SECTION 1: Reflection of Sound Waves, Echoes & SONAR
1. Reflection of Sound Waves
Like light, sound waves also undergo reflection when they strike a large, rigid surface (such as a tall building, cliff, mountain, or wall).
LAWS OF REFLECTION OF SOUND
The angle of incidence ($\angle i$) is equal to the angle of reflection ($\angle r$).
The incident sound wave, the reflected sound wave, and the normal to the reflecting surface at the point of incidence all lie in the same plane.
Requirement for Reflector: Unlike light (which requires a polished smooth surface), sound waves have much longer wavelengths, so reflection occurs from any large obstacle (hard or rough surface like a brick wall, rock face, or hill).
2. Echo and Conditions for its Formation
Echo: An echo is the sound heard after reflection from a distant, rigid obstacle (like a cliff, wall, or mountain) after the original direct sound has ceased.
Principle of Echo Formation: Sound travels a total distance of $2d$ ($d$ to obstacle and $d$ back to ear) in time $t$.
⭐ THREE MANDATORY CONDITIONS FOR FORMATION OF AN ECHO
Persistence of Hearing ($t \ge 0.1\text{ s}$):
The sensation of any sound stays in the human ear for about $0.1\text{ second}$ ($\frac{1}{10}\text{ th}$ of a second).
To hear a distinct echo, the reflected sound wave must reach the ear at least $0.1\text{ s}$ after the original direct sound has stopped.
Minimum Distance of Obstacle ($d_{\text{min}} \ge 17\text{ m}$):
Total distance travelled by sound to the cliff and back = $2d$.
Using speed formula: $\text{Distance} = \text{Speed } (V) \times \text{Time } (t) \implies 2d = V \times t \implies d = \frac{V \times t}{2}$.
Taking speed of sound in air at $20^\circ\text{C}$ as $V = 340\text{ m s}^{-1}$ and minimum time $t = 0.1\text{ s}$:
$$ d_{\text{min}} = \frac{340\text{ m s}^{-1} \times 0.1\text{ s}}{2} = 17\text{ m} $$
(Note: At $0^\circ\text{C}$ where $V = 330\text{ m s}^{-1}$, $d_{\text{min}} = 16.5\text{ m}$; at $344\text{ m s}^{-1}$, $d_{\text{min}} = 17.2\text{ m}$).
If the reflecting surface is closer than $17\text{ m}$, the reflected sound overlaps with the original sound, creating reverberation instead of a clear echo.
Size and Nature of Reflector:
The size of the reflecting obstacle must be large compared to the wavelength of sound.
The reflecting surface must be rigid and hard enough to reflect a significant portion of sound energy.
3. Applications / Uses of Echoes in Nature, Medicine & Technology
Echoes of ultrasonic waves (sound of frequency $f > 20,000\text{ Hz}$) have numerous indispensable applications:
Domain / User
Application Details
Physics Mechanism
Bats (Echolocation)
Bats fly safely in pitch darkness and catch flying insects by emitting high-frequency ultrasonic chirps.
The ultrasonic waves bounce off obstacles or prey. By detecting the time delay and direction of reflected echoes, bats judge exact distance and location.
Dolphins
Dolphins navigate through muddy sea water and locate fish.
They emit ultrasonic clicks and listen for echoes returned from fish or underwater rocks.
Fishermen & Trawlers
Fishermen use Echo Sounders / Fish Finders to locate underwater shoals of fish.
Ultrasonic pulses sent into the sea reflect back from schools of fish; time delay indicates depth of the fish shoal.
High-frequency ultrasonic beams pass into the body. Partial reflections from boundaries of internal organs/tissues are processed into real-time images. Safe and painless.
SONAR
Sound Navigation And Ranging used in ships and submarines.
Measures sea depth, detects submerged icebergs, enemy submarines, and shipwrecks using ultrasonic echoes.
4. SONAR (Sound Navigation And Ranging)
SONAR: An acoustic technique that uses ultrasonic waves to detect underwater objects and measure the depth of the sea floor.
Working of SONAR: Ultrasonic pulses sent from Transmitter ($T$) travel depth $d$ to sea bed and echo back distance $d$ to Receiver ($R$).
SONAR DEPTH FORMULA
If $V$ is the velocity of ultrasonic waves in sea water ($\approx 1400 - 1500\text{ m s}^{-1}$), $d$ is the depth of the sea floor, and $t$ is the total time elapsed between sending the signal and receiving its echo:
$$ \text{Total Distance Travelled} = 2d = V \times t \implies d = \frac{V \cdot t}{2} $$
WHY ULTRASONIC WAVES ARE USED IN SONAR
High Directionality: Ultrasonic waves have very high frequency ($f > 20\text{ kHz}$) and short wavelength, so they can be sent as a narrow, sharp beam that does not spread out over long distances.
Low Absorption: They can travel large distances in sea water without losing much energy.
Inaudible: They do not disturb human ears or ship crew.
When a mechanical body is disturbed from its rest position and released, it can undergo four types of vibrations depending on environment and external forces:
1. Natural (or Free) Vibrations
Natural Vibrations: Periodic vibrations executed by a body when it is displaced from its equilibrium position and released in the absence of any external force or medium resistance (i.e. in vacuum).
Wave pattern of Natural (Free) Vibrations showing constant frequency $f_0$ and constant amplitude $a$.
Natural Frequency ($f_0$): The fixed frequency with which a body vibrates naturally when free. It depends on the size, shape, mass, and elasticity of the body.
Constant Amplitude & Energy: Since there is no friction or air resistance, no energy is lost. The amplitude remains constant indefinitely.
Ideal Concept: Pure natural vibrations occur only in vacuum.
Examples: Simple pendulum oscillating in vacuum, tuning fork struck in vacuum, stretched string plucked in vacuum.
2. Damped Vibrations
Damped Vibrations: Periodic vibrations of a body executed in the presence of a resistive medium (air, liquid), in which the amplitude of vibration continuously decreases with time until vibrations cease altogether.
Wave pattern of Damped Vibrations showing amplitude decaying exponentially due to energy dissipation by air friction.
Energy Dissipation: The vibrating body continuously does work against the frictional force of the surrounding air/medium. Hence, mechanical energy is continuously dissipated as heat.
Decreasing Amplitude: Amplitude decreases exponentially until the body comes to rest.
Frequency: The frequency of damped vibration is slightly smaller than the natural frequency.
Practical Examples:
A simple pendulum oscillating in air eventually stops due to air resistance.
A tuning fork vibrating in air gradually loses sound and stops.
Shock absorbers of motor vehicles damp out road bumps rapidly.
3. Forced Vibrations
Forced Vibrations: Vibrations produced in a body under the influence of an external periodic force acting on it.
Frequency of Vibration: The body does NOT vibrate with its natural frequency; it is forced to vibrate with the frequency ($f$) of the external periodic force.
Amplitude: The amplitude of forced vibration is generally small and depends on the difference between the driving frequency $f$ and natural frequency $f_0$.
Examples:
When the stem of a vibrating tuning fork is pressed firmly against a wooden table top, the table top undergoes forced vibrations at the frequency of the tuning fork.
The paper cone of a loudspeaker undergoes forced vibrations under the action of the alternating current fed into its voice coil.
Soundboards of guitars and pianos vibrate under forced vibrations produced by plucking strings.
4. Resonance — A Special Case of Forced Vibrations
DEFINITION & CONDITION OF RESONANCE
Resonance: Resonance is a special case of forced vibrations in which the frequency of the external applied periodic force becomes exactly equal to the natural frequency of the body ($f_{\text{ext}} = f_0$).
At resonance, the body vibrates with a very large (maximum) amplitude because maximum energy transfer takes place from the driving force to the vibrating system.
Resonance Curve showing dramatic sharp rise in amplitude when driving frequency $f$ matches natural frequency $f_0$.
5. Applications and Practical Examples of Resonance
⭐ FOUR CRITICAL REAL-WORLD EXAMPLES OF RESONANCE
Tuning a Radio / Television Receiver:
Radio broadcasts from different stations travel in air at different electromagnetic wave frequencies.
When you turn the tuning knob of a radio, you alter the natural frequency of its internal electrical circuit ($LC$ circuit).
When the circuit's natural frequency equals the broadcast frequency of a specific station ($f_{\text{circuit}} = f_{\text{station}}$), electrical resonance occurs, and that station's signal is received with maximum amplitude and clear sound.
Soldiers Marching Across a Suspension Bridge:
When troops cross a bridge, they are ordered to break step (stop marching in rhythm).
If soldiers march in step, the periodic frequency of their marching feet might equal the natural frequency of the bridge structure ($f_{\text{march}} = f_{\text{bridge}}$).
This would induce resonance, causing the bridge to vibrate violently with dangerous amplitude, which could lead to structural failure and collapse!
Barton's Pendulums (Resonance in Pendulums of Equal Length):
Four pendulums A, B, C, D are suspended from a common stretched string. Pendulums A and C have equal length, while B is shorter and D is longer.
When pendulum A is set oscillating, its vibrations travel through the string to B, C, and D.
Since pendulum C has the exact same length as A ($l_A = l_C$), its natural frequency matches A ($f = \frac{1}{2\pi}\sqrt{g/l}$). Thus, C vibrates violently with maximum amplitude due to resonance, while B and D oscillate with small amplitudes.
Vehicle Body Rattle at Specific Speeds:
At a certain engine speed (RPM), loose parts or window panes of a car/bus begin to rattle noisily.
This happens because the frequency of engine vibrations matches the natural frequency of that vehicle part, causing resonant vibrations.
6. Comparative Summary: Types of Vibrations
Parameter
Natural Vibrations
Damped Vibrations
Forced Vibrations
Resonance
Environment / Force
Vacuum (No external friction)
Resistive medium (Air, viscous liquid)
External periodic force acting
External periodic force where $f_{\text{ext}} = f_0$
Vibration Frequency
Natural frequency ($f_0$)
Slightly less than natural frequency
Frequency of external force ($f$)
Equal to natural frequency ($f_0$)
Amplitude Behaviour
Remains constant indefinitely
Decays exponentially to zero
Small constant amplitude
Extremely large (maximum) amplitude
Energy Status
No energy loss
Energy dissipated as heat
Energy continuously supplied by driver
Maximum energy transfer rate
SECTION 3: Loudness, Pitch & Quality of Sound
1. Loudness vs Intensity (Subjective vs Objective Nature)
Sound possesses three characteristics: Loudness, Pitch, and Quality.
Property
Nature
Definition & Dependence
Measurement & Units
Intensity ($I$)
Objective Physical Quantity
Amount of sound energy passing per unit area per second normal to propagation. Independent of human ear.
Measurable with instruments. SI Unit: $\text{W m}^{-2}$.
Loudness ($L$)
Subjective Sensation
Sensation produced in human ear. Depends on intensity AND sensitivity of listener's ear ($L \propto a^2$).
Cannot be measured directly by instruments alone. Unit: Decibel ($\text{dB}$).
2. Interdependence of Pitch and Frequency
Pitch: The characteristic that differentiates a shrill note from a grave (flat) note.
PITCH $\iff$ FREQUENCY
$$\text{Higher Frequency } (f) \implies \text{Higher Pitch (Shrill Sound)}$$
$$\text{Lower Frequency } (f) \implies \text{Lower Pitch (Grave Sound)}$$
Musical Instruments Pitch Variation:
Stringed Instruments (Guitar, Violin): Pitch increases with higher tension ($T$), shorter string length ($l$), or thinner wire ($m$).
Wind Instruments (Flute): Pitch increases as the vibrating air column length decreases (opening holes).
Membrane Instruments (Drum, Tabla): Pitch increases with smaller, tighter stretched membranes under high tension.
3. Interdependence of Quality (Timbre) and Waveforms
Quality (Timbre): Enables us to distinguish between two notes of the same loudness and same pitch played by different instruments or different people.
QUALITY $\iff$ WAVEFORM
Quality depends strictly on the waveform (shape of wave), which is governed by the mixture, number, and relative amplitudes of subsidiary overtones present along with the fundamental note.
Tuning Fork: Emits a pure sine wave (single fundamental frequency, monotone).
Piano & Violin: Emit fundamental frequency combined with several overtones, creating distinct complex wave shapes.
Q1. A person standing in front of a vertical cliff fires a gun and hears the echo after $3\text{ seconds}$. Calculate the distance of the person from the cliff. (Take speed of sound in air $V = 340\text{ m s}^{-1}$).
Solution:
Given speed of sound in air, $V = 340\text{ m s}^{-1}$.
Time taken to hear echo, $t = 3\text{ s}$.
Using echo formula:
$$ 2d = V \times t \implies d = \frac{V \times t}{2} $$
$$ d = \frac{340\text{ m s}^{-1} \times 3\text{ s}}{2} = \frac{1020}{2} = 510\text{ m} $$
Distance of the person from the cliff = $510\text{ metres}$.
SOLVED NUMERICAL 2: ECHO BETWEEN TWO PARALLEL CLIFFS
Q2. A man stands between two parallel cliffs and fires a gun. He hears the first echo after $2\text{ s}$ and the second echo after $3\text{ s}$. Calculate the total distance between the two cliffs. (Take speed of sound $V = 330\text{ m s}^{-1}$).
Solution:
Let $d_1$ be the distance from the man to the nearer cliff and $d_2$ be the distance to the farther cliff.
Time for 1st echo from nearer cliff, $t_1 = 2\text{ s}$.
$$ d_1 = \frac{V \times t_1}{2} = \frac{330 \times 2}{2} = 330\text{ m} $$
Time for 2nd echo from farther cliff, $t_2 = 3\text{ s}$.
$$ d_2 = \frac{V \times t_2}{2} = \frac{330 \times 3}{2} = 495\text{ m} $$
Total distance between the two cliffs $D = d_1 + d_2$:
$$ D = 330\text{ m} + 495\text{ m} = 825\text{ m} $$
Distance between cliffs = $825\text{ metres}$.
SOLVED NUMERICAL 3: SONAR SEA DEPTH CALCULATION
Q3. A SONAR device on a research ship sends an ultrasonic signal to the ocean floor and receives the echo after $1.6\text{ seconds}$. If the speed of ultrasonic waves in sea water is $1450\text{ m s}^{-1}$, find the depth of the ocean floor.
Solution:
Given speed of ultrasonic wave in sea water, $V = 1450\text{ m s}^{-1}$.
Time delay, $t = 1.6\text{ s}$.
Using SONAR depth formula:
$$ d = \frac{V \times t}{2} = \frac{1450\text{ m s}^{-1} \times 1.6\text{ s}}{2} $$
$$ d = 1450 \times 0.8 = 1160\text{ m} $$
Depth of the ocean floor = $1160\text{ metres}$ ($1.16\text{ km}$).
SOLVED NUMERICAL 4: MOVING SHIP ECHO PROBLEM
Q4. A ship at rest emits a whistle and receives an echo from a cliff after $4\text{ s}$. The ship then moves $170\text{ m}$ towards the cliff and emits a second whistle. After what time will the second echo be heard? (Take speed of sound $V = 340\text{ m s}^{-1}$).
Solution: Step 1: Initial distance $d_1$ of ship from cliff:
$$ d_1 = \frac{V \times t_1}{2} = \frac{340 \times 4}{2} = 680\text{ m} $$
Step 2: New distance $d_2$ after moving $170\text{ m}$ towards cliff:
$$ d_2 = 680\text{ m} - 170\text{ m} = 510\text{ m} $$
Step 3: Time $t_2$ for second echo:
$$ 2d_2 = V \times t_2 \implies t_2 = \frac{2 d_2}{V} = \frac{2 \times 510}{340} = \frac{1020}{340} = 3\text{ s} $$
The second echo will be heard after $3\text{ seconds}$.
CONCEPTUAL QUESTION 5: RESONANCE CONDITION
Q5. A tuning fork of frequency $256\text{ Hz}$ is held vibrating over the mouth of a glass tube filled with water. As water is gradually drained, loud sound is heard when the air column length is $32\text{ cm}$. Name the phenomenon involved and state the natural frequency of the air column.
Solution: Phenomenon:Resonance (a special case of forced vibrations). Natural Frequency of Air Column: At resonance, the natural frequency of the vibrating air column becomes equal to the driving tuning fork frequency = $256\text{ Hz}$.
ICSE BOARD FORMULA & CONCEPT SUMMARY
$\text{Echo Distance: } d = \frac{V \times t}{2} \quad \implies \quad 2d = V \cdot t$
$\text{Minimum Echo Distance in Air (at } 20^\circ\text{C}): d_{\text{min}} = \frac{340 \times 0.1}{2} = 17\text{ m}$
$\text{Persistence of Hearing: } t = 0.1\text{ s} \ (1/10\text{ th of a second})$
$\text{SONAR Depth: } d = \frac{V_{\text{ultrasonic}} \times t}{2}$
$\text{Wave Velocity: } V = f \cdot \lambda \quad \iff \quad f = \frac{1}{T}$
$\text{Natural Vibrations: Vacuum, Constant Amplitude, Frequency } f_0$
$\text{Damped Vibrations: Medium Friction, Decaying Amplitude, Energy Loss}$
$\text{Forced Vibrations: Driven at External Frequency } f$
VARDAAN COMET