# The Physics of Sound

 Frequency Threshold in Hz in dB SPL 125 45.0 250 25.5 500 11.5 750 8.0 1000 7.0 1500 6.5 2000 9.0 3000 10.0 4000 9.5 6000 15.5 8000 13.0 Answers to Decibel Study Questions 10-12 watts/m2 20 Pa (or, equivalently, 0.0002 dynes/cm2) The threshold of audibility for an average, normal-hearing listener at a particular signal frequency. 3. The threshold of audibility for a particular listener at a particular signal frequency. Consulting the attached figure and table showing the audibility curve for average, normal-hearing listeners, we find that the threshold of audibility at 125 Hz is 45 dBSPL. A listener who barely detected a 125 Hz tone at 55 dBSPL would therefore have hearing loss of 55-45=10 dB; that is, the hearing sensitivity of this listener would be 10 dB worse than normal. Consulting the attached figure and table showing the audibility curve for average, normal-hearing listeners, we find that the threshold of audibility at 1,000 Hz is 7 dBSPL. A listener who barely detected a 1,000 Hz tone at 55 dBSPL would therefore have a hearing loss of 55-7=48 dB; that is, the hearing sensitivity of this listener would be 48 dB worse than normal. The reference for dBHL is the audibility threshold, so this listener would have a 55 dB hearing loss at 125 Hz. There is no need to consult the table. The reference for dBHL is the audibility threshold, so this listener would have a 55 dB hearing loss at 1,000 Hz. There is no need to consult the table. 6 factors of 10 (i.e., 1,000,000 times) more intense than 20 Pa) 6 factors of 10 (i.e., 1,000,000 times) more intense than 10-12 watts/m2 6 factors of 10 (i.e., 1,000,000 times) more intense than a 1,000 Hz tone that is barely audible to an average, normal-hearing listener. a number 20 Pa a sentence The threshold of audibility for an average, normal-hearing listener at a particular signal frequency. 20 dBSL. The reference for the dBSL (SL=sensation level) is the threshold of audibility for a specific listener. So, what we want to know here very simply is where this 90 dBHL tone is in relation to this particular listener’s threshold. This listener has a 70 dB hearing loss at this frequency, so the 90 dBHL tone, which would be 90 dB above a normal-hearing listener’s threshold, is only 20 dB above this particular listener’s threshold. 42 dBSPL: The pressure version of the formula was derived from the intensity version through algebraic manipulations, so they have to be equivalent to one another. The next problem was designed to illustrate how this can be the case. (a) 1,000,000 times (6 factors of 10) more intense than IR. (b) If the intensity ratio is 1,000,000, the pressure ratio has to be the square root of 1,000,000, which is 1,000. (c) dBSPL = 20 log 1,000 = 20 . 3 = 60 dBSPL. This is exactly what we got for the same sound measured in dBIL. It will always be the same. If a sound measures 60 dBIL, that same sound will measure 60 dBSPL. (a) 10,000 times (4 factors of 10) more intense than IR. (b) If the intensity ratio is 10,000, the pressure ratio has to be the square root of 10,000, which is 100. (c) dBSPL = 20 log 100 = 20 . 2 = 40 dBSPL. This is exactly what we got for the same sound measured in dBIL. It will always be the same. If a sound measures 40 dBIL, that same sound will measure 40 dBSPL. See below. The lower of the two marks is 20 dB (2 factors of 10) above the constant reference line of 20 Pa. The higher of the two marks is 20 dB (also 2 factors of 10) above the curvey line, which is the threshold of audibility for the average normal-hearing listener.   1The example of tuning a guitar string is imperfect since the mass of the vibrating portion of the string decreases slightly as the string is tightened. This occurs because a portion of the string is wound onto the tuning key as it is tightened. 2There are some complex periodic signals that have energy at odd multiples of the fundamental frequency only. A square wave, for example, is a signal that alternates between maximum positive amplitude and maximum negative amplitude. The spectrum of square wave shows energy at odd multiples of the fundamental frequency only. Also, a variety of simple signal processing tricks can be used to create signals with harmonics at any arbitrary set of frequencies. For example, it is a simple matter to create a signal with energy at 400, 500, and 600 Hz only. While these kinds of signals can be quite useful for conducting auditory perception experiments, it remains true that most naturally occurring complex periodic signals have energy at all whole number multiples of the fundamental frequency. 3The increase in intensity that would occur as the tuning fork is placed in contact with a hard surface does not mean that additional energy is created. The increase in intensity would be offset by a decrease in the duration of the tone, so the total amount of energy would not increase relative to a freely vibrating tuning fork. 4 Step 4 is the only tricky part of derivation. The reason it works is that squaring a number and then taking a log is the same as taking the log first, and then multiplying the log by 2. For example, note that the two calculations below produce the same result: log 10 1002 = log 10 10,000 = 4 (square first, then take the log) log 10 1002 = (log 10 100) x 2 = 2 x 2 = 4 (take the log, then multiply by 2) 5The standard pressure reference for dBSPL is sometimes given as 0.0002 dynes/cm2 rather than 20 Pa. These two sound pressures are identical, however, in exactly the same sense that 4 quarts and 1 gallon are identical. Likewise, the standard reference for dBIL is often given as 10-16 w/cm2 instead of 10-12 w/m2. These two intensities are also identical. 6A clinical audiometer is an instrument with, among other things, one dial (for each ear) that controls pure-tone frequency and another dial that controls the intensity of the tone. The listener is asked to raise a hand when the tone is barely audible.