Figure 1. Noise on crowded roadways like this one in Delhi makes it hard to hear others unless they shout. In a quiet forest, you can sometimes hear a single leaf fall to the ground.
After settling into bed, you may hear your blood pulsing through your ears. But when a passing motorist has his stereo turned up, you cannot even hear what the person next to you in your car is saying. We are all very familiar with the loudness of sounds and aware that they are related to how energetically the source is vibrating.
In cartoons depicting a screaming person or an animal making a loud noise , the cartoonist often shows an open mouth with a vibrating uvula, the hanging tissue at the back of the mouth, to suggest a loud sound coming from the throat Figure 2. The relevant physical quantity is sound intensity, a concept that is valid for all sounds whether or not they are in the audible range.
Intensity is defined to be the power per unit area carried by a wave. Power is the rate at which energy is transferred by the wave. The intensity of a sound wave is related to its amplitude squared by the following relationship:. We are using a lower case p for pressure to distinguish it from power, denoted by P above. This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its pressure amplitude, the more the air is compressed in the sound it creates.
Figure 2. Graphs of the gauge pressures in two sound waves of different intensities. The more intense sound is produced by a source that has larger-amplitude oscillations and has greater pressure maxima and minima. Because pressures are higher in the greater-intensity sound, it can exert larger forces on the objects it encounters. Sound intensity levels are quoted in decibels dB much more often than sound intensities in watts per meter squared.
Decibels are the unit of choice in the scientific literature as well as in the popular media. The reasons for this choice of units are related to how we perceive sounds. How our ears perceive sound can be more accurately described by the logarithm of the intensity rather than directly to the intensity. In particular, I 0 is the lowest or threshold intensity of sound a person with normal hearing can perceive at a frequency of Hz. Sound intensity level is not the same as intensity.
The units of decibels dB are used to indicate this ratio is multiplied by 10 in its definition. The bel, upon which the decibel is based, is named for Alexander Graham Bell, the inventor of the telephone. That is, the threshold of hearing is 0 decibels. Table 1 gives levels in decibels and intensities in watts per meter squared for some familiar sounds.
One of the more striking things about the intensities in Table 1 is that the intensity in watts per meter squared is quite small for most sounds.
The ear is sensitive to as little as a trillionth of a watt per meter squared—even more impressive when you realize that the area of the eardrum is only about 1 cm 2 , so that only 10 —16 W falls on it at the threshold of hearing! Air molecules in a sound wave of this intensity vibrate over a distance of less than one molecular diameter, and the gauge pressures involved are less than 10 —9 atm.
Another impressive feature of the sounds in Table 1 is their numerical range. Sound intensity varies by a factor of 10 12 from threshold to a sound that causes damage in seconds. You are unaware of this tremendous range in sound intensity because how your ears respond can be described approximately as the logarithm of intensity. Thus, sound intensity levels in decibels fit your experience better than intensities in watts per meter squared. The decibel scale is also easier to relate to because most people are more accustomed to dealing with numbers such as 0, 53, or than numbers such as 1.
For example, a 90 dB sound compared with a 60 dB sound is 30 dB greater, or three factors of 10 that is, 10 3 times as intense. A unit of a logarithmic scale of power or intensity called the power level or intensity level. The decibel is defined as one tenth of a bel where one bel represents a difference in level between two intensities I 1 , I 0 where one is ten times greater than the other.
Thus, the intensity level is the comparison of one intensity to another and may be expressed:. This entire range of intensities can be expressed on a scale of dB. The disturbance then travels from particle to particle through the medium, transporting energy as it moves.
The energy that is carried by the disturbance was originally imparted to the medium by the vibrating string. The amount of energy that is transferred to the medium is dependent upon the amplitude of vibrations of the guitar string. If more energy is put into the plucking of the string that is, more work is done to displace the string a greater amount from its rest position , then the string vibrates with a greater amplitude. The greater amplitude of vibration of the guitar string thus imparts more energy to the medium, causing air particles to be displaced a greater distance from their rest position.
Subsequently, the amplitude of vibration of the particles of the medium is increased, corresponding to an increased amount of energy being carried by the particles. This relationship between energy and amplitude was discussed in more detail in a previous unit.
The amount of energy that is transported past a given area of the medium per unit of time is known as the intensity of the sound wave. The greater the amplitude of vibrations of the particles of the medium, the greater the rate at which energy is transported through it, and the more intense that the sound wave is. As a sound wave carries its energy through a two-dimensional or three-dimensional medium, the intensity of the sound wave decreases with increasing distance from the source.
The decrease in intensity with increasing distance is explained by the fact that the wave is spreading out over a circular 2 dimensions or spherical 3 dimensions surface and thus the energy of the sound wave is being distributed over a greater surface area.
The diagram at the right shows that the sound wave in a 2-dimensional medium is spreading out in space over a circular pattern. Since energy is conserved and the area through which this energy is transported is increasing, the intensity being a quantity that is measured on a per area basis must decrease.
The mathematical relationship between intensity and distance is sometimes referred to as an inverse square relationship. The intensity varies inversely with the square of the distance from the source.
So if the distance from the source is doubled increased by a factor of 2 , then the intensity is quartered decreased by a factor of 4. Similarly, if the distance from the source is quadrupled, then the intensity is decreased by a factor of Applied to the diagram at the right, the intensity at point B is one-fourth the intensity as point A and the intensity at point C is one-sixteenth the intensity at point A.
Since the intensity-distance relationship is an inverse relationship, an increase in one quantity corresponds to a decrease in the other quantity. And since the intensity-distance relationship is an inverse square relationship, whatever factor by which the distance is increased, the intensity is decreased by a factor equal to the square of the distance change factor.
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