Power and Intensity

Music 175: Level • pressure Tamara Smyth, [email protected] The waveform shows how sound varies over Department of Music, . University of California, San Diego (UCSD) • Related to are: April 14, 2020 1. emitted by the source: – a ﬁxed quantity, in Watts (W), – analogous to the wattage rating of a light bulb.

2. measured a distance from the source: – power per unit area carried by wave (W/m2), – inﬂuenced by interference and environment, – analogous to light brightness at diﬀerent positions in a room.

1 Music 175: Sound Level 2

Intensity and Pressure Sound Range of

• Intensity is the power per unit area, • Amplitude (pressure) range of hearing (humans) Power – Threshold of audibility: 0.00002 N/m2 Intensity = , A – Threshold of feeling (or pain!): 200 N/m2 expressed in Watts/square meter (W/m2). • Sound intensity range (humans) • Sound intensity is −12 2 – I0 = 10 W/m (threshold of audibility) – a measure of the power in a sound that actually – 1 W/m2 (threshold of feeling) contacts an area (e.g. eardrum); • The intensity ratio between the that bring – a quantity inﬂuenced by environment pain to our ears and the weakest sounds we can hear surroundings/surfaces and interference from other is more than 1012. sources; • Intensity is related to pressure squared: I = p2/(ρc), where ρ is the density of air (kg/m3), and c is the (m/s).

Music 175: Sound Level 3 Music 175: Sound Level 4 Linear vs logarithmic scales. (dB)

• Human hearing is better measured logarithmically. • The (dB) is a unit named after • On a linear scale, telecommunications pioneer, . • – a change between two values is perceived on the To understand decibels DON’T watch. basis of the diﬀerence between the values; • The decibel is a , used to compare – e.g.: a change from 1 to 2 would be perceived as two quantities such as having the same increase as from 4 to 5. – the power gain of an ampliﬁer; • On a logarithmic scale, – the relative power of two sound sources. – a change between two values is perceived on the • A decibel is deﬁned as one tenth of a bel, basis of the ratio of the two values; 1 B = 10 dB. – e.g.: a change from 1 to 2 would be perceived as having the same increase as a change from 4 to 8. (to convert from B to dB, multiply by 10) • Linear: moving one unit to the right adds 1. • To compare quantities A and B: Linear A log = value (B) 10 B −6 −5 −4 −3 −2 −1 0 1 2 43 65  

Logarithmic A 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 10log = value (dB) 10 B • Logarithmic: moving right one unit multiplies by 10.

Music 175: Sound Level 5 Music 175: Sound Level 6

Comparing Power and Intensity Power and Intensity Levels

• The decibel diﬀerence between two power levels ∆L • Decibels are often used as absolute measurements. is deﬁned in terms of their power ratio W2/W1:

∆L = L2 − L1 = 10log W2/W1 dB.

• Since power is proportional to intensity, the decibel diﬀerence between two levels with intensities I1 and I2 is given by − ∆L = L2 L1 = 10log I2/I1 dB. • There is an implied ﬁxed reference (e.g. the threshold of audibility). • Sound power level of a source: W LW = 10log dB, W0 −12 where W0 = 10 W. • Sound intensity level at a distance from the source

I LI = 10log dB, I0 −12 2 where I0 = 10 W/m .

Music 175: Sound Level 7 Music 175: Sound Level 8 Sound pressure Level (SPL or Lp) Increasing distance from a source

• Recall: intensity is proportional to pressure squared: • Assuming radiation in free space (and equally in all I = p2/(ρc), directions) and a distance r from the source, (where ρc ≈ 400). – intensity decreases by 1/r2

• The sound pressure level Lp (SPL) is equivalent to sound intensity level in dB:

Lp = 10log I/I0 2 = 10log p /(ρcI0).

• Question: If there is a doubling of distance from • The product of ρ and c is often approximated by 400: the source, by what factor will the intensity change? p2 L = 10log p2/(ρcI ) = 10log • p 0 4 × 10−10 Solution: p 2 – Given an intensity at some initial distance: = 10log 2 × 10−5 P I1 = , p 4πr2 = 20log 2 × 10−5 p – doubling the distance from the source yields, = 20log dB. P P 1 p0 I2 = 2 = 2 2 = 2I1, −5 4π(2r) 2 4πr 2 where p0 =2 × 10 is the threshold of hearing for 2 pressure variations. a change in intensity by a factor of of 1/2 .

Music 175: Sound Level 9 Music 175: Sound Level 10

Sound Intensity Level with a Doubling SPL with a Doubling of Distance of Distance

• We should obtain the same result for pressure as with • How does the sound level change with a doubling intensity. of distance? • If intensity decreases by 1/r2, then – intensity will drop by a factor of 1/22 or 2−2 and – pressure decreases by 1/r, I − L = 10log 2 2 – (intensity is proportional to pressure squared). I I  0 • With a doubling of distance, pressure will drop by a I − = 10log + 10log(2 2) factor of 1/2 or 2−1, I0 p − I − L = 20log 2 1 = 10log 20 log(2) p p I0  0  p I − = 20log − 20 log(2) = 10log 20(.3) p I0  0 p I − = 20log − 6 dB. = 10log 6 dB. p I0  0

– doubling the distance from a source causes a decrease of 6 dB in the sound level.

Music 175: Sound Level 11 Music 175: Sound Level 12 Multiple sources Example SPL Levels

• When there are multiple sound sources, the total • The following is taken from Dan Levitin’s, Your Brain power emitted is the sum of the power from each on Music. source. – 0 dB: a mosquito ﬂying in a quiet room, ten feet • By how much would the sound level increase when away from your ears two sources sound simultaneously with equal power? – 20 dB: a recording studio – the sound power level would double, – 35 dB: a typical quiet oﬃce with the door closed 2W and computer oﬀ LW = 10log – 50 dB: typical conversation  W0  W – 75 dB: typical comfortable music listening level = 10log + 10log(2) W0 (headphones) W – 100-105 dB: Classical music concert during loud = 10log +3 dB, W0 passages; the highest level of some portable music players and there would be an increase of 3 dB. – 110 dB: A jackhammer 3 feet away • Similarly, there would be a 3 dB increase in the sound – 120 dB: A jet engine heard on the runway from intensity level measured at some distance away from 300 ft away; typical rock concert the source. – 126-130 dB: Threshold of pain and damage; a • This accounts for most cases; the actual result rock concert by the Who depends on correlation (and interference) of sound – 180 dB: Space shuttle launch sources.

Music 175: Sound Level 13 Music 175: Sound Level 14