Chapter 10 Mechanical and Sound Wave

CHAPTER 10 MECHANICAL AND SOUND WAVES prepared by Yew Sze Ling@Fiona, KML

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Chapter 10 Mechanical and Sound Wave

Curriculum Specification Remarks

Before After Revision

10.1 Properties of Waves

a) Define wavelength and wave number. (C1, C2)

b) Solve problems related to equation of progressive wave,

𝑦(𝑥, 𝑡) = 𝐴 sin(𝜔𝑡 ± 𝑘𝑥) (C3, C4)

c) Discuss and use particle vibrational velocity and wave

propagation velocity. (C1, C2)

d) Discuss the graphs of:

i. displacement – time, y – t

ii. displacement – distance, y – x

(C1, C2)

10.2 Superposition of Waves

a) State the principle of superposition of waves for the

constructive and destructive interference. (C1, C2)

b) Use standing wave equation, 𝑦 = 𝐴 cos𝑘𝑥 sin 𝜔𝑡 (C3, C4)

c) Discuss progressive wave and standing wave. (C1, C2)

10.3 Sound Intensity

a) Define and use sound intensity. (C1, C2)

b) Discuss the dependence of intensity on amplitude and

distance from a point source by using graphical illusion.

(C1, C2)

10.4 Application of Standing Waves

a) Solve problem related to the fundamental and overtone

frequencies for:

i. stretched string.

ii. air-columns (open and closed end).

b) Use wave speed in a stretched string, (C3, C4)

c) Investigate standing wave formed in a stretched string.

(Experiment 6: Standing waves) (C1, C2, C3, C4)

d) Determine the mass per unit length of the string

(Experiment 6: Standing waves) (C1, C2, C3, C4)

10.5 Doppler Effect

a) State Doppler Effect for sound waves. (C1, C2)

b) Apply Doppler Effect equation, for

relative motion between source and observer. Limit to

stationary observer and moving source, and vice versa.

(C3, C4)

𝑣 = √𝑇

𝜇

𝑓𝑎 = (𝑣 ± 𝑣0𝑣 ∓ 𝑣𝑠

) 𝑓


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10.1 Properties of Waves

Waves can occur whenever a system is disturbed from equilibrium and when the disturbance can

travel, propagate, from one region of the system to another. As wave propagates, it carries. This

chapter is about mechanical waves − waves that travel within some material or substance called a medium.

Important terms in wave:

Term Definition Unit

Amplitude, A The maximum displacement from the equilibrium position to the crest

or trough of the wave motion. m

Frequency, f The number of cycles (wavelength) produced in one second. Hz or

s-1

Period, T The time taken for a particle (point) in the wave to complete one

cycle. s

Wavelength, λ

The wavelength λ is the horizontal length of one cycle of the wave.

The wavelength is also the horizontal distance between two successive

crests, two successive troughs, or any two successive equivalent

points on the wave.

m

Wave number,

k The number of cycles (wavelength) produced per unit distance:

rad m-1

Angular

frequency, ω The number of cycles produced per unit time: rad s

-1

A wave which travels continuously in a medium in the same direction without any change in its amplitude is called a progressive wave or a traveling wave.

𝑘 =2𝜋

𝜆

𝜔 =2𝜋

𝑇= 2𝜋𝑓

Wave displacement, x

Definition: Distance of the particle

from the source of disturbance.

Particle displacement, y

Definition: Displacement of the

particle from its equilibrium

position.


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The general wave equation for a sinusoidal progressive wave:

Move to the right (in positive-x direction)

𝑦 = sin(𝜔𝑡 − 𝑘𝑥)

Move to the left (in negative-x direction)

𝑦 = sin(𝜔𝑡 + 𝑘𝑥)

As a wave propagate with a velocity, each particle along the wave is displaced, one after the other,

from its undisturbed position. For example, figure below shows a wave traveling on a rope or cord.

The wave travels to the right along the rope with a constant velocity known as wave propagation

velocity, v. Meanwhile, particles of the rope oscillate back and forth on the table top with a velocity known as particle vibrational velocity, vy.

Wave propagation velocity, v Particle vibrational velocity, vy

The distance travelled by a wave profile per

unit time.

𝑣 = 𝑓𝜆

The velocity is constant.

It is the speed of particle vibrates with

simple harmonic motion around its equilibrium position.

The velocity is changes as the wave passes.

Displacement – time graph, y – t Displacement – distance graph, y – x

The graph shows the displacement of any one

particle in the wave at any particular distance, x

from the origin.

(OR y–t graph shows how displacement (y) varies with time (t) for one particle in a wave.)

From y–t graph, we can obtain:

1. Amplitude, A

2. Period, T

The graph shows the displacement of all the

particles in the wave at any particular time, t.

(OR y-x graph shows how displacement (y)

varies with distance (x) for all particles in a wave at one instant.)

From y–x graph, we can obtain:

1. Amplitude, A

2. Wavelength, λ

Direction of propagation

𝑣𝑦 =𝑑𝑦

𝑑𝑡

𝑣 =𝜆

𝑇

T λ


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10.2 Superposition of Waves

When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.

Constructive Interference Destructive Interference

The vertical displacements of the two pulses are

in the same direction, and the amplitude of the

combined waveform is greater than that of either

pulse.

The vertical displacement of one of the pulse has

a negative displacement, the two pulses tend to

cancel each other when they overlap, and the

amplitude of the combined waveform is smaller than that of either pulse.

A standing (stationary) wave is another interference effect that can occur when two waves overlap. It is called a “standing wave” because it does not appear to be traveling.

It is formed when two waves which are travelling in opposite directions, and which have the

same wavelength, frequency and amplitude are superimposed.

The nodes (N) are places that do not vibrate at all, and the antinodes (AN) are places where

maximum vibration occurs. Distance between 2 consecutive nodes or antinodes = λ/2. Distance

between consecutive nodes and antinodes = λ/4.

Standing waves can arise with transverse waves, such as those on a guitar string, and also with longitudinal sound waves, such as those in a flute.

N N N

AN AN AN AN

AN AN AN AN

Progressive

wave 1

Progressive

wave 2

Standing

wave


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The general wave equation for a standing wave :

𝑦 = 𝐴 cos𝑘𝑥 sin 𝜔𝑡

where

A is the maximum amplitude of standing wave formed. A = 2a, where a is the amplitude of progressive wave.

A cos kx

Determine the amplitude for any point along the standing wave.

It is called the amplitude formula.

Its value depends on the distance, x

sin ωt

Determine the time for antinodes and nodes will occur in the standing wave.

Difference between progressive wave and stationary wave:

10.3 Sound Intensity

Sound wave is a longitudinal wave which requires a medium for its propagation.

Sound waves are produced by vibrating objects.

Vibrating objects disturb the air molecules, producing alternating high-pressure regions

(compressions) & low-pressure regions (rarefactions) which form sound wave.

Sound intensity is perceived by the ear as loudness.

Sound waves carry energy that can be used to do work. The amount of energy transported per second by a sound wave is called the power of the wave.

When a sound wave leaves a source, the power spreads out.


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The power passes perpendicularly through surface 1 and then surface 2 which has the larger area.

The greater the distance from the source, the larger the area over which a given amount of sound energy is spread, and thus the lower its intensity.

The sound intensity I is defined as the sound power P that passes perpendicularly through a

surface divided by the area A of that surface.

The unit of sound intensity is power per unit area (W m-2

).

Dependence of intensity on amplitude Dependence of intensity on distance

Since

Thus, intensity is directly proportional to the squared of amplitude.

Since

Thus, intensity is inversely proportional to the squared of distance.

𝐼 =𝑃

𝐴

𝐼 =𝑃

𝐴𝑟𝑒𝑎 𝑃 =

𝐸

𝑡 and 𝐸 =

1

2𝑘(𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒)2 𝐼 =

𝑃

𝐴𝑟𝑒𝑎 and 𝐴 = 4𝜋𝑟2

𝐼 ∝ 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒2 𝐼 ∝1

𝑟2


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10.4 Application of Standing Waves

In musical instrument, the source of sound is set into vibration by striking, plucking, bowing, or

blowing. Standing waves are produced and the source vibrates at its natural resonant frequencies.

The vibrating source is in contact with the air (or other medium) and pushes on it to produce sound

waves that travel outward. The frequencies of the waves are the same as those of the source, but the

speed and wavelengths can be different. Important terms:

Fundamental frequency : The lowest resonant frequency of a vibrating object.

Harmonic : An integer (whole number) multiple of the fundamental frequency

of a vibrating object

Overtone : Any resonant frequency above the fundamental frequency. An

overtone may or may not be a harmonic

Stretched String (Example: guitar, violin, etc)

When a guitar string is plucked, a wave is produced in the

string; this wave is reflected and re-reflected from the ends

of the string, making a standing wave. This standing wave

on the string in turn produces a sound wave in the air, with a

frequency determined by the properties of the string such as

the mass, m and the length of the string, l.

Mode Figure Relationship

between l and λ Summary

Fundamental

(1st Harmonic)

𝑙 =𝜆

2

Harmonic n Mode

1st 1 Fundamental

2nd

2 1st overtone

3rd

3 2nd

overtone

General relationship between l

and λ:

𝑙 =𝑛𝜆

2

Wave velocity general equation:

𝑣 = 𝑓𝜆

Wave velocity on string:

𝑣 = √𝑇

𝜇

where

𝜇 =𝑚

𝑙

Frequency

𝑓𝑛 =𝑛

2𝑙√𝑇

𝜇= 𝑛𝑓1

1st Overtone

(2nd

Harmonic)

𝑙 = 𝜆

2nd

Overtone

(3rd

Harmonic)

𝑙 =3

2𝜆


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Open-end air-column (Example: flute)

An open tube has antinodes at both ends since the air is free to move at open ends. There must be at

least one node within an open tube if there is to be a standing wave at all. For open-end air-column,

all harmonic exist.



Fundamental

(1st Harmonic)

𝑙 =𝜆

2

Harmonic n Mode

1st 1 Fundamental

2nd

2 1st overtone

3rd

3 2nd

overtone


and λ:

𝑙 =𝑛𝜆

2


𝑣 = 𝑓𝜆

where v is the speed of sound in air.

Frequency

𝑓𝑛 =𝑛𝑣

2𝑙= 𝑛𝑓1

1st Overtone

(2nd

Harmonic)

𝑙 = 𝜆

2nd

Overtone

(3rd

Harmonic)

𝑙 =3

2𝜆

Closed-end air-column (Example: clarinet)

For a closed-end air-column, there is always a node at the closed end because the air is not free to

move and an antinode at the open end where the air can move freely. For closed-end air-column,

ONLY odd harmonic exist



Fundamental

(1st Harmonic)

𝑙 =𝜆

4

Harmonic n Mode

1st 1 Fundamental

3rd

3 1st overtone

5th

5 2nd

overtone


and λ:

𝑙 =𝑛𝜆

4


𝑣 = 𝑓𝜆

1st Overtone

(3rd

Harmonic)

𝑙 =3

4𝜆


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2nd

Overtone

(5th Harmonic)

𝑙 =5

4𝜆

where v is the speed of sound in air.

Frequency

𝑓𝑛 =𝑛𝑣

4𝑙= 𝑛𝑓1

10.5 Doppler Effect

Doppler Effect is the change in frequency or pitch of the sound detected by an observer because the

sound source and the observer have different velocities with respect to the medium of sound propagation

General equation of Doppler Effect:

Two cases for Doppler Effect

Case 1: Stationary observer, Moving source


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Subcase 1: Source moves towards observer (approaches)

Subcase 2: Source moves away from observer (recedes)

The following graph is applied for the case of stationary observer where the source is moving towards, passing by and moving away from the observer.

Part A: When the source moves towards the observer, the apparent frequency, fo is greater

than the source frequency, fs and its value is constant.

Part B: At the moment source crosses observer, the apparent frequency, fo is equal to the source frequency, fs.

Part C: When the source moves away from the observer, the apparent frequency, fo is less

than the source frequency, fs and its value is constant.

Case 2: Moving observer, Stationary source

𝑓0 = (𝑣

𝑣 − 𝑣𝑠)𝑓 𝑠

𝑓0 = (𝑣

𝑣 + 𝑣𝑠) 𝑓 𝑠


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Subcase 1: Observer moves towards source (approaches)

Subcase 2: Observer moves away from source (recedes)

The rules of using the general equation for Doppler Effect

vS OR vO in the same direction with v (speed of sound) → “‒”

vS OR vO opposite direction with v (speed of sound) → “+”

𝑓0 = (𝑣 + 𝑣0𝑣

) 𝑓 𝑠

𝑓0 = 𝑣 − 𝑣0𝑣

𝑓 𝑠


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Exercise

Properties of Waves

1. A fisherman notices that wave crests pass the bow of his anchored boat every 3.0 s. He

measures the distance between two crests to be 7.0 m. How fast are the waves traveling?

2. Using the data in the graphs that accompany this problem, determine the speed of the wave.

3. A progressive wave is represented by the equation

𝑦 = 4.0 sin 2𝜋 4𝑡 +𝑥

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where x and y are in cm, and t is in second. Determine

a) the wave speed

b) the vibrational velocity at time = 0 for the particle at

i. x = 0

ii. x = 3.0 cm

c) What is the conclusion that can be made based on the values of velocity in question (a) and (b)?

4. The displacement-distance graph of a progressive wave of frequency 20 Hz moving in the

negative x – direction as shown in figure below.

Deduce the equation for the wave.

5. Figure below shows the displacement-time graph which represents the simple harmonic

motion of a particle in a progressive wave with wavelength 0.5 m.

Deduce the equation for the wave.

6. A progressive wave is represented by the equation y = 2 sin (2πt + 4πx), where x and y are in

cm and t in second. Sketch

a) the displacement-time graph of the particle at x = 0 cm

b) the displacement-distance graph of the particle at t = 0 s


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Superposition of Waves

1. Transverse waves travel along a stretched string at speed 80 m s-1

and frequency 200 Hz.

Stationary waves are produced on the string. Determine the distance between

a) 2 consecutive nodes.

b) A node and the adjacent antinodes.

2. A stationary wave is represented by the following expression

𝑦 = 5 cos𝜋𝑥 sin 𝜋𝑡

where y and x in centimetres and t in seconds. Determine the three smallest vale of x (x > 0)

that corresponds to

a) nodes

b) antinodes

Sound Intensity

1. A loudspeaker radiates sound waves uniformly in all directions. At a distance 3 m the

intensity of the sound is 0.85 W m-2

. Determine

a) the power of loudspeaker,

b) the sound intensity at distance 6 m from the source.

2. A rocket in a fireworks display explored high in the air. The sound spreads out uniformly in

all directions. The intensity of the sound is 2.0×10-6

W m-2

at a distance 120 m from the explosion. Find the distance from the source at which the intensity is 0.80×10

-6 W m

-2.

3. If the amplitude of a sound wave is made 3.5 times greater, by what factor will the intensity

increase?

Application of Standing Waves

1. If a violin string vibrates at 440 Hz as its fundamental frequency, what are the frequencies of

the first three harmonics?

2. A guitar string is 92 cm long and has a mass of 3.4 g. The distance from the bridge to the

support post is l = 62 cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first two overtones?

3. An organ pipe is 116 cm long. Given speed of sound is 331 m s-1

. Determine the fundamental

and first three audible overtones if the pipe is

a) closed at one end

b) open at both ends

4. The frequency of the first overtone produced by an open pipe is equal to the frequency of the

overtone produced by a closed pipe of length 0.6 m. Determine the length of the open pipe.

5. A variable length air column is placed just below a vibrating wire that is fixed at both ends.

The length of the air column open at one end is gradually increased until the first position of

resonance is observed at 34.0 cm. The wire is 120 cm long and is vibrating in its third

harmonic. Calculate the speed of transverse waves in the wire. Given speed of sound is 331 m s

-1.


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6. A tuning fork is set into vibration above a vertical open tube filled with

water. The water level is allowed to drop slowly. As it does so, the air

in the tube above the water level is heard to resonate with the tuning

fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

7. A stretched wire of length 80.0 cm and mass 15.0 g vibrates transversely. Waves travel along

the wire at speed 220 m s-1

. Two antinodes can be found in the stationary waves formed in between the two fixed ends of the wire.

a) Sketch and label the waveform of the stationary wave.

b) Determine

i. the wavelength of the progressive wave which move along the wire,

ii. the frequency of the vibration of the wire,

iii. the tension in the wire.

Doppler Effect

1. The whistle from a stationary policeman at a junction emits sound of frequency 1000 Hz. If

the speed of sound is 330 m s-1

, what is the frequency of the sound heard by a passenger

inside a car moving with a speed of 20 m s-1

.

a) Towards the junction?

b) Away from the junction ?

2. A train moving at constant speed 20 m s-1

towards a stationary observer standing on the

station platform produces a loud sound signal at frequency 500 Hz. Determine the frequency of sound heard by the observer when the train

a) Towards the observer

b) Moves away from the observer.

Given speed of sound in air = 340 m s-1

3. A bat flying at 5 m s-1

towards a wall emits a chirp at 50 kHz. If the wall reflects this sound

pulse, what is the frequency of the echo received by the bat? Given speed of sound is

331 m s-1

.

4. Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each

other essentially head-on. One driver hears the horn of the other truck at a frequency that is

1.14 times the frequency he hears when the trucks are stationary. The speed of sound is 343

m s-1

. At what speed is each truck moving?

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Chapter 10 Mechanical and Sound Wave

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