Acoustic Standing Waves and resonance

Leacture

Dr.khitam Y. Elwasife

Acoustics

is the science that deals with the study of all mechanical waves in gases, liquids, and

solids including vibration, sound, ultrasound and infrasound .

A scientist who works in the field of acoustics is an acoustician while someone

working in the field of acoustics technology may be called an acoustical engineer.

The application of acoustics is present in almost all aspects of modern society with

the most obvious being the audio and noise control industries

When waves are combined in systems with boundary conditions, only certain allowed frequencies can exist.

We say the frequencies are quantized.

Quantization is at the heart of quantum mechanics.

The analysis of waves under boundary conditions explains many quantum phenomena.

Quantization can be used to understand the behavior of the wide array of musical instruments that are based on strings and air columns.

Waves can also combine when they have different frequencies.

Quantization

Waves vs. Particles

Waves are very different from particles.

Particles have zero size. Waves have a characteristic size – their

wavelength.

Multiple particles must exist at

different locations.

Multiple waves can combine at one point in

the same medium – they can be present at

the same location.

Introduction

Superposition Principle

Waves can be combined in the same location in space.

To analyze these wave combinations, use the superposition principle:

If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves.

Waves that obey the superposition principle are linear waves.

For mechanical waves, linear waves have amplitudes much smaller than

their wavelengths.

Superposition and Interference

Two traveling waves can pass through each other without being destroyed or

altered.

A consequence of the superposition principle.

The combination of separate waves in the same region of space to produce a

resultant wave is called interference.

The term means waves pass through each other.

Superposition Example

Two pulses are traveling in opposite

directions (a).

The wave function of the pulse

moving to the right is y1 and for the

one moving to the left is y2.

The pulses have the same speed but

different shapes.

The displacement of the elements is

positive for both.

When the waves start to overlap (b),

the resultant wave function is y1 + y2.

Superposition Example,

When crest meets crest (c) the

resultant wave has a larger amplitude

than either of the original waves.

The two pulses separate (d).

They continue moving in their original directions.

The shapes of the pulses remain unchanged.

This type of superposition is called constructive interference.

Destructive Interference Example

Two pulses traveling in opposite directions.

Their displacements are inverted with respect to each other.

When these pulses overlap, the resultant pulse is y1 + y2.

This type of superposition is called destructive

interference.

Constructive interference occurs when the displacements caused by the two pulses are in the same direction.

The amplitude of the resultant pulse is greater than either individual pulse.

Destructive interference occurs when the displacements caused by the two pulses are in opposite directions.

The amplitude of the resultant pulse is less than either individual pulse.

Types of Interference

Radiation from a monopole source

A monopole is a source which radiates sound equally well in all directions. The simplest example of a monopole source would be a sphere whose radius alternately expands and contracts sinusoidally. The monopole source creates a sound wave by alternately introducing and removing fluid into the surrounding area. A boxed loudspeaker at low frequencies acts as a monopole. The pattern for a monopole source is shown in the figure at right.

.

A dipole source consists of two monopole

sources of equal strength but opposite

phase and separated by a small distance

compared with the wavelength of sound.

While one source expands the other source

contracts. The result is that the fluid (air)

near the two sources sloshes back and

forth to produce the sound. A sphere which

oscillates back and forth acts like a dipole

source, as does an unboxed loudspeaker .

A dipole source does not radiate sound in

all directions equally. The pattern shown at

right; there are two regions where sound is

radiated very well, and two regions where

sound cancels.

.

Radiation from a dipole source

Radiation from a r quadrupole source

Superposition of Sinusoidal Waves

Assume two waves are traveling in the same direction in a linear medium, with the same frequency, wavelength and amplitude.

The waves differ only in phase:

y1 = A sin (kx - wt)

y2 = A sin (kx - wt + f)

y = y1+y2 = 2A cos (f /2) sin (kx - wt + f /2)

The resultant wave function, y, is also sinusoida.l

The resultant wave has the same frequency and wavelength as the original waves.

The amplitude of the resultant wave is 2A cos (f / 2) .

The phase of the resultant wave is f / 2.

Sinusoidal Waves with Constructive-destructive Interference

When f = 0, then cos (f/2) = 1

The amplitude of the resultant wave is 2A.

The crests of the two waves are at the same location in space.

The waves are everywhere in phase.

The waves interfere constructively.

In general, constructive interference occurs when cos (Φ/2) = ± 1.

That is, when Φ = 0, 2π, 4π, … rad

When Φ is an even multiple of π

When f = p, then cos (f/2) = 0

Also any odd multiple of p

The amplitude of the resultant wave is 0.

See the straight red-brown line in the figure.

The waves interfere destructively

Sinusoidal Waves, General Interference

When f is other than 0 or an even multiple of p, the amplitude of the resultant is between 0 and 2A. The wave functions still add

The interference is neither constructive nor destructive.

Constructive interference occurs when f = np where n is an even integer (including 0).

Amplitude of the resultant is 2A

Destructive interference occurs when f = np where n is an odd integer. Amplitude is 0

General interference occurs when 0 < f < np

Amplitude is 0 < Aresultant < 2A

Interference in Sound Waves

Sound from S can reach R by two different

paths.

The distance along any path from speaker

to receiver is called the path length, r.

The lower path length, r1, is fixed.

The upper path length, r2, can be varied.

Whenever Dr = |r2 – r1| = n l, constructive

interference occurs.a maximum in sound

intensity is detected at the receiver.

Whenever Dr = |r2 – r1| = (nl)/2 (n is odd),

destructive interference occurs. No sound is

detected at the receiver.

A phase difference may arise between two

waves generated by the same source when

they travel along paths of unequal lengths.

Standing wave

When a sound wave hits a wall, it is partially absorbed and partially reflected. A person

far enough from the wall will hear the sound twice. This is an echo. In a small room

the sound is also heard more than once, but the time differences are so small. This

is known as reverberation.

Music is the Sound that is produced by instruments or voices. To play most musical

instruments you have to create standing waves on a string or in a tube or pipe.

The pitch of the sound is equal to the frequency of the wave. The higher the frequency,

the higher is the pitch.

Wind instruments produce sounds by means of vibrating air columns. To play a wind

instrument you push the air in a tube with your mouth. The air in the tube starts to

vibrate with the same frequency as your lips. Resonance increases the amplitude of

the vibrations, which can form standing waves in the tube. The length of the air

column determines the resonant frequencies. The mouth or the reed produces a

mixture if different frequencies, but the resonating air column amplifies only the

natural frequencies. The shorter the tube the higher is the pitch. Many instruments

have holes, whose opening and closing controls the effective pitch.

Standing Waves--Free/Open Ends This shows longitudinal standing waves, such as in sound waves in a pipe open at both ends. The top wave is the fundamental or first harmonic. It has displacement antinodes at each end, and a displacement node in the middle. When the displacement is large and to the right at one end, it is large and to the left at the other, and they're separated by one node. In other words, it is one-half wavelength from one end to the other. The next two waves show the next two harmonics, the second and the third. The pipe holds, respectively, two and three half-wavelengths, and shorter wavelengths correspond to higher frequencies--the frequencies are two and three times the fundamental.

L

Type of air column

Closed at both ends

Closed at one end

open at the other

Open at both ends

The longest standing wave in a tube of length L with two open ends has displacement antinodes (pressure nodes) at both ends. It is called the fundamental.

The longest standing wave in a tube of length L with

two open ends has displacement antinodes

(pressure nodes) at both ends. It is called the

fundamental.

The next longest standing wave in a tube of length L with two open ends is the second harmonic.

It also has displacement antinodes at each end.An integer number of half wavelength have to fit into the tube of length L. L = nλ/2, λ = 2L/n, f = v/λ = nv/(2L). For a tube with two open ends all frequencies fn = nv/(2L) = nf1, with n equal to an integer, are natural frequencies.

The quality of a sound depends on the relative

intensities of the waves with the natural

frequencies. It depends on the spectrum of the

sound. The sound quality of a musical

instrument is called its timbre.

A sinusoidal sound wave of frequency f is a

pure tone. A note played by a musical

instrument is not a pure tone. Its wave

function is not sinusoidal, i.e. it is not of the

form ∆P(x,t) = ∆Pmaxsin(kx - ωt + φ). The wave

function is a sum of sinusoidal wave functions

with frequencies nf, (n = 1, 2, 3, ...,) with

different amplitudes, which decrease as n

increases. The harmonic waves with different

frequencies which sum to the final wave are

called a Fourier series. Breaking up the

original sound wave into its sinusoidal

components is called Fourier analysis.

Standing Waves in air column

Sound wave in a pipe with one closed and one open end (stopped pipe)

21

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

equilibrium position of particles

instantaneous position of

particles

sine curve showing

instantaneous displacement of

particles from equilibrium

instantaneous pressure

distribution

time averaged pressure

fluctuations

Enter t/T

Normal modes in a pipe with an open and a closed end (stopped pipe)

,...)5,3,1(4

4 n

n

LornL n

n ll

,...)5,3,1(4

nL

vnfn

Standing waves in air column

23

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

position along column

CP 523

25

Problem

What are the natural frequencies of vibration for a human ear?

Why do sounds ~ (3000 – 4000) Hz appear loudest?

Solution

Assume the ear acts as pipe open at the atmosphere and closed at the eardrum.

The length of the auditory canal is about 25 mm.

Take the speed of sound in air as 340 m.s-1.

L = 25 mm = 0.025 m v = 340 m.s-1

For air column closed at one end and open at the other

L = l1 / 4 l1 = 4 L f1 = v / l1 = (340)/{(4)(0.025)} = 3400 Hz

When the ear is excited at a natural frequency of vibration large amplitude

oscillations (resonance) sounds will appear loudest ~ (3000 – 4000) Hz.

Standing Waves

Assume two waves with the same amplitude, frequency and wavelength, traveling in opposite directions in a medium.

The waves combine in accordance with the waves in interference model.

y1 = A sin (kx – wt) and

y2 = A sin (kx + wt)

They interfere according to the superposition principle.

The resultant wave will be y = (2A sin kx) cos wt.

This is the wave function of a standing wave.

There is no kx – wt term, and therefore it is not a traveling wave.

In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves.

yright Asin 2px

l ft

A sin 2px

l

cos2p ft cos 2p

x

l

sin2p ft

yleft Asin 2px

l ft

A sin 2px

l

cos2p ft cos 2p

x

l

sin2p ft

yright yleft 2Asin 2px

l

cos2p ft

Note on Amplitudes

There are three types of amplitudes used in describing waves.

The amplitude of the individual waves, A

The amplitude of the simple harmonic motion of the elements in the

medium,

2A sin kx, A given element in the standing wave vibrates within the

constraints of the envelope function 2 A sin k x.

The amplitude of the standing wave, 2A

Standing Waves, Definitions

A node occurs at a point of zero amplitude.

These correspond to positions of x where

An antinode occurs at a point of maximum displacement, 2A.

These correspond to positions of x where

0,1, 2, 3,2

nx n

l

1, 3, 5,4

nx n

l

Nodes and Antinodes

The diagrams above show standing-wave patterns produced at various times by two waves of equal amplitude traveling in opposite directions.

In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c).

Standing Waves in a String

Consider a string fixed at both ends, The string has length L.

Waves can travel both ways on the string.

Standing waves are set up by a continuous

superposition of waves incident on and reflected

from the ends. There is a boundary condition on

the waves. The ends of the strings must necessarily be nodes.

They are fixed and therefore must have zero displacement.

The boundary condition results in the string having a set of natural patterns of oscillation, called normal modes.

Each mode has a characteristic frequency.

This situation in which only certain frequencies of oscillations are allowed is called quantization.

The normal modes of oscillation for the string can be described by imposing the requirements that the ends be nodes and that the nodes and antinodes are separated by λ/4.

We identify an analysis model called waves under boundary conditions.

Note the stationary outline that results from the superposition of two identical waves traveling in opposite directions.

The amplitude of the simple harmonic motion of a given element is 2A sin kx.

This depends on the location x of the element in the medium.

Each individual element vibrates at w

Standing Wave Example

Standing Waves in a String

Consecutive normal modes add a loop at each step.

The section of the standing wave from one

node to the next is called a loop.

The second mode (b) corresponds to to l = L.

The third mode (c) corresponds to l = 2L/3.

This is the first normal mode that is consistent with

the boundary conditions.

There are nodes at both ends.

There is one antinode in the middle.

This is the longest wavelength mode:

½l1 = L so l1 = 2L

The section of the standing wave between nodes is

called a loop.

In the first normal mode, the string vibrates in one

loop.

Standing Waves on a String, Summary

The wavelengths of the normal modes for a string of length L fixed at both ends are ln = 2L / n n = 1, 2, 3, …

n is the nth normal mode of oscillation

These are the possible modes for the string:

The natural frequencies are

Also called quantized frequencies

ƒ2 2

n

v n Tn

L L

The fundamental frequency corresponds to n = 1.

It is the lowest frequency, ƒ1

The frequencies of the remaining natural modes are integer multiples of the fundamental frequency.

ƒn = nƒ1

Frequencies of normal modes that exhibit this relationship form a harmonic series.

The normal modes are called harmonics.

Waves on a String, Harmonic Series

Resonance,

Because an oscillating system exhibits a large amplitude when driven at any of its

natural frequencies, these frequencies are referred to as resonance frequencies.

If the system is driven at a frequency that is not one of the natural frequencies, the

oscillations are of low amplitude and exhibit no stable pattern.

Resonance occurs when a system is able to store and easily transfer energy between

two or more different storage modes .However, there are some losses from cycle to

cycle, called damping. When damping is small, the resonant frequency is

approximately equal to the natural frequency of the system, which is a frequency of

unforced vibrations. Some systems have multiple, distinct, resonant frequencies.

Resonance phenomena occur with all types of vibrations or waves there is mechanic

cal resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic

resonance(NMR), electron spin resonance (ESR) and resonance of quantum wave

functions. Resonant systems can be used to generate vibrations of a specific

frequency (e.g., musical instruments),

Resonance

• When we apply a periodically varying force to a

system that can oscillate, the system is forced to

oscillate with a frequency equal to the frequency of

the applied force (driving frequency): forced

oscillation. When the applied frequency is close to a

characteristic frequency of the system, a

phenomenon called resonance occurs.

• Resonance also occurs when a periodically varying force

is applied to a system with normal modes.

•of the applied

force is close to one of normal modes of the system,

resonance

occurs.

35

Why does a tree howl? The branches of trees vibrate because of the wind.

The vibrations produce the howling sound.

N A

Length of limb L = 2.0 m

Wave speed in wood v = 4.0103 m.s-1

Fundamental L = l / 4 l = 4 L

v = f l

f = v / l = (4.0 103) / {(4)(2)} Hz

f = 500 Hz

Fundamental mode of vibration

Problem

The sound waves generated by the

fork are reinforced when the length

of the air column corresponds to one

of the resonant frequencies of the

tube. Suppose the smallest value of

L for which a peak occurs in the

sound intensity is 9.00 cm.

Problem

Resonance

Lsmalles t= 9.00 cm

(a) Find the frequency of the

tuning fork.

-1 2

11 345 m.s 9.00 10 mn v L

1 2

1

345 Hz 985 Hz

4 4(9.00 10

vf

L

(b) Find the wavelength and the next two water levels giving resonance.

2

14 4(9.00 10 ) m 0.360 mLl

m. 450.02/ m, 270.02/ 2312 ll LLLL

Beats and interference

Wave interference is the phenomenon that occurs when two waves meet while

traveling along the same medium. The interference of waves causes the medium

to take on a shape that results from the net effect of the two individual waves

upon the particles of the medium.

if two upward displaced pulses having the same shape meet up with one another

while traveling in opposite directions along a medium, the medium will take on

the shape of an upward displaced pulse with twice the amplitude of the two

interfering pulses. This type of interference is known as constructive

interference

If an upward displaced pulse and a downward displaced pulse having the same

shape meet up with one another while traveling in opposite directions along a

medium, the two pulses will cancel each other's effect upon the displacement of

the medium and the medium will assume the equilibrium position. This type of

interference is known as destructive interference

The diagrams below show two waves - one is blue and the other is red -

interfering in such a way to produce a resultant shape in a medium; the

resultant is shown in green. In two cases (on the left and in the middle),

constructive interference occurs and in the third case (on the far right,

destructive interference occurs.

An animation below shows two sound waves interfering constructively in order to produce very large oscillations in pressure at a variety of anti-nodal locations. Note that compressions are labeled with a C and rarefactions are labeled with an R.

Beats

Two sound waves with different but close frequencies give rise

to BEATS s1 x,t sm cosw1tConsider

s2 x,t sm cosw2tw1 w2

s s1 s2 2sm cos w t coswt

cos cos 2cos1

2 cos

1

2

w 1

2w1 w2 Very

small w

1

2w1 w2

≈w1≈w2

s 2sm cos w t coswt

On top of the almost same frequency, the amplitude takes

maximum twice in a cycle: cosw’t = 1 and -1: Beats Beat frequency fbeat: fbeat f1 f2

Sum and Difference Frequencies

When you superimpose two sine waves of different frequencies, you get components at

the sum and difference of the two frequencies. This can be shown by using a sum

rule from trigonometry. For equal amplitude sine waves

The first term gives the phenomenon of beats with a beat frequency equal to the

difference between the frequencies mixed. The beat frequency is given by

since the first term above drives the output to zero (or a minimum for unequal

amplitudes) at this beat frequency. Both the sum and difference frequencies are

exploited in radio communication, forming the upper and lower sidebands and

determining the transmitted bandwidth.

When you say that the beat frequency is f1-f2 rather than (f1-f2)/2, that requires some

explanation. For the difference frequency you can just say that you get a minumum

when the modulating term reaches zero, which it does twice per cycle, so that the

number of minima per second is f1-f2.

The beat frequency

refers to the rate at which the volume is heard to be oscillating from high to low volume.

For example, if two complete cycles of high and low volumes are heard every second,

the beat frequency is 2 Hz. The beat frequency is always equal to the difference in

frequency of the two notes that interfere to produce the beats. So if two sound waves

with frequencies of 256 Hz and 254 Hz are played simultaneously, a beat frequency of 2

Hz will be detected. A common physics demonstration involves producing beats using

two tuning forks with very similar frequencies. If a tine on one of two identical tuning

forks is wrapped with a rubber band, then that tuning forks frequency will be lowered. If

both tuning forks are vibrated together, then they produce sounds with slightly different

frequencies. These sounds will interfere to produce detectable beats. The human ear is

capable of detecting beats with frequencies of 7 Hz and below.

The diagram below shows several two-point source interference patterns.

The crests of each wave is denoted by a thick line while the troughs are

denoted by a thin line. Subsequently, the antinodes are the points where

either the thick lines are meeting or the thin lines are meeting. The nodes

are the points where a thick line meets a thin line

How is Sound Converted Into Electrical Signals?

All types of terminal devices used in electronic communications operate by

converting data into electrical signals. Telephones convert sound into electricity and

electricity into sound as they send signals over long distances.

Let's think about the microphone. Sound (vibrations in the air) causes vibration in

an acoustic membrane which applies distortion to a piezoelectric element which

generates electricity. The greater the vibration in the membrane, the greater the

distortion in the element, and the higher the voltage produced

The microphone uses audio vibrations to create distortion in a piezoelectric

element, where they are converted into electrical signals.

Crystals which acquire a charge when compressed, twisted or distorted are said to

be piezoelectric. This provides a convenient transducer effect between electrical

and mechanical oscillations

Radio frequencies occupy the range from a few hertz to 300 GHz, although

commercially important uses of radio use only a small part of this

spectrum. Other types of electromagnetic radiation, with frequencies above

the RF range, are infrared, visible light, ultraviolet,X-rays and gamma rays.

Since the energy of an individual photon of radio frequency is too low to

remove an electron from an atom, radio waves are classified as non-ionizin

radiation.