave propagation in solids involves the use of the concepts and principles of 3-dimensional stress and strain analysis. But we start by outlining related concepts concerning the vibration of a single degree of freedom mass-on-a-spring system. 2 Mass on a spring Consider a mass m attached by a massless elastic spring to a motionless vertical wall and resting on top of a horizontal frictionless surface. In the equilibrium situation, the distance between the centre of mass and the wall is x0. The mass is pulled back over a distance ¢x by a force F . Then, for a stationary mass, F = s ¢x, (2.1) where s is the sti®ness of the spring. After release, we can de¯ne the excursion x(t) around the equilibrium by m xÄ = ¡s x (2.2) or d2x + !2 x = 0, (2.3) dt2 0 where r s ! = (2.4) 0 m is the natural (or resonance) frequency of the system in radians per time unit, and !0 = 2¼f0, (2.5) 7 8 where f0 is the resonance frequency in cycles per time unit. Consider the following solution of (2.3): x(t) = a0 cos(!0t) + a1 sin(!0t). Since max(x) = ¢x when t = 0, a0 = ¢x. Also, sincex _ = 0 when t = 0, a1 = 0. Here, the period is given by r 1 2¼ m T0 = = = 2¼ . (2.6) f0 !0 s 3 Wave equation in fluid Consider an in¯nitesimal element of length dx and cross-section dS. Let's assume the element is rigid in all directions except for the x-direction. Also, let's de¯ne a longitudinal compressive sound wave traveling in the x-direction. Suppose that the centre of the element is displaced by a distance u as³ a result´ of a sound pressure p. Then, the displacement of the boundaries is u + @u dx and ³ ´ @x 2 @u dx u ¡ @x 2 , respectively. The di®erence in volume, therefore, is "Ã ! Ã !# @u dx @u dx @u u ¡ ¡ u + dS = ¡ dx dS. (3.1) @x 2 @x 2 @x The volumetric strain @u dx dS @u ¢ = " + " + " = ¡ = ¡ . (3.2) x y z @x dx dS @x By de¯nition, the bulk modulus of elasticity · is given by stress p p · = = = ¡ (3.3) strain ¢ @u=@x or @u p = ¡· . (3.4) @x 9 10 The di®erence in deformation is caused by a di®erence in the sound pressure is 1 1 p(u ¡ 2 dx) at one side of the element and p(u + 2 dx) at the other side. Hence, the force acting on the element is expressed by · 1 1 ¸ @p ¡ p(u + dx) ¡ p(u ¡ dx) dS = ¡ dx dS. (3.5) 2 2 @x This force accelerates the mass ½ dx dS with ½ being the density. The accelera- tion equals the second time derivative of the displacement in the x-direction u. Therefore, the following balance must hold: @p @2u ¡ dx dS = ½ dx dS, (3.6) @x @t2 which gives the equation of motion @p @2u = ¡½ . (3.7) @x @t2 Substituting (3.4) for p into (3.7) results in @2u @2u ¡· = ¡½ (3.8) @x 2 @t2 or @2u · @2u = , (3.9) @t2 ½ @x 2 which is the displacement wave equation of a periodic fluctuation u in the x- direction at a speed s · c = . (3.10) ½ Taking the partial derivative to x of (3.7) gives @2p @3u = ¡½ , (3.11) @x 2 @x @t2 whereas taking the second derivative to t of (3.4) gives @2p @3u = ¡· . (3.12) @t2 @x @t2 Combining (3.11) and (3.12) gives the linear wave equation @2p · @2p @2p = = c2 . (3.13) @t2 ½ @x 2 @x 2 4 Sound speed in air For an adiabatic change in an ideal gas PV γ = constant, (4.1) where P is the absolute pressure, V the volume of an element, and γ the ratio of speci¯c heats Cp . Di®erentiating to V gives Cv dP V γ + γPV γ¡1 = 0. (4.2) dV Dividing this through by V γ¡1 gives V ¡ dP = γP. (4.3) dV The volumetric strain is dV ¢ = ¡ (4.4) V and the stress associated with it is p = dP. (4.5) 11 12 Hence, using (3.3), (4.3) can be written as · = γP. (4.6) Therefore, the speed of sound in air s s · γP c = = . (4.7) ½ ½ In an ideal gas, PV = nRT , (4.8) where n is the amount of substance, R is the gas constant, and T is the absolute m ¹ temperature. Replacing n by M and R by MR, in which M is the molar mass and R¹ the speci¯c gas constant, we obtain PV = mR¹T = ½V R¹T , (4.9) or P = R¹T , (4.10) ½ Thus, for an ideal gas, q c = γR¹T . (4.11) 5 Solutions of the 1-D wave equation Consider the wave equation (3.13) @2p @2p = c2 (5.1) @t2 @x 2 and a solution of the form p = f1(ct ¡ x) + f2(ct + x). (5.2) Then, @2p = c2 (f 00 + f 00) (5.3) @t2 1 2 and @2p = f 00 + f 00. (5.4) @x 2 1 2 For the displacement equation (3.9), a similar solution is considered: u = g1(ct ¡ x) + g2(ct + x). (5.5) If we concentrate on the wave traveling in the positive direction (progressive wave), described by g1, for the moment ignoring g2, at t = 0, u0 = g1(¡x). Also, at t = 1, u1 = g1(c ¡ x). The displacement u1 caused by p1 must have 13 14 the same form as the displacement u0 caused by p0, only at a distance 1c further on. Hence, g1(¡x0) = g1(c ¡ x1) (5.6) and thus ¡x0 = c ¡ x1 (5.7) or x1 = x0 + c.

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