Waves and Sound

Sound is a mechanical wave produced by vibrating bodies.

When the air vibrations reach the ear, they cause the eardrum to vibrate; this produces nerve impulses that are interpreted by the brain.

All matter transmits sound to some extend, but a material medium is needed between the source and the receiver to propagate sound.

Properties of Sound

Types of wavesLongitudinal waves: The vibration of the particles of the medium is along the direction of the wave’s motion.

Transverse waves: The vibration of the particles of the medium is perpendicular to the direction of the wave’s motion.

Properties of Sound

Two important characteristics of sound are intensity, which is determined by the magnitude of compression and rarefaction in the propagating medium, and frequency, which is determined by how often the compressions and rarefactions take place.

Frequency is measured in cycles per second, which is designated by the unit hertz (Hz) after the scientist Heinrich Hertz.

1 Hz = 1 cycle per second

The distance between two least-density points is called the wavelength.

The speed of sound v is given by

fv

Energy transported by sound

The pressure variations due to the propagating of a sinusoidal sound wave is of the form

where is the maximum pressure change, and f is the frequency of the sound. The amount of energy transmitted by a sinusoidal sound wave per unit time through each unit area perpendicular to the direction of sound propagation is called the intensity I and is given by

Here is the density of the medium, and v is the speed of the sound propagation.

v

PI

2

20

0P

)2sin(0 ftPP

Fundamental and Harmonics

In most cases, the sound wave patterns produced by instruments and voices are highly complex.

The lowest frequency in the wave form is called the fundamental, and the higher frequencies are called harmonics. It is the harmonic content of the sound that differentiates one sound from another.

Fourier decomposition

However, complex wave shapes can be analyzed into simple sinusoidal waves of different frequencies by Fourier Analysis.

Mathematical representation of a traveling wave

+ : the wave moves to the right

)sin()22

sin(),( 00 tkxPftxPtxP

numberwave2

k

frequencyangular2 f

- : the wave moves to the left

Reflection and refraction

Law of reflection: the angle of reflection equals the angle of incidence.

When a wave traveling one medium crosses a boundary into a medium where its speed is different, the transmitted wave may move in a different direction than the incident wave. The phenomenon is known as refraction.

1

2

1

2

sin

sin

v

v

Law of refraction:

When sound traveling in air is incident perpendicular to a water surface, only about 0.1% of the sound energy enters the water; 99.9% is reflected.

Diffraction

Waves have a tendency to spread as they propagate through a medium. As a result, when a wave encounters an obstacle, it spreads into the region behind the obstacle. This phenomenon is called diffraction.

The longer the wavelength, the greater is the spreading of the wave. Significant diffraction into the region behind the obstacle occurs only if the size of the obstacle is smaller than the wavelength.

Interference

When waves travel simultaneously in the same medium, the total disturbance in the medium is at each point the vectorial sum of the individual disturbances produced by each wave. This phenomenon is called interference.

Interference

)cos()sin(2)sin()sin( tkxAtkxAtkxA

Constructive interference: Waves are in phase.

Destructive interference: Waves are out of phase.

Standing wave: Two waves of the same frequency and magnitude traveling in opposite directions. The resultant wave pattern is stationary in space.

Beats – interference in time

)2

cos()2

sin(2)sin()sin( 212121 ttAtAtA

)2

sin()]2

cos(2[ 2121 ttA

Characteristics of sound

The speed of sound is different in different materials. In air at 0°C and 1 atm, sound travels at a speed of 331 m/s. (Table 16.1)

In liquids and solids, which are much less compressible, the speed is larger.

The speed of sound in air increases in temperature:

s/m)60.0331( Tv

Hearing and the ear

The ear is much more sensitive to pressure variations than any other part of the body.

2.5 cm

Air inside Liquid

inside

Hearing and the ear

The ear is much more sensitive to pressure variations than any other part of the body.

Uncoil the cochlea

3.5 cm

Frequency and Pitch

The human ear is capable of detecting sound at frequencies between about 20 and 20,000 Hz, and most sensitive to frequencies between 200 and 4,000 Hz.

The sensation of pitch increases with frequency. The frequency of middle C is 256 Hz. There is, however, no simple mathematical relationship between and frequency.

Test your ear

Intensity and Loudness

Logarithmic intensity (in dB) = )log(100I

I

where I is sound intensity in ; is the intensity of a chosen reference level, normally chosen to be

Threshold of hearing:

Threshold of pain:

1610 2cm/W2cm/W410

1610 2cm/W0I2cm/W

Bats and Echoes

The Vespertilionidae family of bats emit short chirps as they fly. The chirps last about 3 msec with a time interval between chirps of about 70 msec. Each chirp starts at a frequency of about 100,000 Hz and falls to about 30,000 Hz at the end.

As the bat comes closer to the object, both the duration of and the spacing between chirps decrease, allowing the bat to localize the object more accurately.

Experiments have shown that with echo location bats can avoid wire obstacles with diameters down to about 0.1 mm, but they fail to avoid finer wires.

Ultrasonic waves

With special electronically driven crystals, it is possible to produce mechanical waves at very high frequencies, up to millions of cycles per second. These waves are called ultrasonic waves. Because of their short wavelength, ultrasonic waves can be focused onto small areas and can be imaged much as visible light.

Ultrasound is now routinely used to destroy kidney and gall stones.

Ultrasound is now routinely used to destroy kidney and gall stones.

Stone size less than 2 mm

Doppler effect

The frequency of sound detected by an observer depends on the relative motion between the source and the observer. This phenomenon is called the Doppler effect. If the observer is stationary and the source is in motion, the frequency f’ detected by the observer is given by

svv

vff

'

Where f is the frequency in the absence of motion, v is the speed of sound, and is the speed of the source. The minus sign in the denominator is to be used when the source is approaching the observer, and the plus sign when the source is receding.

sv

Shock waves

When a source of sound moves at subsonic speeds (less than the speed of sound), the pitch of the sound is altered as we have seen (the Doppler effect). But if a source of sound moves faster than the speed of sound, a more dramatic effect known as a shock wave occurs.

•一人搭超音速火車，途中向其站在地面的女友喊說 : 「我愛妳」。請問他女友聽到的是「我愛妳」 ，還是「妳愛我」？

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