ACOUSTICS

CHAPTER 1

Fundamentals of Acoustics and sounds

Sound

• Sound is a result of wave created by vibrating objects, propagated through a medium.

• It is also pressure fluctuations through an elastic medium.

• Sound waves require medium to propagate

Fundamentals of Acoustics and sounds

ACOUSTICS

Study of sound, its production, transmission or propagation through fluid and solid media.

MECHANICAL WAVES

A disturbance travelling through medium transports energy from one location to another.

TYPES OF MECHANICAL WAVES

Transverse wave Longitudinal waves

Rarefaction Compression

Speed of longitudinal wave

The speed of the wave in solid rod depends on the mechanical properties of the rod.

Shear waves

Shear waves exist in solids and very viscous liquids. There is no change in volume and density of materials.

Sound waves in solids are composed of compression waves (just as in gases and liquids), but there is also a different type of sound wave called a shear wave, which occurs only in solids. These different types of waves in solids usually travel at different speeds.

Properties of a Sound Wave

Sound speed in air: 340m/s and in water 1480m/s

Sound Frequency

Infrasound range

• Frequencies below audible range

Audible range

• Frequency range of hearing• Humans: 20-20 000Hz• Child: 15-40 000 Hz• Dog: 20-45 000 Hz• Whale: 1000- 123 000 Hz

Ultrasound range

• Frequencies above audible range.

• Medical applications 2.5 – 40 MHz

Speed of sound

• The speed of sound is the displacement travelled per unit time by a sound wave propagating through an elastic medium.

• It depends on the mechanical properties of a medium.

• The speed of sound varies from substance to another.• Stiffer materials have greater bulk modulus and therefore greater speed of sound.

• Sound travels faster in liquids and non-porous solids than it does in air. Its speed in water (1,484 m/s), and in iron (5,120 m/s).

• Speed depends on : compressibility - Density C = √ B/r

 

Question

Which of the following actions will increase the speed of sound in air? • (a) decreasing the air temperature • (b) increasing the frequency of the sound • (c) increasing the air temperature • (d) increasing the amplitude of the sound wave• (e) reducing the pressure of the air.

Problem

• (a) If a solid bar of aluminum 1.00 m long is struck at one end with a hammer, a pulse propagates in the bar. Find the speed of sound in the bar, which has a Young’s modulus of 7.0 X 1010 Pa and a density of 2.7 X 103 kg/m3.

• (b) Calculate the speed of sound in ethyl alcohol, which has a density of 806 kg/m3 and bulk modulus of 1.0 X 109 Pa.

• (c) Compute the speed of sound in air at 400C.

CHAPTER 2

The nature of sound propagation

• When energy passes through a medium resulting in a wave motion with different types of waves:

• Transverse • Longitudinal• Rotational

• Torsional

Plane wave

• In one-dimensional wave equation, in x-direction:

The displacement after time “t”:y (x , t)= A sin (wt-kx)

Complex waves

• The complex periodic waveform is a sum of harmonically related waves.

• The harmonic relation is that the frequency of one harmonic is the twice that of the other.

• Sound pressure waves are complex and periodic

Sound pressure can be given by the equation:

P(t) = ∑ An sin(nwt + fn) = ∑ Cneiwt

• This equation constitutes a form of Fourier series. Fourier characterizes the complex waves.

• When 2 or more sound waves superimposed, they can combine: their amplitude add algebraically at any point.

Standing waves

standing waves generate when a sound wave is superposed upon another wave in the same frequency but in different direction.

P1(t)+p2(t) = A1sin(2pft-kx)+ A2sin(2pft+kx)

P1(t)+p2(t) = 2A1 cos2pft . sin kx

At f1: L = ½ l

At f2: L = l

At f3: L =3/2 l

Standing wavesStanding waves can be set up in a stretched string by connecting one end of

the string to a stationary clamp and connecting the other end to a vibrating

object, such as the end of a tuning fork, or by shaking the hand holding the

string up and down at a steady rate. Traveling waves then reflect from the

ends and move in both directions on the string. The incident and reflected

waves combine according to the superposition principle. If the string vibrates

at exactly the right frequency, the wave appears to stand—hence its name,

standing wave. A node occurs where the two traveling waves always have the

same magnitude of displacement but the opposite sign, so the net

displacement is zero at that point. There is no motion in the string at the

nodes, but midway between two adjacent nodes, at an antinode, the string

vibrates with the largest amplitude.

• Since: v = l1f1

• Then: f1 = v / l1

• Where at f1: L = ½ l1 l1 =2L f1= v/2L

• where F is the tension in the string and mis its mass per unit length.

Example (1) {1} The high E string on a certain guitar measures 64.0 cm in length and has a fundamental frequency of 329 Hz. When a guitarist presses down so that the string is in contact with the first fret, the string is shortened and the frequency becomes 349 Hz. (a)How far is the first fret from the nut?

Example (2)

• (a) Find the frequencies of the fundamental, second, and third harmonics of a steel wire 1.00 m long with a mass per unit length of 2.00 x 103 kg/m and under a tension of 80.0 N.

• (b) Find the wavelengths of the sound waves created by the vibrating wire for all three modes. Assume the speed of sound in air is 345 m/s.

Doppler effect

If a car, the frequency of the sound you hear is higher as the vehicle approaches you and lower as it moves away from you. This is one example of the Doppler effect.

Example

• A train moving at a speed of 40.0 m/s sounds its whistle, which has a frequency of 5.00 x 102 Hz. Determine the frequency heard by a stationary observer as the train approaches the observer. The ambient temperature is 24.0C.

Root mean square sound pressure

= Pm/√2

• The sound pressure is an oscillation of a pressure above and

below the atmospheric pressure.

• It is detected by human ear as low as (20 m Pa).

• The threshold of pain is 40 000 000 m Pa.

• It is convenient to use the decibel scale.

Sound Intensity

• The sound power radiated by the source.

• W or P = ∫ I. ds

• I = • This is the inverse square law.

w

Power and Intensity

Intensity• It is analogous to the brightness

of light.• It is a power passing through a

unit area (W/m-2).• The range of human hearing :

• 10-12 – 1 watt/m-2

Power• It is analogous to the power of

light measured in watts.

Decibel B = log (I/I0)

1 B = 10 dB• The decibel is a logarithmic scale used to compare two as power

gain of 2 sources.

• DL = 10 Log (W2/W1) dB

• The decibel is the difference between 2 power levels.

Sound Power and Intensity level

• The sound power level is expressed using the threshold of audibility W0 = 10-12 as a reference:

Lw = 10 log W/W0

Intensity & Intensity Level

• Don’t confuse intensity with intensity level. • Intensity is a physical quantity with units of watts per meter

squared.

• intensity level, or decibel level, is a convenient mathematical transformation of intensity to a logarithmic scale.

Sound pressure level

• The intensity “I” is:I = P2/(rc)

The sound pressure level Lp is equivalent to sound intensity :

Lp = 10 Log (I/I0) = 20 Log(p/p0)

Multiple sources

• How much the sound wave increases when 2 sound sources are used simultaneously?

• The sound power level would double:L = 10 Log (2W/W0) = 3dB+10Log(W/W0)

• So the increase is 3dB in the sound power level.

Example

• A noisy grinding machine in a factory produces a sound intensity of 1.00 X 105 W/m2. Calculate:

• (a) the decibel level of this machine.• (b) the new intensity level when another identical machine is

added to the factory.• (c) A certain number of additional such machines are put into

operation along side these two. When all the machines are running at the same time the decibel level is 77.0 dB. Find the sound intensity.

Energy density

• Kinetic energy and potential energy are involved in sound propagation.

• Interchange between these two energies occurs from the compression and rarefaction and the motion of the propagated medium particles.

Acoustic impedance

• When a pressure is applied to a molecule it will exert pressure on the adjacent molecule.

• The pressure propagates through medium and depends on the particle speed and the properties of the medium.

Z = p/u• Acoustic impedance depends on the elasticity of the medium.