Chapter 1 Fundamentals of Sound Waves -1

Transcription

1 Chapter 1 Fundamentals of Sound Waves -1

2 Sound Sections What is Sound? Producing a Sound Wave Characteristics of Sound Waves The Speed of Sound Intensity of Sound Waves

3 What is Sound? Sound is such a common part of everyday life that we rarely appreciate all of its functions. It provides enjoyable experiences such as listening to music or to the singing of birds. Sound waves are the most common example of longitudinal waves. They travel through any material medium with a speed that depends on the properties of the medium.

4 Producing a Sound Wave A tuning fork can be used as an example of producing a sound wave. A tuning fork will produce a pure musical note As the tines vibrate, they disturb the air near them As the tine swings to the right, it forces the air molecules near it closer together This produces a high density area in the air This is an area of compression (C)

5 As the tine moves toward the left, the air molecules to the right of the tine spread out This produces an area of low density This area is called a rarefaction (R).

6 As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork A sinusoidal curve can be used to represent the longitudinal wave Crests correspond to compressions and troughs to rarefactions

7 Sound is also a pressure wave. Changes in air pressure in relation to static air pressure is called sound. In air sound propagates as longitudinal wave motion

8 Example: Sound A sound wave is a a) transverse wave caused by the compression of particles b) transverse wave that passes through a vacuum c) longitudinal wave produced by the motion of water d) longitudinal wave caused by the vibration of a medium

9 Representation of Sound Waves Sound wave is a sinusoidal wave which is the simplest example of a periodic continuous wave. Figure shows a periodic sinusoidal waveform. B C S P T Q

10 Amplitude, A is defined as the maximum displacement from the equilibrium position to the crest or trough of the wave motion. Frequency, f is defined as the number of cycles (wavelength) produced in one second. Its unit is hertz (Hz) or s 1. Period, T is defined as the time taken for a particle (point) in the wave to complete one cycle. In this period, T the wave profile moves a distance of one wavelength,. Its unit is second (s). T 1 f

11 Wavelength, is defined as the distance between two consecutive particles (points) which have the same phase in a wave. From the Figure, o Particle B is in phase with particle C. o Particle P is in phase with particle Q o Particle S is in phase with particle T The S.I. unit of wavelength is meter (m). Wave number, k is defined as k 2 The S.I. unit of wave number is m 1.

12 Wave speed, v is defined as the distance travelled by a wave profile per unit time. Figure 10.8 shows a progressive wave profile moving to the right. v It moves a distance of in time T hence v distance time v T and T 1 f v f

13 The S.I. unit of wave speed is m s 1. The value of wave speed is constant but the velocity of the particles vibration in wave is varies with time, t Displacement, y is defined as the distance moved by a particle from its equilibrium position at every point along a wave. y = Acos(wt + φ)

14 Types of Sound Waves i. Audible waves are waves that lie within the range of sensitivity of the human ear. They can be generated in a variety of ways, such as by musical instruments, human vocal cords, and loudspeakers. Audible range: 20 Hz < f <20 khz or W/m 2 < I < 1 W/m 2 ii. Infrasonic waves are waves having frequencies below the audible range (f < 20 Hz). Elephants can use infrasonic waves to communicate with each other, even when separated by many kilometers. Whales, elephants, hippopotamuses, rhinoceros, giraffes, okapi, and alligators iii. Ultrasonic waves are waves having frequencies above the audible range (f > 20 khz). You may have used a silent whistle to retrieve your dog.

15 Applications of Sound Waves Can be used to produce images of small objects. Widely used as a diagnostic and treatment tool in medicine May use piezoelectric devices that transform electrical energy into mechanical energy Reversible: mechanical to electrical Ultrasonic ranging unit for cameras

16 Characteristics of Sound Waves Pitch: This is the characteristic of a note which enables us to differentiate a high note from a low note. It depends on the frequency of the sound wave. The higher the frequency, the more high-pitched a sound is perceived. Loudness: This is determined by the amplitude of the sound waves. It is a sensation in the mind of the individual observer. The louder the sound, the higher the amplitude. So, amplitude is also a way of measuring the energy has. The higher the energy, the higher the amplitude resulting a louder sound. Intensity: Intensity is the average amount of sound power transmitted through a unit area in a specified direction. The unit of intensity is watts per square meter.

17 Example: Sound The pitch of a musical note is characterized by the sound s (A) amplitude (B) frequency (C) wavelength (D) speed

18 The Speed of Sound Sound can travel through any kind of matter, but not through a vacuum. The speed of sound is different in different materials; in general, it is slowest in gases, faster in liquids, and fastest in solids. The speed depends somewhat on temperature, especially for gases.

19 Example: Sound Which of the following is a true statement? (A) Sound waves are transverse pressure waves. (B) Sound waves can not travel through a vacuum. (C) Sound waves and light waves travel a the same speed. (D) When the frequency of a sound wave increases its wavelength also increases.

20 The Speed of Sound The sound speed c in a fluid whose density is ρ and has a volume elasticity κ is given by In the case of a gas, the pressure variations associated with the sound wave are adiabatic, therefore, when the ratio of specific heats under conditions of constant pressure and constant volume is γ where P 0 is the atmospheric pressure. In air at 0 C and 1 atm of pressure, γ =1.41, P= Nm 2 (or Pa)=1013 mbar, ρ =1.29 kgm 3 and c=331.5 ms 1.

21 The Speed of Sound Rearranging these formulas leads to c P 0 0 Alternatively, c RT M c Speed of sound P 0, 0 - Pressure and Density - Ratio of specific heats R Universal Gas Constant T Temperature in 0 K M Molecular weight Speed of Light: 299,792,458 m/s Speed of sound 344 m/s

22 The Speed of Sound in Solid Since ρ varies with temperature while the atmospheric pressure remainssubstantially constant, at t C the sound speed is c m / s c m/ s Therefore, 340ms 1 is generally used for calculation at normal temperatures.also, the effect due to humidity is negligible. The sound speed of a longitudinal wave in a solid, whose density is ρ and Young s modulus E, is expressed in the same form as follows:, c = E ρ This equation can be used to find the Young s modulus E of a material,from the measured value of c under suitable conditions

23 The sound speed in a solid is much larger than in air except in the case of rubber, which is used as a special building material.

24 Example: Sound As the temperature of the air increases, the speed of sound through the air (A) increases. (B) decreases. (C) does not change. (D) increases if the atmospheric pressure is high.

25 Example Sound of a frequency f travels through air with a wavelength λ. If the frequency of the sound is increased to 2f, its wavelength in air will be (A) 2 (B) (C) 2 (D) 2

26 Example Find the speed of sound in water, which has a bulk modulus of 2.1x10 9 N/m 2 at a temperature of 0 C and a density of 1.0 x10 3 kg/m 3.

27 Exercise A copper alloy has a Young s Modulus of 1.1x10 11 Pa and a density of 8.92 x10 3 kg/m3. What is the speed of sound in a thin rod made of this alloy?

28 Exercise A lightning flash is seen in the sky and 8.2 seconds later the boom of thunder is heard. The temperature of the air is 12 0 C. a) What is the speed of sound in air at that temperature? b) How far away is the lightning strike?

29 Sound Measurement Provides definite quantities that describe and rate sound Permit precise, scientific analysis of annoying sound (objective means for comparison) Help estimate Damage to Hearing Powerful diagnostic tool for noise reduction program: Airports Factories, Homes Recording studios Highways, etc.

30 Quantifying of Sound rms Sound Pressure Physically, the rms value is indicative of the energy density of the disturbance. Mathematically, the rms value is obtained by squaring the sound pressures at any instant of time and then integrating over the sample time and averaging the results. The rms value is then the square root of this time average: p rms p 1/ T T 0 p 2 t dt 1/ 2 where the overbar refers to the time-weighted average and T is the time period of the measurement

31 Sound Pressure Level (SPL or L p ) Magnitudes of sound pressure affecting the ear vary from 2 x 10-5 Pa at the threshold, up to 200 Pa in the region of instantaneous damage. This may be compared with normal atmospheric pressure of 10 5 Pa. Because of this inconveniently large order of values involved and also because the ear response is not directly proportional to pressure, a different scale is used, pressure.

32 The sound pressure level then is a logarithmic ratio L p defined as: where p rms = the sound pressure of interest (in Pa) and p ref = reference sound pressure (in Pa) usually chosen as the limit of hearing of 20 μpa. NOTE: (P log 10 x n ) = P (n) log 10 x The unit for the sound pressure level, SPL or L p, is the decibel (db)

33 The L P is measured against a standard reference pressure, p ref = p o = 2 x 10-5 N/m 2 which is equivalent to zero decibels. The relationship between sound pressure and sound pressure level (with 20 μpa as the reference sound pressure) is shown in below Table. A scale showing some common sound pressure level is shown in the following Figure.

34 db SCALE Human ear responded logarithmically to power difference. Because of the wide range of intensities that human ear can detect, it is convenient to use a logarithmic scale. Alexander Graham Bell invented a unit Bel to measure the ability of people to hear. Power Ratio of 2 = db of 3 Power Ratio of 10 = db of 10 Power Ratio of 100 = db of 20 In acoustics, multiplication by a given factor is encountered most W 1 =W 2 *n So, Log 10 W 1 = Log 10 W 2 + Log 10 n Thus, if the two powers differ by a factor of 10 (n=10), the

35 Decibel 10Log 10 W 1 = 10Log 10 W Log 10 n to avoid fractions, now we have above quantities in decibel, 10 db=1 Bel decibels are thus another way of expressing ratios Electrical Power W V R 2 Sound Power 20Log 10 V 1 = 20Log 10 V Log 10 n (1/2) W 2 P r r - acoustic impedance 20Log 10 P 1 = 20Log 10 P Log 10 n (1/2)

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37 Relative scales of sound pressure levels

38 Example The RMS pressure of a sound is 200 Pa. What is the sound pressure level (SPL)? (Reference pressure 2 x 10-5 Pa)

39 Exercise What is the sound pressure level in decibels of a sound whose intensity is 0.01 W/m 2?

40 Example What is the increase in sound pressure level (in db) if the intensity is doubled?

41 Combining Sound Pressure Levels Since we re dealing with the logarithmic heritage in SPL, adding the decibels is the same as multiplying them. For example, adding 0 db (20µPa) noise with 0 db to it, you ll get a 6.02 db noise. Two approaches: 1) addition through formula 2) addition through graphical solution

42 For skeptics, this can be demonstrated by converting the db to SPL, adding them and converting back to db. Thus, the addition of these Sound Pressure Level is denoted by: L p = 20 log 10 (P/P o ) db L pt = 10 log 10 [ Σ (10) Lpi/10 ] db

43 A graphical solution for this type of problem is provided as in Figure. For noise pollution work, results should be reported to the nearest whole number. For equal decibel values, a shortcut method can be applied: total SPL L /10 n n LP i Pi 10log 10 log 10 n 10 log 10 (n) n 10 log 10 (n)

44 Figure: Graph for solving decibel addition problems

45 Example Three SPL s 68 db, 79 db and 75 db, what is SPL of combination? 68 db 75 db = db 79 db = db

46 Alternatively, this can be solved by converting the readings to SPL, adding them and convert back to SPL: L W 10log 10 log 80.7dB 10 (68/10) ,365,173 75/10 79 /10 10

47 Sound Power Travelling waves of sound pressure transmit energy in the direction of propagation of waves magnitude of the displacement times component of force in the direction of the displacement The rate at which this work is done: WORK Sound Power Level is defined as : = (SOUND) POWER, Watt where, L w W W o SWL = L w = 10Log W W 0 db = Sound Power level in db = Sound Power in watt = Reference sound power=10-12 watt The standard reference power watt is the threshold power of our hearing.

48 Please note that, direct reading of sound power is not possible. Sound Level meter functions by measuring sound pressure or the difference in pressure due to vibration of air molecule, compared to 1 atm. For example, when we speak, our voice vibrates the air molecule causing the pressure to increase and a sound power is generated. What is measured by the instrument is the pressure. Peak Power output: Female Voice 0.002W, Male Voice 0.004W, A Soft whisper 10-9 W, An average shout 0.001W Large Orchestra 10-70W, Large Jet at Takeoff 100,000W 15,000,000 speakers speaking simultaneously generate 1HP

49 Exercise Given L w = 90 db. What is the sound power in watt? W o = Reference sound power= watt

50 Sound Intensity

51 Intensity of Sound Waves The intensity of sound from a point source diminishes with distance: I 1 r 2

52 Intensity of Sound Waves Sound intensity (I) is defined as the time-weighted average sound power per unit area normal to the direction of propagation of the sound wave I W A I = Sound Intensity (watt/m 2 ) W A = Sound Power (watt) = area (sphere) normal to source = 4πr 2, r is distance from source The higher the area, the lower will be the sound heard at a distance from origin. Therefore, sound intensity is reduced proportionately with increase in coverage area.

53 Intensity is related to sound pressure in the following manner: where, I = Intensity (watt/m 2 ) p rms I p rms c = root mean square pressure, Pa ρ = density of medium (kg/m 3 ) c = speed of sound in medium (m/s) 2

54 Both the density of air and speed of sound are a function of temperature. Given the temperature and the pressure, the density of air may be determined from Standard Table. Sound Intensity Level can be written as: SIL = L I = 10Log I I 0 db where, L I I = Sound Intensity Level (db) = Sound Intensity (watt/m 2 )=1x10 12

55 Example: Sound Intensity The intensity of a point source at a distance d from the source is I. The intensity of the sound at a distance 2d from the source? (A) I 4 (B) 2I (C) 4 I (D) I 2

56 Example: Sound Intensity The intensity of a sound increases when there is an increase in the sound s (A) frequency (B) wavelength (C) amplitude (D) speed

57 Various Intensities of Sound Threshold of hearing Faintest sound most humans can hear About 1 x W/m 2 Threshold of pain Loudest sound most humans can tolerate About 1 W/m 2 The ear is a very sensitive detector of sound waves It can detect pressure fluctuations as small as about 3 parts in 10 10

58 Various Intensity Levels Threshold of hearing is 0 db Threshold of pain is 120 db Jet airplanes are about 150 db Table lists intensity levels of various sounds Multiplying a given intensity by 10 adds 10 db to the intensity level

59 Example An outside loudspeaker (considered a small source) emits sound waves with a power output of 100 W. (a) Find the intensity 10.0 m from the source. (b) Find the intensity level, in decibels, at this distance. (c) At what distance would you experience the sound at the threshold of pain, 120 db?

60 Answer

61 Answer

62 Exercise he sound level 25 m from a loudspeaker is 71 db. What is the rate at hich sound energy is being produced by the loudspeaker, assuming it to e an isotropic source?

63 Exercise : Sound radiates in a hemi-sphere from a rock band. If the sound level is 100 db at 10 m, then find the sound level at 4 m.

64 COMBINATION OF SEVERAL SOURCES Total Intensity produced by several sources I T =I 1 + I 2 + I 3 + Usually, intensity levels are known (L 1, L 2, ) I LT 10Log 10 T 12 I L 10Log LT L1 L2 L Log

65 If intensity levels of each of the N sources is same, LT COMBINATIONS OF SOURCES L Log N 10 L 10LogN L Thus for 2 identical sources, total Intensity Level is 10Log2 i.e., 3dB greater than the level of the single source T 1 For 2 sources of different intensities: L 1 and L 2 L 1 =60dB, L 2 =65.5dB L T =66.5dB L 1 =80dB, L 2 =82dB L T =84dB

66 Addition of sound levels

67 Example Let s say it is warm in our studio and a fan is brought in to augment the air conditioning (A/C) system. If both fan and the A/C are turned off, a very low noise level prevails, low enough to be neglected in the calculation. If the A/C alone is running, the sound-pressure level at a given position is 55 db. If the fan alone is running, the sound-pressure level is 60 db. What will be the sound-pressure level if both are running at the same time?

68 If the combined level of two noise sources is 80 db and the level with one of the sources turned off is 75 db, what is the level of the remaining source? In other words, combining the 78.3 db level with the 75 db level gives the combined level of 80 db.

69 The Human Ear Outer Ear: Pinna and auditory canal concentrate pressure on to drum Middle Ear: Eardrum, Small Bones connecting eardrum to inner ear Inner Ear: Filled with liquid, cochlea with basilar membrane respond to stimulus of eardrum with the help of thousands of tiny, highly sensitive hair cells, different portions responding different frequencies of sound. The movement of hair cells is conveyed as sensation of sound to the brain through nerve impulses Masking takes place at the membrane; Higher frequencies are masked by lower ones, degree depends on freq.difference and relative magnitudes of the two sounds

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71 This sensitivity dependence on frequency is also dependent on SPL!!!! SOUND BITS Unless there is a 3 db difference in SPL, human beings can not distinguish the difference in the sound Sound is perceived as doubled in its loudness when there is 10dB difference in the SPL. (Remember 6dB change represents doubling of sound pressure!!) Ear is not equally sensitive at all frequencies: highly sensitive at frequencies between 2kHz to 5kHz less at other freq.

72 The Response of Human Ear Loudness Level (Phon) Equal to numerical value of SPL at 1000Hz 0Phon: threshold of hearing Phon is useful for comparing two different frequencies for equal loudness But, 60Phon is still not twice as loud as 30Phon Equal Loudness Contours for pure tones, Free Field conditions Doubling of loudness corresponds to increase of 10Phon

73 D-weighting for Aircraft Noise Weighting Characteristics A-weighting: 40Phon equal loudness level contour C-weighting: 90Phon equal loudness level contour

74 Loudness vs. Frequency Each equal-loudness contour is identified by its value at 1,000 Hz, and the term loudness level in phons is thus defined. For example, the equal-loudness contour passing through 40-dB sound-pressure level at 1,000 Hz is called the 40- phon contour. Loudness is a subjective term; sound-pressure level is strictly a physical term. Loudness level is also a physical term that is useful in estimating the loudness of a sound (in units of sones) from sound-level measurements. The shapes of thee qual-loudness contours contain subjective information because they were obtained by a subjective comparison of the loudness of a tone to its loudness at 1,000 Hz

75 LOUDNESS INDEX Direct relationship between Loudness Level P (Phons) and Loudness Index S (Sones) S 2 P Sones is twice as loud as 4 Sones

76 Hearing Damage Potential to sound energy depends on its level and to the duration of exposure. Equivalent Continuous Sound Level (L eq ) N 10 Leq 10Log10 t j10 db j 1 L j t j : Fraction of total time duration for which SPL of L j was measured Total time interval considered is divided in N parts with each part has constant SPL of L j Leq 10Log dB 8 8

77 Exercise : Noise from a building site is caused by five items of plant. The periods of operation of each item of plant during the working day and the noise level each produces at a noise sensitive property at the boundary of the site are shown below. Calculate the equivalent continuous noise level over a twelve-hour working day. Compressor 83 db(a) operating for 5 h Excavator 85 db(a) operating for 2 h Dumper truck 76 db(a) operating for 6 h Pump 75 db(a) operating for 7 h Pile-driver 88 db(a) operating for 1.5 h

78 FREQUENCY & FREQUENCY BANDS Frequency of sound ---- as important as its level Sensitivity of ear Sound insulation of a wall Sound absorption of a material Attenuation of silencer Resonane all vary with freq. <20Hz 20Hz to 20000Hz > 20000Hz Infrasonic Audio Range Ultrasonic

79 Frequency Composition of Sound Pure tone Musical Instrument For multiple frequency composition sound, frequency spectrum is obtained through Fourier analysis

80 Amplitude (db) Complex Noise Pattern A produced by exhaust of Jet Engine, water at base of Niagara Falls, hiss of air/steam jets, etc 1 f 1 Frequency (Hz) No discrete tones, infinite frequencies Better to group them in frequency bands total strength in each band gives measure of sound Octave Bands commonly used (Octave: Halving / doubling)

81 OCTAVE BANDS An octave is defined as a 2:1 ratio of two frequencies. For example, middle C (C4) on the piano has a frequency close to 261 Hz. The next highest C (C5) has a frequency of about 522 Hz. As the ratio of 2:1 is defined as the octave, its mathematical expression is: in which: f U f L = 2 n f U = the frequency of the upper edge of the octave interval. f L = the frequency of the lower edge of the octave interval. n = the number of octaves.

82 Example If 446 Hz is the lower edge of a 1 3 octave band, what is the frequency of the upper edge?

83 Exercise What is the lower edge of a 1 3 octave band centered on 1,000 Hz? The f 1 is 1,000 Hz but the lower edge would be 1 6 octave lower than the 1 3 octave, so n = 1 6:

84 Octave Band Filters The human ear is sensitive to sounds having frequencies in the range from about 16Hz to 16 khz. Because it is not practical to measure the sound level at each of the 15,984 frequencies in this range, acoustic measuring instruments generally measure the acoustic energy included in a range of frequencies. The frequency interval over which measurements are made is called the bandwidth. In acoustics, the bandwidths are often specified in terms of octaves, where an octave is a frequency interval such that the upper frequency is twice the lower frequency.

85 Octave Filters

86 Instruments for analysing Noise Constant Bandwidth Devices f f U L 2 n n=1 for octave, Proportional Bandwidth Devices n=3 for 1/3 rd octave If we divide each octave into three geometrically equal subsections, i.e., f f U L f f f 2 c U L Absolute Bandwidth = f U - f L = f L % Relative Bandwidth = (f U -f L / f c ) = 70.7% These bands are thus called 1/3 rd octave bands with % relative bandwidth of 23.1% f 2 U 1/10 For 1/10 th Octave filters, % relative bandwidth of 5.1% fl f f U L 1/3 2

87 Octave and 1/3 rd Octave band filters mostly to analyse relatively smooth varying spectra If tones are present, 1/10 th Octave or Narrow-band filter be used

88 Measuring Sound-Pressure Level A sound level meter is designed to give readings of sound-pressure level; sound pressure in decibels referred to the standard reference level, 20 μpa. Sound level meters usually offer a selection of weighting networks designated A, B, and C having frequency responses shown in figure. Network selection is based on the general level of sounds to be measured (background noise? jet engines?), such as: For sound-pressure levels of db...use network A. For sound-pressure levels of db...use network B. For sound-pressure levels of db...use network C. These network response shapes are designed to bring the sound level meter readings into closer conformance to the relative loudness of sounds.

89 A, B, and C weighting response characteristics for sound level meters. (ANSI S )

90 Originally, the A-scale was designed to correspond to the response of the human ear for a sound pressure level of 40 db at all frequencies. The B-scale was designed to correspond to the response of the human ear for a sound pressure level of 70 db at all frequencies. The C-scale was approximately flat (constant) for frequencies between 63 Hz and 4000 Hz. The B-scale is rarely used at present. The A-scale is widely used as a single measure of possible hearing damage, annoyance caused by noise, and compliance with various noise regulations. The sound levels indicated by the A-scale network are denoted by LA, and the units are designated dba.

91 If the sound pressure level spectrum is measured or calculated for each octave band, the A-weighted sound level may be calculated, using the A weighting factors (CFA) from Table 2-4: L A = 10log (L p+cfa)/10 where the summation is carried out for all octave bands. The A-scale conversion process is illustrated in the following example. CFA: conversion factor to A-scale; CFC: conversion factor to C-scale.

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93 Exercise : The measured octave band sound pressure levels around a punch press are given in Table 2-5. Determine the A-weighted sound level and the overall sound pressure level.

94 The A-weighted sound level is calculated The overall sound pressure level is obtained by adding the individual unweighted octave band sound pressure levels given in Table 2-5, using decibel addition,

95 We note that the A-weighted sound level is lower than the overall sound pressure level in this problem. The reason for this difference is that the sound energy is more predominant in the lower octave bands, such as the 250 Hz band.

96 Sound Field Definitions (see ISO 12001) Free field The free field is a region in space where sound may propagate free from any form of obstruction. Near field The near field of a source is the region close to a source where the sound pressure and acoustic particle velocity are not in phase. In this region the sound field does not decrease by 6 db each time the distance from the source is increased (as it does in the far field). The near field is limited to a distance from the source equal to about a wavelength of sound or equal to three times the largest dimension of the sound source (whichever is the larger).

97 Far field The far field of a source begins where the near field ends and extends to infinity. Note that the transition from near to far field is gradual in the transition region. In the far field, the direct field radiated by most machinery sources will decay at the rate of 6 db each time the distance from the source is doubled. For line sources such as traffic noise, the decay rate varies between 3 and 4 db. Direct field The direct field of a sound source is defined as that part of the sound field which has not suffered any reflection from any room surfaces or obstacles. Reverberant field reverberant field of a source is defined as that part of the sound field radiated by a source which has experienced at least one reflection from a boundary of the room or enclosure containing the source.

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