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Changing intensity levels of a sound

Show that if one sound is twice as intense as another, it has a sound level about 3 dB higher.

Strategy

We are given that the ratio of two intensities is 2 to 1, and are then asked to find the difference in their sound levels in decibels. We can solve this problem by using of the properties of logarithms.

Solution

  1. Identify knowns:
    The ratio of the two intensities is 2 to 1, or
    I 2 I 1 = 2.00 .

    We wish to show that the difference in sound levels is about 3 dB. That is, we want to show:
    β 2 β 1 = 3 dB.

    Note that
    log 10 b log 10 a = log 10 ( b a ) .
  2. Use the definition of β to obtain
    β 2 β 1 = 10 log 10 ( I 2 I 1 ) = 10 log 10 2.00 = 10 ( 0.301 ) dB .

    Thus,
    β 2 β 1 = 3.01 dB .

Significance

This means that the two sound intensity levels differ by 3.01 dB, or about 3 dB, as advertised. Note that because only the ratio I 2 / I 1 is given (and not the actual intensities), this result is true for any intensities that differ by a factor of two. For example, a 56.0-dB sound is twice as intense as a 53.0-dB sound, a 97.0-dB sound is half as intense as a 100-dB sound, and so on.

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Check Your Understanding Identify common sounds at the levels of 10 dB, 50 dB, and 100 dB.

10 dB: rustle of leaves; 50 dB: average office; 100 dB: noisy factory

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Another decibel scale is also in use, called the sound pressure level    , based on the ratio of the pressure amplitude to a reference pressure. This scale is used particularly in applications where sound travels in water. It is beyond the scope of this text to treat this scale because it is not commonly used for sounds in air, but it is important to note that very different decibel levels may be encountered when sound pressure levels are quoted.

Hearing and pitch

The human ear has a tremendous range and sensitivity. It can give us a wealth of simple information—such as pitch, loudness, and direction.

The perception of frequency is called pitch    . Typically, humans have excellent relative pitch and can discriminate between two sounds if their frequencies differ by 0.3% or more. For example, 500.0 and 501.5 Hz are noticeably different. Musical notes    are sounds of a particular frequency that can be produced by most instruments and in Western music have particular names, such as A-sharp, C, or E-flat.

The perception of intensity is called loudness    . At a given frequency, it is possible to discern differences of about 1 dB, and a change of 3 dB is easily noticed. But loudness is not related to intensity alone. Frequency has a major effect on how loud a sound seems. Sounds near the high- and low-frequency extremes of the hearing range seem even less loud, because the ear is less sensitive at those frequencies. When a violin plays middle C, there is no mistaking it for a piano playing the same note. The reason is that each instrument produces a distinctive set of frequencies and intensities. We call our perception of these combinations of frequencies and intensities tone quality or, more commonly, the timbre    of the sound. Timbre is the shape of the wave that arises from the many reflections, resonances, and superposition in an instrument.

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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