Sound - Longitudinal Waves• Sound is sourced by a vibrating object. A

tuning fork when struck has tines that vibrate back and forth. Movement to the right compresses air molecules ( high density), compression, to the left low density rarefaction. Continued vibrations cause compressions and rarefactions similar to that represented by a sinusoidal curve. Sound in air moves at the RMS of the molecules.

Sound Wave Characteristics

• Audible waves- longitudinal waves with a frequency range 20- 20,000Hz

• Infrasonic waves - longitudinal below the range of hearing ex. Earth quakes

• Ultrasonic waves- longitudinal above the range of hearing ex. Dog whistle

Application of Ultrasonic Waves• These short wavelengths can produce

images by the difference in reflected and absorbed waves ex. Liver and spleen

• Can be used to measure the speed of blood flow by comparison of scattered waves with the incident wave

• By passing an alternating current through a crystal can produce a vibration same as voltage frequency. Incident and reflected waves compared in sonograms

Applications Continued

• Pr =( (I- r)/ (I+ r))2x100

• About 23Hz waves are used to destroy kidney stones and brain tumors

• Ultrasound waves are use as range finders in cameras

Speed of Sound

• In a fluid the speed depends on the bulk modulus ( ability to be compressed)ß, and the equilibrium density . v =√(ß/)

• In a solid speed is given by v = √(Y/) where Y is Young’s modulus This valid only for thin rods.

• Sound speed depends on temperature. In air v = (331m/s) (√T/273k) Where 331 is speed at 0 degrees C

Energy and Intensity of Sound

• The intensity of a wave is given by the rate of energy flow divided by the surface area.

• I = 1/A(ΔE/Δt) = P/A w/m2

• Threshold of hearing I = 1x10-12 w/m2

• Threshold of pain I = 1 w/m2

Intensity in Decibels• The sensation of loudness is an approximate

logarithmic in the human ear. The decibel level ß = 10 log (I/Io) where Io = 1x10-12

w/m2 reference level I = intensity ß = intensity in decibels. Multiplying a given intensity by 10 increases the decibel rating by 10

• Threshold of pain = 10 log(1/ 1x10-12 ) =120

• prolonged exposure to 90 decibels can damage ears

Spherical and plane Waves

• A spherical wave is produced by a source whose radius changes periodically. It is assumed that the intensity is the same in all directions. As area of a sphere is 4r2

• I = ave Power/A =Pave/ 4r2

• I varies to 1/r2 the ratio of two intensities

I1/I2 = r22/r12

Spherical Waves

• Spherical waves are represented by wave fronts ( arcs), the distance between the arcs is the wavelength. Radial lines pointing out from the source are called rays. When considering spherical waves far from the source the rays are almost parallel and small parts of the fronts are considered to approximate plane waves.

The Doppler Effect

• The intensity of sound increases as one moves closer to the source and decreases as one move away from the source. Doppler effect.

• Let vo = speed of observer

vs = speed of source fs = frequency of source

s = wavelength of source

v = speed of sound

Doppler cont.

• When moving towards a source additional wave fronts are encountered

then fo = fs(v+vo/v)

• When moving away from the source the frequency of wave fronts decrease

then fo = fs(v-vo/v)

Doppler cont.• When a source is moving towards a

stationary object

then fo = fs(v/v-vs)

• When the source is moving away from a stationary object

then fo = fs(v/v+vs)

• When both observer and source are moving

then fo = fs(v+vo/v-vs)

Shock Waves

• When the source velocity exceeds the wave velocity a shock wave is produced. The shock wave consists of a cone shaped wave front. Mach Number = vs/v

so sn

Interference of Sound Waves

• Consider a sound source split into two paths and the reunited.

in out

r1

r2

Interference cont.

• A) If r1 and r2 are equal or if one of the r’s is a whole wavelength longer than the other then constructive interference will occur. The sound out will be louder. r1-r2= n

• B) If r1 and r2 differ in length by a half multiple of the wavelength then destructive interference will occur. Sound could be totally canceled. r1-r2= (n+1/2)

• Where n = 0,1,2,3,…….

Standing Waves

• If a string is fixed at one end and vibrated at the other end at the right frequency so that the incident and reflected waves superposition, the resultant wave can appear to be stationary, hence a standing wave.

• When two traveling waves have equal but opposite magnitudes certain point along the medium have a zero net displacement.

Nodes and Antinodes

• Points with zero net displacement are called nodes (N). The distance between nodes is ½ the wavelength of the wave.

• Midway between two adjacent nodes are antinodes (A). Antinodes have maximum diplacement.

Fundamental Frequency

• Consider a string of length L fixed at both ends. The ends are nodes. If the center of the string is displaced and released it becomes the antinode. The distance between N and A is always λ/4. As there are two segments, N-A and A-N the length of the string L = 2(λ/4) = λ/2

• As frequency f = v/λ then f = v/2L as v = √f/µ then f = 1/2L(√F/µ)• The lowest possible frequency of vibration is the

fundamental frequency or First Harmonic

Harmonics

• First harmonics only have nodes at the ends and have the pattern N-A-N

• Second harmonics ( First Overtone) consist of three nodes having the pattern with four segments, N-A-N-A-N. Each segment either N-A or A-N is λ/4 and L = 4(λ2/4) = λ2

• Frequency of the second harmonic f2 = v/λ2 = v/L = 2(v/2L) = 2f1 The frequency of the

second harmonic is twice the fundamental harmonic

Harmonics Series

• Further insertion of nodes produce subsequent harmonics, third harmonic

(second overtone) etc. All harmonics are integer multiples of the fundamental harmonic fn = nf1 = n/2L(√F/µ)

• f1, f2, f3, …….. Is called the harmonic series

Forced Vibration and Resonance

• If a system of some natural frequency fo is pushed back and forth by a periodic force with frequency f, the system will vibrate at frequency f. This is called forced vibration.

• The amplitude is maximum when the driving force equals the natural frequency fo, the resonant frequency. When this occurs the system is in resonance.

Beats

• Beats occur when two sources of different frequencies vibrate causing alternating constructive and destructive interference.

• The constructive interference produces a loud sound alternating with a quieter one. The Louder sound is called the beat.

Sound Quality

• A tuning fork produces a sound based on one harmonic. Musical instruments produce sounds that are mixture of many harmonics. These mixtures of harmonics produce complex wave patterns and the unique sounds particular to each instrument.