Figure 1 A sinusoidal relation between displacement and time. The period (Pr) is the time it takes to complete one cycle, and A is the peak amplitude. Period is 10 msec, frequency is 100 Hz, peak amplitude is 10, and starting phase is 0°.

period and describes how often per unit of time the object oscillates. Starting phase describes the relative starting point of the vibration.

A powerful theorem, Fourier's theorem, states that any arbitrary physical vibratory pattern consists of a sum of sinusoidal vibrations (Fig. 1). Sinusoidal vibration is a basic unit of vibration, and thus of sound. It is therefore not surprising that much of what is known about hearing has been derived from the study of sinusoidal vibrations and sounds. The sinusoidal relationship is written as A(t)=A sin (2pft + 0), where A(t) is the instantaneous amplitude, A is the peak amplitude, f is frequency, t is time, and 0 is the starting phase.

1. Intensity

The amplitude of vibration can be expressed as the instantaneous amplitude [A(t)], peak amplitude (A), peak-to-peak amplitude, or the root mean square amplitude. When an object vibrates it can produce pressure, which is proportional to amplitude. A vibrating object can also produce work. Work is measured in units of power or energy. Sound intensity is the measure ofsound magnitude in units ofpower or energy. Thus, sound magnitude can be expressed in units of amplitude (displacement), pressure, or intensity (power or energy).

The dynamic range of sound intensity over which the human auditory system functions is approximately 1013. In order to deal with this large dynamic range, sound intensity is often converted to decibels (dB) by dB=10 log10 (/1//0) or equivalently 20 logw (P1/P0) since sound intensity (I) is proportional to pressure (p) squared. Therefore, in decibels the dynamic range of hearing is 130 dB (10 x log101013=130 dB). It is common to use 20 (mPa) as the referent pressure (p0) in calculating decibels since the threshold for hearing sinusoidal sounds is approximately 20 mPa. When this reference is used, the sound is measured in dB sound pressure level (SPL); therefore, a sound level of 50 dB SPL is a sound that is 50 dBs more intense than 20 mPa, (i.e., 50 dB higher than the level of the softest sound humans can detect).

2. Frequency

Frequency (f) is measured in hertz (Hz), where f in Hz=1/Pr, when Pr is the period of vibration and is measured in seconds (Fig. 1). A vibration that oscillates with a frequency of n Hz goes through n complete vibratory cycles in 1 sec. Many animals can hear over a range of sinusoidal frequencies from a few hertz to tens of thousands of hertz. The human range of hearing is approximately 20-20,000 Hz.

3. Starting Phase

The starting phase (0) of vibration is measured in angular units of degrees or radians. When a sinusoid completed one cycle, it has gone through 360° (2p radians) since sinusoidal motion can be derived from the projection of a point on a rotating circle. The sinusoidal motion that begins at zero amplitude at time zero, as shown in Fig. 1, is defined as having zero starting phase. All other possible starting phases are relative to this zero starting phase condition. Humans are only sensitive to changes in starting phase in certain situations.

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