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Home , Acoustic resonance, Superposition principle

Acoustic Standing and

Leacture

Dr.khitam Y. Elwasife

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

Quantization

When waves are combined in systems with boundary conditions, only certain allowed can exist. . We say the frequencies are quantized. . Quantization is at the heart of . 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. Waves vs. Particles

Waves are very different from particles. Particles have zero size. Waves have a characteristic size – their .

Multiple particles must exist at Multiple waves can combine at one point in different locations. 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 combinations, use the superposition principle: If two or more traveling waves are moving through a medium, the resultant value of the wave 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 much smaller than their .

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 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 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. Types of 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. 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 at low frequencies acts as a monopole. The pattern for a monopole source is shown in the figure at right. .

Radiation from a dipole source A dipole source consists of two monopole sources of equal strength but opposite 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 r quadrupole source

Superposition of Sinusoidal Waves

Assume two waves are traveling in the same direction in a linear medium, with the same , 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 . In a small room the sound is also heard more than once, but the time differences are so small. This is known as . 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 by means of vibrating air columns. To play a 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 , 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. L The top wave is the fundamental or first . It has displacement antinodes at each end, and a displacement 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 , the second and the third. The pipe holds, respectively, two and three half-wavelengths, and shorter Type of air column wavelengths correspond to higher frequencies--the Closed at both ends frequencies are two and three times the Closed at one end open at the other fundamental. Open at both ends

The longest 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 The longest standing wave in a tube of length L with of length L with two open ends is the two open ends has displacement antinodes second harmonic. (pressure nodes) at both ends. It is called the fundamental. 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 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 . Breaking up the original sound wave into its sinusoidal components is called . Standing Waves in air column

Sound wave in a pipe with one closed and one open end (stopped pipe) 14 equilibrium position of particles 13 12 instantaneous position of particles 11 10 sine curve showing 9 instantaneous displacement of 8 particles from equilibrium 7 6 instantaneous pressure 5 distribution 4 3 2 time averaged pressure 1 fluctuations 0 Enter t/T -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

21 Standing waves in air column Normal modes in a pipe with an open and a closed end (stopped pipe)

l 4L v L  n n or l  (n  1,3,5,...) f  n (n  1,3,5,...) 4 n n n 4L 120

100

80

60

40

20

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 position along column

23 CP 523

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 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.

25

Standing Waves

Assume two waves with the same amplitude, frequency and wavelength, traveling in   x   yright  Asin 2p   ft  opposite directions in a   l   medium.   x   x    A sin 2p  cos2p ft  cos 2p  sin2p ft   l   l  The waves combine in   accordance with the waves in   x   yleft  Asin 2p   ft  interference model.   l     x   x   y = A sin (kx – wt) and  A sin 2p  cos2p ft  cos 2p  sin2p ft  1   l   l   y2 = A sin (kx + wt)  x  yright  yleft  2Asin 2p  cos2p ft They interfere according to the  l  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. 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. nl . These correspond to positions of x where xn0,1, 2, 3, 2

An antinode occurs at a point of maximum displacement, 2A. . These correspond to positions of x where nl xn1, 3, 5, 4 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. Standing Wave Example

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 Waves in a String This is the first 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.

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. 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 v n T ƒ n n 22LL Also called quantized frequencies Waves on a String, Harmonic Series

The 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. 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 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. Problem Why does a tree howl? The branches of trees vibrate because of the wind. The vibrations produce the howling sound. Length of limb L = 2.0 m Wave speed in wood v = 4.0103 m.s-1

Fundamental L = l / 4 l = 4 L

N A v = f l

f = v / l = (4.0 103) / {(4)(2)} Hz

f = 500 Hz

Fundamental mode of vibration 35 Problem Resonance 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.

Lsmalles t= 9.00 cm

(a) Find the frequency of the n1 v  345 m.s-1 L  9.00  10 2 m . 1 v 345 f1  2 Hz  985 Hz 4L1 4(9.00 10 (b) Find the wavelength and the next two water levels giving resonance. 2 l 4L1  4(9.00  10 ) m  0.360 m

L2  L1  l / 2  0.270 m, L3  L2  l / 2  0.450 m. 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. 1 1 cos  cos  2cos   cos    2 2 Beats Two sound waves with different but close frequencies give rise to BEATS Consider s1x,t  sm cosw1t w1  w2 s2x,t  sm cosw2t

s  s1  s2  2sm cosw t coswt 1 1 ≈w ≈w Very w   w w  w  w w  1 2 small 2 1 2 2 1 2 s  2sm cosw t coswt On top of the almost same frequency, the amplitude takes maximum twice in a cycle: cosw’t = 1 and -1: Beats frequency f : beat 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 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 , 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.