Document Sample

THE LIFE AND TIMES OF THE DECIBEL The measurement of sound amplitude is complicated by the fact that sound may be considered either in terms of its pressure or its intensity. Intensity is the amount of power acting on an area (i.e., against a surface), and, in terms of sound, is measured as watts/cm2 (where a watt is the unit of power). Sound pressure on the other hand is the amount of force acting on an area, and, again in terms of sound, is measured in Pascals where a Pascal represents 1 newton/m2 (where a newton is the unit of force). Since watts/cm2 and Pascals can both to measure sound amplitude, it follows that intensity (I) and pressure (p) must somehow be related. In fact, I = p 2. Remember this relationship - we‟ll be referring to it again later.1 Let‟s consider two pure tones (frequency needn‟t be considered - the explanation that follows is valid for all sound frequencies). The first pure tone is hereby designated as the „quiet‟ (Q) sound - it‟s the quietest sound that the average human ear can hear. The second pure tone is henceforward to be known as the „loud‟ (L) sound - it‟s the loudest sound that the average human ear can tolerate without pain. The table below shows the measurements for these two sounds in terms of both intensity and pressure 2. Q L Intensity (in watts/cm2) 10-16 10-2 Pressure (in Pascals) 2 10-5 2 102 Table 1. Intensity and pressure value for Q and L In common speech, sound pressure for low amplitude sounds is usually referred to in terms of millionths of a Pascal or micro-Pascals (P). Thus, the 2 10-5 P above is more commonly referred to as 20 micro- Pascals (= 20 P). Sounds with even greater pressures are measured in milli-Pascals or even directly in Pascals. For example, 2 102 P is the same as 200 P. The next stage to the birth of the decibel is to consider the ratio of the amplitude of the L sound to that of the Q sound (i.e., L Q. We need to do this for both intensity and for pressure. Intensity L 10 2 watts / cm2 10 2 16 16 10 2 10 16 10 216 1014 Q 10 watts / cm2 10 Equation 13 Thus the L sound is 1014 times as intense as the Q sound (in long form, that‟s 100,000,000,000,000 or 100 trillion times as intense). Pressure L 2 102 Pascals 102 5 5 102 105 10 25 107 Q 2 10 Pascals 10 Equation 2 7 This indicates that the L sound has 10 (10,000,000 or 10 million) times as much pressure than the Q sound. 1 For you mathematical purists out there, intensity does not, strictly speaking, equal sound pressure squared, but is proportional to pressure squared. For the sake of simplicity, I‟m ignoring that nicety. 2 For anyone unfamiliar or uncomfortable with the exponential notation used in the following table, there are some notes at the end explaining it as well as arithmetic operations involving exponentials. 3 Note: the watts/cm2 divides out of the equation since it is in both the numerator and denominator of the equation. Anything divided by itself equals 1 and thus cancels out of the equation. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 1 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Why this exercise in arithmetic? Well, first remember that dividing one quantity by another is mathematically equivalent to comparing the one to the other. These sound amplitude ratios therefore compare the L sound to the Q sound in terms of their intensity and their pressure respectively. These ratios demonstrate the wide range of sound amplitudes that the ear can process effectively - this is known as the „dynamic range‟ of hearing. However, the numbers involved in the measurement of hearing and with the dynamic range of hearing are very small or very large for the most part - not at all comfortable numbers to understand or have a feel for. Who of us really comprehends what 10 -16 signifies, or 100 trillion or 20 millionths? Certainly not the engineers and physicists. Accordingly, they exercised their minds to establishing a way of expressing these very large and small numbers in a way that is more convenient and easier to understand. Convenience they succeeded with. You can be the judge of ease. The method they settled on did, nonetheless, depends on comparing one sound to another. The first sound is the one they wish to measure. I‟ll call this the test sound. The second, the comparison sound is known as the reference sound. In other words, their method of expressing sound measurement does not give an absolute measurement of pressure or intensity. Rather it give a relative measurement, a measurement that tells how much more (or less) sound intensity or sound pressure the test sound has relative to the reference sound. This is a very effective way of dealing with sound amplitude (it really is) … (really and truly) … (believe me) … (please), but which has been a curse to generations of students (me included) ever since. Let‟s see how it‟s done. First of all, we already know how to compare one sound intensity (or pressure) to another sound intensity (or pressure) - divide the intensity (or pressure) of the sound of interest (the test sound) by the reference sound‟s intensity (or pressure). That‟s what we did in equations 1 and 2 above where L was the test sound and Q was the reference sound. Intensity In acoustics and audiology, the intensity of the Q sound is very commonly (but by no means always) used as the reference in measuring the intensity of a test sound. We shall use it here. In fact, we already did in equation 1 where we calculated the ratio L/Q = 1014. This is one of those uncomfortable numbers - a „1‟ with a string of fourteen zeros after it. But can we find a more comfortable number with 10 14 - hopefully one for which we have more of a natural „feel‟. Of course we can - what about the 14! But can one just pull the exponent away from the base number 10 like that. Of course we can - that‟s exactly what the mathematical function known as a logarithm does. In other words: log 1014 = 14 Equation 3 Some dyslexic boffin decided to name the result of this operation on sound intensity - he named it the Bel after Alexander Graham Bell. Thus, the L sound is 14 Bels more intense than the Q sound. Notice the last sentence carefully - its wording is critical. Often it gets shortened to “the L sound‟s intensity is 14 Bels”. However, that abbreviated form is meaningless unless the listener already knows what the reference sound is that L is being compared to. An important point number one to remember: THE COMPARISON OF THE TEST SOUND’S INTENSITY TO A REFERENCE SOUND’S INTENSITY IS ALWAYS IMPLICIT WHENEVER THE Bel IS USED. This will also be true of the decibel when we reach it. Remember also that while the Q sound frequently is used as the reference sound, it isn‟t always. The reference intensity can be and often is the intensity of some other sound - the choice of the Q sound as reference is completely arbitrary. Let‟s look at the general equation for the Bel, and at one other example: 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 2 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL It Intensity in Bels log Ir Equation 4 where It and Ir refer to the intensities of the test and reference sounds respectively. As an example, the average intensity of conversational level speech at about three feet from the speaker is about 10-10 watts/cm2. Applying the above equation, and using the Q sound intensity again as reference: 10 10 watts / cm 2 Intensity of conversation log 10 16 watts / cm 2 log 10 10 1016 log10 6 6 Bels Equation 5 Thus, conversation speech has, on average, an intensity of 6 Bels with reference to the Q sound, the quietest sound the average ear can detect. Let‟s reconsider the dynamic intensity range of the ear, this time as measured in Bels. It is, of course, 14 Bels as calculated in equation 1. Having put us through all this misery, one would think the physicists would be happy. No way. They now decided that, whereas 10 14 was too large a number with which to meaningfully describe the ear‟s dynamic range, 14 on the other hand was too small! So, totally arbitrarily, they decided to multiply Bels by 10. The result of this, and our ultimate destination, is the decibel, or dB. Consequently, the dynamic intensity range of the ear is 140 dB relative to Q, or 10 log (IL IQ) = 10 log (1014) = 10 14 = 140 dB. Similarly, conversational speech is, on average, 60 dB relative to Q. The general equation for dB is: It Intensity in dB = 10 log Ir Equation 6 Pressure Do you remember that equation from earlier: I = p 2. I warned you it would come back to haunt this discussion. In calculating dB for pressure (I‟m going to leave Bels for pressure as and exercise for you), we must take this equation into account and use one additional feature of logarithms. If we let n stand for any number we choose, then: log n 2 2 log n Equation 7 Recall to the general equation for intensity in dB (equation 6). We can rewrite equation 6 substituting p2 wherever there‟s and I as follows: 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 3 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL p2 Pressure in dB 10 log 2 t pr 2 p 10 log t pr p 2 10 log t pr p 20 log t pr Equation 8 Equation 8 is the general equation for pressure in dB. Let‟s put some numbers in. Earlier, in equation 2, we calculated that the dynamic pressure range of the ear (L Q) is 107. If we let 2 102 Pascals (from Table 1) = pt, and if we let 2 10-5 (also from Table 1) = pr, then apply the general equation, we get: 20 log 107 20 7 140 dB Equation 9 Finally everything has come together! Through this tortuous route, we have learned that the dynamic range of the ear is 140 dB whether measured as intensity or as pressure. Without doing the calculation, it should come as not surprise that the average pressure level of conversation is 60 dB re 20 P („re 20 P‟ simply indicates the reference pressure used in the measurement). You‟ll be pleased to hear that, for our purposes and for most of the time, sound pressure rather than sound intensity. Hence, for us, equation 8 is more relevant than equation 6.4 Let‟s review and summarize the most crucial things so far discussed about the decibel. 1. It was introduced to transform the very small and very large values that arise from the measurement of sound into more manageable numbers. 2. The decibel is NOT a unit of sound pressure or sound intensity. A sound of interest is always compared mathematically to an arbitrary reference sound by dividing its pressure (or intensity) by the pressure (or intensity) of the reference sound. 3. A frequently used reference sound is the Q sound, the lowest pressure (or intensity) sound that the average human ear can hear. Remember, however, that other reference pressures are also used for specific purposes. 4. First, in order for the decibel to have any meaning, it is essential that the reference pressure (or intensity) be explicitly stated. Beware of any articles, whether scholarly or popular, that fail to mention the reference pressure when reporting decibel measurements. With regard to point 4, there are a number of conventions used to declare the reference pressure (or intensity). For example, you will very often encounter „dB SPL‟. This stands for „decibels Sound Pressure Level‟. This is a shorthand way of saying that the reference sound is the Q sound and hence the reference pressure is 20 P. (For sound intensity, the equivalent is „dB IL‟ or „dB Intensity Level‟ where 10-16 watts/cm2 is the reference intensity.) 4 If you decide to take the instrumentation module next year, be warned - equation 6 will again surface if only momentarily. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 4 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Another very common shorthand method of declaring the reference pressure is „dBA‟ or „dBC‟.5 The „A‟ and „C‟ refer to settings on a sound level meter used for measuring sound. Since sound level meters are normally calibrated to use the Q sound as reference, dBA and dBC can be considered shorthand for dBA SPL and dBC SPL respectively. Of the two, we are most likely to encounter and/or use the dBA setting. „dB HL‟ and possible „dB SL‟ are other declaration types you will encounter in studying audiology. We won‟t get into these here (they‟ll rear their heads in audiology) other than to issue the warning that for HL and SL, the Q sound is not the reference. (HL stands for „Hearing Level‟ and SL for Sensitivity Level.) Otherwise, when the Q sound is not the reference, and when dB HL and dB SL are not being used, the reference should be written out. This is usually done in the format I used earlier, viz., dB re n P (or dB re n watts/cm2 for intensity). There are now one or two burning questions you are asking yourselves. Aren‟t there? So let‟s conclude by briefly answering them. First, what if the pressure of the test sound is less than that of the reference sound?. For example, consider a sound of 2 P (most of us wouldn‟t be able to hear it, but it is a sound nonetheless). First, let‟s compare it to the Q sound which has more pressure: p t 2 10 6 P 5 10 6 105 10 1 p r 2 10 P Equation 10 Converting this ratio to dB: 20 log (10-1) = 20 (-1) = -20 dB SPL Equation 11 So if a sound has less pressure (or intensity) than the reference sound, its measurement in dB will be a negative number. Or to do the obvious and turn the statement around, negative decibels signify a sound pressure (or intensity) that is less than that of the reference sound. Next question: what does 0 dB mean? Well, if positive numbers mean that pt > pr, and if negative numbers mean that pt < pr, then intuitively, one might conclude that 0 dB means that p t = pr. Why? If pt = pr, then their ratio must = 1 and log (1) = 0 and 20 0 = 0 dB irrespective of the reference pressure. Question no. 3 arises from the fact that so far, all the sounds discussed have had pressure ratios that were obvious powers of 10. Very neat, but also very unrealistic in practical terms. What, for example, if a sound has twice the pressure of the reference. As a concrete example, take a sound that is 40 P. The ratio with the Q sound by definition must be 2, but let‟s work it out anyway: p t 4 10 5 P 5 2 10 5 105 2 100 2 1 2 p r 2 10 P Equation 12 and the pressure in decibels: 20 log (2) 5 Yes, there is also a „dBB‟ and even a „dBD‟, but you are less likely to encounter these so I‟ll ignore them here. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 5 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Equation 13 Uh oh - previously, we‟ve just been able to casually pull off the exponent from 10 (recall that log (10 n) = n). Here there ain‟t no exponent of 10 to pull off - or is there? Is there a number, call it n, such that 10n = 2? Of course there is - it just happens to be 0.301029995664… . In other words, 100.301029995664… = 2. So let‟s redo equation 13 as: 20 log ( 100.301029995664…) 20 log (100.3) = 20 .3 = 6 dB SPL Equation 14 I‟ll leave the arithmetic for you, dear readers, but can you see that if you double the sound pressure of any sound, this will be the same as increasing it by 6 dB. For example, if a sound has a pressure of 37 dB SPL, and if you double its pressure, it will then have a pressure of 37 + 6 = 43 dB SPL. One the other hand, if we were to halve the pressure, this equates to reducing its pressure by 6 dB, i.e., by subtracting 6 dB. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 6 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Exponentials Mindful that some readers may be completely new to exponentiation, let‟s start at the beginning. Exponents are a shorthand method of indications repeated multiplication of a number by itself. So 10 10 is multiplying 10 two times, and is signified by 102 (aka „10 squared‟ and „10 to the 2nd power‟). 10 10 = 102 = 100 Equation 15 Similarly, 10 10 10 10 = 104 = 10,000 (or 10 to the 4th power) Equation 16 and 2 2 2 2 2 2 2 2 2 = 29 = 512 (or 2 to the 9th power) Equation 17 Since the discussion of decibels mainly uses powers of 10, I‟ll stick to these. Exponents really come into their element with very large (or, as we shall see, with very small numbers). Thus, 100,000,000,000,000,000,000 is more easily representated as 1020 (10 to the 20th power). A brief table of exponentiated 10‟s follows: 100 = 1 101 = 10 102 = 100 103 = 1,000 104 = 10,000 105 = 100,000 106 = 1,000,000 etc.6 How would 7,000 be represented exponentially? We could use 10 3.84509804. More usually, however, it is written in the form 7 103 (i.e., 7 1000). Similarly, 93,000,000 (the distance of the earth from the sun in miles) would be 93 106 (93 1,000,000). It could also be written as 930 105 (930 100,000) or 9.3 107 (9.3 10,000,000) - all three forms equal 93,000,000. Multiplication of exponential numbers The multiplication of exponentiated numbers is quite simple. Take the example 1,000 1,000,000 = 1,000,000,000 or 109. Let‟s look at it: 103 106 = 103+6 = 109 Equation 18 6 As a side-track, do you know what a „googol‟ is? Seriously - look it up in a dictionary! It‟s 10100. Written out, a googol is a 1 followed by 100 zeros. Even larger is the „googolplex‟ or 10 googol. Written out, it would be 1 with a googol of zeros following. I‟ve seen a googolplex written out - it takes several hundred pages of small print. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 7 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL In other words, to multiply 10n by 10m, you simply add the exponents m + n. The general rule for the multiplication of exponential number is: 10n 10m = 10n+m Equation 19 A slightly more complicated example would be to multiply 4,700 300. This is done as follows: 4700 300 4.7 10 3 3 10 2 4.7 3 10 3 10 2 14.1 10 32 14.1 105 1,410,000 Equation 20 Easy isn‟t it? If in doubt, try working it out longhand - that should convince you. By the way, the brackets used above aren‟t really necessary - they can be omitted. Division of exponential numbers Division is no less simple. For example, divide 10,000,000 by 1,000. The quotient is obviously 10,000 or 104. Let‟s look at it exponentially: 10 7 3 10 7 10 3 10 7 3 10 4 10,000 10 Equation 21 So, in division, take the denominator (i.e., the divisor), change the sign of its exponent, then multiply it times the numerator (dividend) using the addition-of-exponents rule from above. In general form, the division rule is: 10 n m 10 n 10 m 10 n m 10 n m 10 Equation 22 Try the following division using exponents yourself: 4,800,000 2,400 2,000 One further example is interesting and important - what is 100 200? The answer is obviously 1, but let‟s look at the exponential division: 10 2 2 10 2 10 2 10 2 2 10 0 10 Equation 23 0 So, 10 = 1. Similarly, any number raised to the zero power = 1. Thus, 5 0 = 4,7650 = 1,000,0000 = n0 = 1. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 8 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Negative exponents In the discussion of the decibel, there are many examples of negative exponents, e.g., 10 -16 watts/cm2. What does this number mean? A little thought about the division rule should indicate the truth. Let‟s work backwards - take 10-2. This can be rewritten as follows: 10 2 1 10 2 since any number multiplied by 1 equals itself Equation 24 By applying the division rule in reverse: 10 0 1 1 10 2 10 0 10 2 2 0.01 10 100 Equation 25 -2 Note the change of the sign of 10 when it moves to the denominator. It can be seen, then, that a number with a negative exponent is merely a fraction less than one. Another brief table may be of help: 10-1 = 0.1 10-2 = 0.01 10-3 = 0.001 10-4 = 0.0001 10-5 = 0.00001 etc. Frequency and exponential numbers Exponentiation also typically finds a role in graphing frequency. The frequency range that the human ear can hear is approximately from 20Hz to 20,000Hz. This is called the frequency bandwith of hearing. As we saw with the ear‟s dynamic range, the frequency bandwidth is also very broad. You will frequently encounter graphs that plot sound frequency against sound amplitude (e.g., audiologists use the „audiogram‟ which is a graph of a person‟s hearing sensitivity). Commonly such charts will use a logarithmic axis for frequency. Below is a typical chart (since it‟s the chart itself that‟ of interest for the moment, there‟s no data plotted on it): 120 110 Relative amplitude (dB SPL) 100 90 80 70 60 50 40 30 20 10 0 10 100 1,000 10,000 100,000 Frequency (Hz) 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 9 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL Note the labels on the horizontal axis showing frequency - the numbers progress exponentially as 10, 100, 1,000, 10,000, 100,000 or 101, 102, 103, 104, 105. This use of a logarithmic scale for the axis allows a very wide bandwidth to be displayed on a relatively small graph. (Imagine how wide this page would have to be if the axis labels increased by a contant 100Hz - there would have to be 1,000 ticks on the x axis). One other point about the graph - why do you think the vertical axis is labelled „Relative amplitude‟? Hint: if you don‟t know, think about the meaning of the decibel again. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 10 of 11 02/09/10 THE LIFE AND TIMES OF THE DECIBEL APPENDIX - THE PHYSICAL PATH TO THE DECIBEL Pressure Force: a mass of approximately 100g can produce a force of 1 Newton (N). Pressure: when a force acts (pushes) against a surface or area (A), the result is pressure. Mathematically, this is represented as: f p A For a force of 1N acting against a surface of 1m2 will produce a pressure of 1 Pascal (Pa) or: 1N N 1Pa or, more simply, 2 Pa 1m 2 m Intensity Work: to move a mass of 10.2kg a distance of 1cm requires 1 joule (J) of work, or: 1J 10.2kgcm Power: work (w) done in a given time (t) is power (P), or: w P t 1J applied for 1 second produces 1 watt (W) of power, or: 1J J 1W or, more simply, W sec sec Intensity: If power is applied to a surface, the result is intensity (I), or: P I A 1W applied to a surface of area 1cm2 produces an intensity of 1W/cm2. 9ce626d5-189f-47bf-af92-436325323c5d.doc: dc Page 11 of 11 02/09/10

DOCUMENT INFO

Shared By:

Tags:
sound pressure, frequency range, air pressure, the wave, sound wave, sound field, fundamental frequency, sound waves, sound level, amplitude response, sound amplitude, frequency components, sound power, frequency response, tuning fork

Stats:

views: | 10 |

posted: | 9/2/2010 |

language: | English |

pages: | 11 |

OTHER DOCS BY knm75792

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.