An Introduction to Acoustics

by Randall, Robert H.

• ISBN13: 9780486442518
• ISBN10: 0486442519
• Format: Paperback
• Copyright: 2005-04-26
• Publisher: Dover Publications

Summary

This undergraduate-level text opens with an overview of fundamental particle vibration theory, and it proceeds to examinations of waves in air and in three dimensions, interference patterns and diffraction, and acoustic impedance, as illustrated in the behavior of horns. Supplementary sections include four appendixes and answers to problems. 1951 edition.

Table of Contents

INTRODUCTION	1	(8)
I-1 Sound vs acoustics	2	(1)
I-2 Vibrating bodies	2	(1)
I-3 Frequency	3	(1)
I-4 Amplitude	3	(1)
I-5 Waves	3	(1)
I-6 Wavelength. Frequency in the wave	4	(1)
I-7 The principle of superposition	4	(1)
I-8 Energy density. Intensity in the wave	4	(1)
I-9 Sound "quality"	5	(1)
I-10 The use of electrical analogs	5	(1)
I-11 Waves in solids	5	(1)
I-12 Experimental technique	6	(1)
I-13 Applied acoustics	6	(1)
I-14 Systems of units	7	(2)
CHAPTER 1. FUNDAMENTAL PARTICLE VIBRATION THEORY	9	(28)
1-1 Simple harmonic motion of a particle	9	(1)
1-2 Energy in SHM	10	(1)
1-3 Combinations of SHM's along the same straight line	11	(2)
1-4 Two collinear SHM's whose frequencies differ by a small amount Beats	13	(2)
1-5 Mathematical vs audible beats	15	(1)
1-6 Combinations of more than two SHM's of different frequencies	15	(1)
1-7 Fourier's theorem	16	(1)
1-8 Determination of the Fourier coefficients	16	(2)
1-9 Even and odd functions	18	(1)
1-10 Convergence	19	(1)
1-11 Application of the Fourier analysis to empirical functions	20	(1)
1-12 Damped vibrations of a particle	20	(1)
1-13 Case I. Large frictional force	21	(1)
1-14 Case II. Small frictional force	22	(1)
1-15 Case III. Critical damping	23	(1)
1-16 Forced vibrations	24	(1)
1-17 The differential equation	25	(1)
1-18 The steady state solution for forced vibrations	25	(2)
1-19 Velocity and displacement resonance	27	(1)
1-20 The amplitude at resonance	28	(1)
1-21 Phase relationships	29	(1)
1-22 Energy transfer in forced oscillations	30	(1)
1-23 Some applications of the theory of forced vibrations	31	(2)
1-24 The importance of the transient response	33	(1)
1-25 Superposition of SHM's mutually perpendicular	33	(4)
CHAPTER 2. PLANE WAVES IN AIR	37	(18)
2-1 Introduction	37	(1)
2-2 Dilatation and condensation	38	(1)
2-3 Bulk modulus	39	(1)
2-4 Significant variables in the field of sound	40	(1)
2-5 The differential equation for plane waves	40	(2)
2-6 Physical significance of the particle displacement	42	(1)
2-7 Solution of the wave equation	42	(2)
2-8 Disturbances of a periodic nature	44	(1)
2-9 The wavelength	44	(1)
2-10 Graphical representation	45	(1)
2-11 Waves containing more than one frequency component	45	(1)
2-12 Alternate forms for the steady state solution to the wave equation	46	(1)
2-13 Phase relationships	47	(1)
2-14 Energy in the wave	48	(1)
2-15 Kinetic energy	48	(1)
2-16 Potential energy	48	(2)
2-17 Total energy density in the wave	50	(1)
2-18 Sound intensity	50	(1)
2-19 Units of intensity	51	(1)
2-20 The decibel	51	(1)
2-21 Intensity "level"; pressure "level"	52	(3)
CHAPTER 3. WAVES IN THREE DIMENSIONS	55	(19)
3-1 Waves in three dimensions. The equation of continuity	55	(1)
3-2 Application of Newton's second law	56	(1)
3-3 The differential equation for waves in three dimensions	57	(1)
3-4 The differential equation for spherical waves	58	(1)
3-5 The solution of the differential equation	58	(1)
3-6 The velocity potential, Φ	59	(2)
3-7 Application of the function Φ The "pulsing sphere"	61	(1)
3-8 Intensity for spherical waves	62	(1)
3-9 The "strength" of a source	63	(1)
3-10 Sources equivalent to a pulsing sphere	64	(1)
3-11 Limitations on the use of the "strength of source" concept	65	(1)
3-12 Extension of the "strength of source" concept	66	(1)
3-13 The double source	66	(1)
3-14 Examples of the double source	67	(1)
3-15 Radiation from a double source as a function of frequency	68	(1)
3-16 Quantitative analysis of the double source	68	(2)
3-17 Comparison of total power radiated by different types of sources	70	(1)
3-18 Practical double sources. The principle of the baffle	71	(3)
CHAPTER 4. INTERFERENCE PATTERNS. DIFFRACTION	74	(25)
4-1 Definition of interference for wave motion	74	(1)
4-2 Diffraction	74	(1)
4-3 Diffraction in acoustics and in light	74	(1)
4-4 Importance of the ratio of wavelength to dimension	75	(1)
4-5 The single slit pattern. Simplifying assumptions	75	(1)
4-6 Application of Huygens' Principle	76	(1)
4-7 Vector method of determining the acoustic pressure at point a	77	(1)
4-8 Essential geometry and equations	78	(1)
4-9 The variation of pm, with theta	79	(1)
4-10 The variation of (pm)² with the polar angle	80	(1)
4-11 Representation of intensity distribution on a polar graph	80	(2)
4-12 General significance of the diffraction pattern for a single slit	82	(1)
4-13 Openings of other shapes. The short rectangle or the square	82	(1)
4-14 Diffraction pattern for a circular aperture	83	(2)
4-15 Practical examples of the diffraction pattern for a circular aperture	85	(1)
4-16 Multiple slits and openings	86	(1)
4-17 Diffraction effects around the edges of obstructions	86	(1)
4-18 Fresnel laminar zones	87	(2)
4-19 The Fresnel integrals. The spiral of Cornu	89	(2)
4-20 Use of the Cornu spiral to determine the diffraction pattern for a straight edge	91	(1)
4-21 Direct graph of intensity	92	(1)
4-22 The shape of the diffraction pattern, as a function of λ	93	(1)
4-23 Diffraction of waves around obstacles of various contours lying in a field of sound	94	(2)
4-24 Diffraction effects for an acoustic piston set in a circular plate of finite size	96	(1)
4-25 General conclusions on diffraction	97	(2)
CHAPTER 5. ACOUSTIC IMPEDANCE. BEHAVIOR OF HORNS	99	(29)
5-1 The principle of analog	99	(1)
5-2 Types of analogies	99	(2)
5-3 Sound radiation and acoustic impedance	101	(1)
5-4 Elements of complex notation as applied to electrical circuits	101	(2)
5-5 Specific acoustic impedance	103	(1)
5-6 Specific acoustic impedance for plane waves	104	(2)
5-7 Analogous acoustic impedance	106	(1)
5-8 Specific acoustic impedance for spherical waves	106	(1)
5-9 The Helmholtz resonator	107	(3)
5-10 The resonance frequency	110	(1)
5-11 The behavior of horns	111	(1)
5-12 Radiation into a cylindrical tube closed at one end	112	(2)
5-13 Force on the piston. Total radiation impedance	114	(1)
5-14 Tube open at one end	114	(2)
5-15 Horns	116	(1)
5-16 The conical horn	117	(1)
5-17 Transmission coefficient for a horn	118	(1)
5-18 The exponential horn	119	(2)
5-19 Transmission coefficient for an exponential horn. Comparison with the conical horn	121	(1)
5-20 Effect of reflections upon horn behavior	122	(1)
5-21 The horn as an impedance matching device	122	(1)
5-22 The "hornless" or direct-radiator loudspeaker. Specific acoustic impedance at the surface	122	(3)
5-23 Distribution pattern for energy leaving horns or direct radiators	125	(1)
5-24 General significance of acoustic impedance for radiation	125	(3)
CHAPTER 6. LONGITUDINAL WAVES IN DIFFERENT GASES. WAVES IN LIQUIDS AND SOLIDS	128	(23)
6-1 Isothermal and adiabatic bulk modulus for an ideal gas	128	(1)
6-2 Factors affecting the velocity of longitudinal waves in gases	129	(2)
6-3 Experimental determination of the velocity of sound waves in gases	131	(1)
6-4 Transmission of longitudinal waves through gases as related to kinetic theory	132	(2)
6-5 Waves of large amplitude	134	(3)
6-6 Miscellaneous open-air effects	137	(1)
6-7 Acoustic focusing devices. Mirrors and lenses	138	(3)
6-8 Attenuation of longitudinal waves in gases	141	(1)
6-9 The Doppler effect	142	(1)
6-10 Practical importance of the Doppler effect	143	(2)
6-11 Transmission of longitudinal waves through liquids	145	(1)
6-12 Experimental measurement of c for liquids	145	(1)
6-13 Attenuation effects in liquids	146	(1)
6-14 Longitudinal waves in solids	147	(2)
6-15 The measurement of c as a means of studying the elastic properties of solids	149	(1)
6-16 Dissipation within solids	149	(2)
CHAPTER 7. STATIONARY WAVES. VIBRATING SOURCES. MUSICAL INSTRUMENTS	151	(33)
7-1 Introduction	151	(1)
7-2 The ideal string	152	(1)
7-3 The differential equation	153	(1)
7-4 The solution	154	(1)
7-5 The string of limited length	154	(1)
7-6 Reflection at one end of a string	155	(1)
7-7 Simultaneous reflection at both ends of a string	156	(1)
7-8 Vibrating string fixed at both ends	157	(1)
7-9 Interpretation of the stationary wave equation	158	(1)
7-10 Other end conditions. Both ends free	159	(3)
7-11 Vibrating string, one end fixed, one end free	162	(1)
7-12 Initial conditions	163	(1)
7-13 Other initial conditions	164	(2)
7-14 Bowing. Relaxation oscillations	166	(1)
7-15 Vibration of membranes. Stationary waves in two dimensions	166	(2)
7-16 Longitudinal stationary waves in bars	168	(1)
7-17 Transverse waves in bars	169	(2)
7-18 The tuning fork	171	(1)
7-19 The vibration of plates	172	(1)
7-20 Stationary air waves in pipes	172	(1)
7-21 Vibrations in a pipe closed at both ends	173	(1)
7-22 Vibration of an open organ pipe	174	(1)
7-23 Reflection and acoustic impedance	175	(1)
7-24 Frequencies of vibration of a "closed" organ pipe	176	(1)
7-25 General features of stringed instruments. The violin	176	(3)
7-26 The piano	179	(1)
7-27 The wind instruments. Excitation of an organ pipe	180	(1)
7-28 Wind instruments of the reed type	181	(1)
7-29 Wind instruments as radiating sources of sound	182	(2)
CHAPTER 8. REFLECTION AND ABSORPTION OF SOUND WAVES	184	(18)
8-1 Introduction	184	(1)
8-2 Reflection of longitudinal waves at a boundary between two ideal elastic media, each infinite, in extent	184	(2)
8-3 Relative magnitudes of the particle velocities	186	(1)
8-4 Relative phases	187	(1)
8-5 Magnitudes and phases of the acoustic pressures	187	(1)
8-6 Practical implications	188	(2)
8-7 The effect of partial reflection upon the stationary wave pattern	190	(2)
8-8 The absorption coefficient	192	(1)
8-9 Specific acoustic impedance of a boundary surface	193	(1)
8-10 Both media perfectly elastic and infinite in extent	193	(1)
8-11 Boundaries for which zn is reactive	193	(2)
8-12 Specific acoustic impedance at positions of discontinuity in the tube cross section	195	(1)
8-13 Specific acoustic impedance at the surface of absorbing materials	196	(1)
8-14 The relation between zn, and the absorption coefficient for plane air waves of normal incidence	196	(1)
8-15 Other absorption coefficients	197	(2)
8-16 Use of panel resonance	199	(1)
8-17 Absorbing "layers." The effect of thickness	199	(1)
8-18 Good reflectors and good absorbers	200	(2)
CHAPTER 9. SPEECH AND HEARING	202	(28)
9-1 Importance of the subjective element in acoustics	202	(1)
9-2 The vocal apparatus	202	(1)
9-3 The speech process	203	(2)
9-4 The vocoder	205	(1)
9-5 Energy distribution in speech as a function of frequency	206	(1)
9-6 Intelligibility of speech as related to frequency band width	207	(1)
9-7 Miscellaneous voice properties	208	(1)
9-8 Artificial voices	208	(1)
9-9 The hearing process	208	(1)
9-10 The structure of the ear	209	(1)
9-11 The organ of Corti	210	(1)
9-12 Mechanical properties of the cochlea: Resonance theory of Helm-holtz	211	(1)
9-13 Other resonance theories	211	(1)
9-14 The organs of sensation	212	(1)
9-15 Frequency perception	213	(1)
9-16 Hearing data for the normal ear	214	(1)
9-17 Threshold of audibility	214	(1)
9-18 Loudness and loudness level	215	(3)
9-19 Differential intensity level sensitivity of the ear	218	(1)
9-20 Pitch vs frequency	218	(1)
9-21 Differential frequency sensitivity of the ear	219	(1)
9-22 Shift in pitch (or apparent frequency) at high intensities	220	(1)
9-23 Masking	221	(1)
9-24 Sum and difference frequencies	222	(1)
9-25 The response of the ear to a harmonic series	223	(1)
9-26 The importance of the transient period to sound quality. Phase effects	224	(1)
9-27 Binaural effects	225	(1)
9-28 Hearing defects	226	(1)
9-29 Musical intervals. Scales	226	(2)
9-30 Consonance and dissonance	228	(2)
CHAPTER 10. SOUND MEASUREMENTS. EXPERIMENTAL ACOUSTICS	230	(30)
10-1 Precise acoustic measurement	230	(1)
10-2 Free-space measurements. Anechoic rooms	230	(3)
10-3 Reverberant chambers	233	(1)
10-4 Standard sound sources. The thermophone	233	(1)
10-5 The pistonphone	234	(1)
10-6 The electrostatic actuator	235	(1)
10-7 Measurements in a field of sound. The Rayleigh disk	236	(2)
10-8 Other absolute detection methods	238	(1)
10-9 Detectors requiring calibration. Microphones	239	(1)
10-10 Microphones	240	(6)
10-11 Relative sensitivities of different types of microphones	246	(1)
10-12 The calibration of microphones	246	(1)
10-13 The reciprocity method for calibrating microphones	247	(2)
10-14 Measurement of frequency in a wave	249	(1)
10-15 Complex wave analysis	250	(2)
10-16 Noise. The continuous acoustic spectrum	252	(2)
10-17 The measurement of acoustic impedance	254	(4)
10-18 Conclusion	258	(2)
CHAPTER 11. REPRODUCTION OF SOUND	260	(31)
11-1 Introduction	260	(1)
11-2 The general problem	261	(1)
11-3 An ideal transducer	262	(2)
11-4 Early types of transducers	264	(2)
11-5 Transducer with electromagnetic drive	266	(1)
11-6 "Blocked" vs "motional" impedance	266	(1)
11-7 Motional impedance and mechanical impedance	267	(1)
11-8 Motional impedance and acoustic radiation	268	(2)
11-9 Behavior of the transducer in a vacuum tube circuit	270	(2)
11-10 Behavior of the cone vs the acoustic piston	272	(1)
11-11 Acoustic coupling problems	273	(1)
11-12 Back of cone completely enclosed	274	(1)
11-13 Loudspeaker cabinet with open back	275	(1)
11-14 The acoustic phase inverter	276	(2)
11-15 The half wavelength pipe	278	(1)
11-16 The use of horns	279	(1)
11-17 High frequency radiation problems. Multiple loudspeakers	280	(2)
11-18 Effect of room resonances	282	(1)
11-19 Electrical equalization circuits	283	(1)
11-20 Transducers for disk phonograph records	284	(1)
11-21 Differences in transfer behavior	285	(1)
11-22 Mechanical impedance of moving parts. Tracking	286	(2)
11-23 Conclusion	288	(3)
CHAPTER 12. MISCELLANEOUS APPLIED ACOUSTICS	291	(28)
12-1 The acoustic properties of rooms	291	(1)
12-2 An ideal reverberant room	292	(1)
12-3 Rate of disappearance of energy from the ideal reverberant room	292	(1)
12-4 The steady state energy density	293	(1)
12-5 The transient equations	294	(2)
12-6 Reverberation time	296	(1)
12-7 Partially live rooms	297	(1)
12-8 Determination of αs	297	(1)
12-9 Effect of varying frequency	298	(1)
12-10 Absorbing surfaces of limited area	298	(1)
12-11 Computation of αs from zn	299	(1)
12-12 Effect of room resonances. Steady state	299	(1)
12-13 Normal modes of vibration. The transient period	300	(1)
12-14 Transmission of wave energy through partitions	301	(1)
12-15 Acoustic filters	302	(5)
12-16 Ultrasonics	307	(1)
12-17 Ultrasonic sources	308	(1)
12-18 Piezoelectric generators	309	(2)
12-19 Detectors of ultrasonic waves	311	(1)
12-20 Coupling between transducer and medium	312	(1)
12-21 Undersea signaling and ranging	313	(1)
12-22 Diffraction of light by liquids carrying ultrasonic waves	314	(1)
12-23 Testing of materials with ultrasonic waves	315	(1)
12-24 Other industrial applications of ultrasonic waves	316	(1)
12-25 Biological effects of ultrasonic waves	317	(1)
12-26 Acoustics in relation to other branches of physics	317	(2)
APPENDIX I. THE INTRODUCTION OF T11E VELOCITY POTENTIAL, Φ, INTO THE DIFFERENTIAL EQUATION FOR SPACE WAVES	319	(2)
APPENDIX II. THE RELATIONSHIP BETWEEN THE VELOCITY POTENTIAL Φ AND THE CONDENSATION, s	321	(2)
APPENDIX III. TABLE OF FRESNEL INTEGRALS	323	(1)
APPENDIX IV. DERIVATION OF THE EXPRESSION, U = eic/4 (Eq. 12-1)	324	(2)
LIST OF SYMBOLS	326	(3)
REFERENCES	329	(2)
INDEX	331	(6)
ANSWERS TO PROBLEMS	337