22-4-2015 1FCI

Prof. Nabila M. Hassan

Faculty of Computers and InformationFayoum University

2014/2015

22-4-2015 2FCI

Chapter 4 - Superposition and Standing Waves:

• Superposition and Interference:

• Interference of Sound Waves:

• Standing Waves:

• Standing Waves in String Fixed at Both Ends:

• Resonance:

• standing Waves in Air Columns:

• Beats:

22-4-2015 3FCI

Superposition of Sinusoidal Waves

Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude

The waves differ in phase Where : sin a +sin b = 2 cos [(a-b)/2] sin [(a+b)/2]

y1 = A sin (kx – ωt) & y2 = A sin (kx – ωt + )

y = y1+y2 = 2A cos (/2) sin (kx – ωt + /2)

The resultant wave function, y, is also sinusoidalThe resultant wave has the same frequency and

wavelength as the original wavesThe amplitude of the resultant wave is 2A cos (/2) The phase of the resultant wave is /2

22-4-2015 4FCI

Sinusoidal Waves with Constructive Interference

If = 0, 2, 4,… (even multiple of ), then: cos(/2) = ±1 y = ± 2A(1) sin(kx – ωt + 0/2)

y = ± 2A sin(kx – ωt )The amplitude of the resultant wave is ± 2A

The crests of one wave coincide with the crests of the other waveThe waves are everywhere in phaseThe waves interfere constructively

22-4-2015 5FCI

Sinusoidal Waves with Destructive Interference:

If = , 3, 5,… (odd multiple of ), then: cos(/2) = 0 y = 2A(0)sin(kx – ωt + /2)

The amplitude of the resultant wave is 0Crests of one wave coincide with troughs of the other wave

The waves interfere destructively

22-4-2015 6FCI

Sinusoidal Waves, General Interference

When is other than 0 or an even multiple of , the amplitude of the resultant is between 0 and 2A.The wave functions still addThe interference is neither constructive nor destructive.

22-4-2015 7FCI

Summary of Interference

Constructive interference occurs when = 0Amplitude of the resultant is 2A

Destructive interference occurs when = n where n is an odd integerAmplitude is 0

General interference occurs when 0 < < n Amplitude is 0 < Aresultant < 2A

22-4-2015 8FCI

Interference in Sound Waves

Sound from S can reach R by two different paths

The upper path(r2) can be varied (r1 is fixed)

Whenever

Δr = |r2 – r1| = n (n = 0, 1, …)constructive interference occurs

Whenever

Δr = |r2 – r1| =(n + ½) (n = 0, 1, …) destructive interference occurs

22-4-2015 9FCI

Standing WavesThe student will be able to:

Define the standing wave.Describe the formation of standing waves.Describe the characteristics of standing waves.

22-4-2015 10FCI

Standing Waves on a String, SummaryThe wavelengths of the normal modes for a string of

length L fixed at both ends are n = 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 2n

v n Tn

L L

22-4-2015 11FCI

Waves on a String, Harmonic Series

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.22-4-2015 12FCI

Objectives: the student will be able to:

- Define the resonance phenomena.

- Define the standing wave in air columns.

- Demonstrate the beats

22-4-2015 13FCI

4 - Resonance

A system is capable of oscillating in one or more normal modes

If a periodic force is applied to such a

system, the amplitude of the resulting

motion is greatest when the frequency

of the applied force is equal to one of

the natural frequencies of the system

22-4-2015 14FCI

Because an oscillating system exhibits a large amplitude when driven at any of its natural frequencies, these frequencies are referred to as resonance frequencies!!!

The resonance frequency is symbolized by ƒo

Resonance ,

22-4-2015 15FCI

Example:

22-4-2015 16

An example of resonance.

If pendulum A is set into oscillation,

only pendulum C, whose length matches

that of A, eventually oscillates with large

amplitude, or resonates.

The arrows indicate motion in a plane

perpendicular to the page

FCI

ResonanceA system is capable of oscillating in one or more normal modes.Assume we drive a string with a vibrating blade.

If a periodic force is applied to such a

system, the amplitude of the resulting

motion of the string is greatest when

the frequency of the applied force

is equal to one of the natural

frequencies of the system.

This phenomena is called resonance.

22-4-2015 17FCI

Standing Waves in Air Columns

Standing waves can be set up in air columns as

the result of interference between longitudinal

sound waves traveling in opposite directions.

The phase relationship between the incident and

reflected waves depends upon whether the end of

the pipe is opened or closed.

Waves under boundary conditions model can be

applied.

22-4-2015 18FCI

Standing Waves in Air Columns, Closed End

A closed end of a pipe is a displacement node in the

standing wave.

The rigid barrier at this end will not allow longitudinal

motion in the air.

The closed end corresponds with a pressure antinode.

It is a point of maximum pressure variations.

The pressure wave is 90o out of phase with the

displacement wave.

22-4-2015 19FCI

1-Standing Waves in a Tube Closed at One End

The closed end is a displacement node.

The open end is a displacement antinode.

The fundamental corresponds to ¼

The frequencies are ƒn = nƒ = n (v/4L)

where n = 1, 3, 5, …

In a pipe closed at one end, the natural

frequencies of oscillation form a

harmonic series that includes only odd

integral multiples of the

fundamental frequency.

22-4-2015 20FCI

Standing Waves in Air Columns, Open End

The open end of a pipe is a displacement antinode in the

standing wave.

As the compression region of the wave exits the open

end of the pipe, the constraint of the pipe is removed

and the compressed air is free to expand into the

atmosphere.

The open end corresponds with a pressure node.

It is a point of no pressure variation.

22-4-2015 21FCI

2 -Standing Waves in an Open TubeBoth ends are displacement antinodes.The fundamental frequency is v/2L.

This corresponds to the first diagram.

The higher harmonics are ƒn = nƒ1 = n (v/2L) where n = 1, 2, 3, …

In a pipe open at both ends, the natural frequencies of oscillation form a harmonic series that includes all integral multiples of the fundamental frequency.

22-4-2015 22FCI

Notes About Musical Instruments

As the temperature rises:

Sounds produced by air columns become sharp

Higher frequency

Higher speed due to the higher temperature

Sounds produced by strings become flat

Lower frequency

The strings expand due to the higher temperature.

As the strings expand, their tension decreases.

22-4-2015 23FCI

Quiz:A pipe open at both ends resonates at a fundamental

Frequency fopen. When one end is covered and the pipe

is again made to resonate, the fundamental frequency

is fclosed. Which of the following expressions describes

how these two resonant frequencies compare?

(a) fclosed = fopen (b) fclosed = 1/2 fopen (c) fclosed = 2 fopen (d)

fclosed = 3/2 fopen

22-4-2015 24FCI

(b). With both ends open, the pipe has a fundamental frequency given by Equation : fopen = v/2L.

With one end closed, the pipe has a fundamental frequency given by:

22-4-2015 FCI 25

Resonance in Air Columns, ExampleA tuning fork is placed near the top of the tube.When L corresponds to a resonance frequency of the pipe, the sound is louder.The water acts as a closed end of a tube.The wavelengths can be calculated from the lengths where resonance occurs.

22-4-2015 26FCI

Beats and Beat FrequencyBeating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies.The number of amplitude maxima one hears per second is the beat frequency.It equals the difference between the frequencies of the two sources.The human ear can detect a beat frequency up to about 20 beats/sec.

22-4-2015 27FCI

Consider two sound waves of equal amplitude traveling through a medium with slightly different frequencies f1 and f2 .

The wave functions for these two waves at a point that we choose as x = 0

Using the superposition principle, we find that the resultant wave function at this point is

22-4-2015 28FCI

Beats, EquationsThe amplitude of the resultant wave varies in time according to

Therefore, the intensity also varies in time. The beat

frequency is ƒbeat = |ƒ1 – ƒ2|.

22-4-2015 29

Note that a maximum in the amplitude of the resultant sound wave is detected when,

FCI

This means there are two maxima in each period of the resultant wave. Because the amplitude varies with frequency as ( f1 - f2)/2, the number of beats per second, or the beat frequency f beat, is twice this value. That is, the beats frequency

22-4-2015 30

For example, if one tuning fork vibrates at 438 Hz and a second one

vibrates at 442 Hz, the resultant sound wave of the combination

has a frequency of 440 Hz (the musical note A) and a beat

frequency of 4 Hz. A listener would hear a 440-Hz sound wave go

through an intensity maximum four times every second.

FCI

1. The sound waves that humans cannot hear are those with frequenciesa. from 20 to 20,000 Hz.b. below 20 Hz.c. above 20,000 Hz.d. both B and C

2. Sound travels in air by a series of a. compressions.b. rarefactions.c. both compressions and rarefactions.d. pitches.

Assessment Questions

22-4-2015 31FCI

3. Sound travels faster in a. a vacuum compared to liquids.b. gases compared to liquids.c. gases compared to solids.d. solids compared to gases.

4. The speed of sound varies with a. amplitude.b. frequency.c. temperature.d. pitch.

Assessment Questions

22-4-2015 32FCI

5- When an object is set into vibration by a wave having a frequency that matches the natural frequency of the object, what occurs is a. forced vibration.b. resonance.c. refraction.d. amplitude reduction.

6- The phenomenon of beats is the result of sound a. destruction.b. interference.c. resonance.d. amplification.

Assessment Questions

22-4-2015 33FCI

Non-sinusoidal Wave PatternsThe wave patterns produced by a musical instrument are the result of the superposition of various harmonics.The human perceptive response to a sound that allows one to place the sound on a scale of high to low is the pitch of the sound.

Pitch vs. frequency Frequency is the physical measurement of the number

of oscillations per second. Pitch is a psychological reaction to the sound. Frequency is the stimulus and pitch is the response.

The human perceptive response associated with the various mixtures of harmonics is the quality or timbre of the sound.

22-4-2015 34FCI

Quality of Sound – Tuning ForkA tuning fork produces

only the fundamental frequency.

22-4-2015 35FCI

Quality of Sound – Flute

The same note played on a flute sounds differently.The second harmonic is very strong.The fourth harmonic is close in strength to the first.

22-4-2015 36FCI

Analyzing Non-sinusoidal Wave PatternsIf the wave pattern is periodic, it can be represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series.Any periodic function can be represented as a series of sine and cosine terms.

This is based on a mathematical technique called Fourier’s theorem.

A Fourier series is the corresponding sum of terms that represents the periodic wave pattern.If we have a function y that is periodic in time, Fourier’s theorem says the function can be written as:

ƒ1 = 1/T and ƒn= nƒ1

An and Bn are amplitudes of the waves.

( ) ( sin2 ƒ cos2 ƒ )n n n nn

y t A t B t

22-4-2015 37FCI

Fourier Synthesis of a Square WaveIn Fourier synthesis, various harmonics are added together to form a resultant wave pattern.Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f.In (a) waves of frequency f and 3f are added. In (b) the harmonic of frequency 5f is added. In (c) the wave approaches closer to the square wave when odd frequencies up to 9f are added.

22-4-2015 38FCI

Summary: 1- When two traveling waves having equal amplitudes and frequencies superimpose , the resultant waves has an amplitude that depends on the phase angle φ between the resultant wave has an amplitude that depends on the two waves are in phase , two waves . Constructive interference occurs when the two waves are in phase, corresponding to rad, Destructive interference occurs when the two waves are 180o out of phase, corresponding to rad.

2- Standing waves are formed from the superposition of two sinusoidal waves having the same frequency , amplitude , and wavelength but traveling in opposite directions . the resultant standing wave is described by

22-4-2015 39FCI

3- The natural frequencies of vibration of a string of length L and fixed at both ends are quantized and are given by

Where T is the tension in the string and µ is its linear mass density .The natural frequencies of vibration f1, f2,f3,…… form a harmonic series.

4- Standing waves can be produces in a column of air inside a pipe. If the pipe is open at both ends, all harmonics are present and the natural frequencies of oscillation are

22-4-2015 40FCI

If the pipe is open at one end and closed at the other, only the odd harmonics are present, and the natural frequencies of oscillation are

5- An oscillating system is in resonance with some driving force whenever the frequency of the driving force matches one of the natural frequencies of the system. When the system is resonating, it responds by oscillating with a relatively large amplitude.

6- The phenomenon of beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies.

22-4-2015 41FCI