Sound is a form of energy that produces the sensation of hearing in our ears. Sound is produced when a material body vibrates, and it reaches us through the vibrations of the particles of the surrounding medium. Thus, sound strictly requires a material medium for its propagation and cannot travel through a vacuum.
The mechanical vibrations of the sounding body set the particles of the adjacent medium into vibration. These vibrations travel outward in the form of waves with a definite velocity that depends on the density ($d$) and elasticity ($E$) of the medium. When these mechanical wave disturbances strike our ear-drums, the sensation of sound is registered by the brain.
Range of Audibility: Audible, Infrasonic & Ultrasonic Waves
The human ear is sensitive only to a limited range of sound frequencies:
Range of Audibility (Audible Sound): $20\text{ Hz}$ to $20,000\text{ Hz}$ ($20\text{ kHz}$). Normal human ears can detect sound waves within this frequency spectrum.
Age-Related Hearing Loss: The audibility range of a person decreases with advancing age because the elasticity and sensitivity of the auditory apparatus decrease for both low and high frequencies.
Infrasonic Sound (Infrasound): Sound waves having frequencies below $20\text{ Hz}$ ($f < 20\text{ Hz}$). Produced by earthquakes, volcanic eruptions, and ocean waves. Inaudible to humans.
Ultrasonic Sound (Ultrasound): Sound waves having frequencies above $20,000\text{ Hz}$ ($f > 20\text{ kHz}$). Produced by bats, dolphins, quartz crystals. Inaudible to humans.
Crucial Principle: Both ultrasonic and infrasonic waves are completely inaudible to human beings, but in any given medium, they travel with the exact same speed as audible sound waves!
FIVE FUNDAMENTAL WAVE PARAMETERS
Amplitude ($a$): When a sound wave travels through a medium, the maximum displacement of the particles of the medium on either side of their mean equilibrium position is called the amplitude of the wave. SI unit: metre ($\text{m}$).
Time Period ($T$): The time taken by a vibrating particle of the medium to complete one full oscillation or vibration is called the time period of the wave. SI unit: second ($\text{s}$).
Frequency ($f$ or $\nu$): The number of complete vibrations made by a particle of the medium in one second is called the frequency of the wave. SI unit: hertz ($\text{Hz}$ or $\text{s}^{-1}$).
$$\text{Relation: } f = \frac{1}{T}$$
Note: The frequency of a wave is equal to the frequency of the vibrating source producing it. The frequency depends ONLY on the source and NEVER changes when the wave enters a different medium!
Wavelength ($\lambda$): The linear distance travelled by a wave disturbance during the time period ($T$) in which a medium particle completes one full vibration. Alternatively, it is the distance between two successive compressions or crests. SI unit: metre ($\text{m}$).
Wave Velocity ($V$): The linear distance travelled by the wave in one second through the medium. SI unit: $\text{m s}^{-1}$.
$$\text{Master Wave Equation: } V = \frac{\text{Distance}}{\text{Time}} = \frac{\lambda}{T} \implies V = f\lambda$$
Note: Wave velocity $V$ depends on the physical properties of the medium (elasticity and density), NOT on the source.
ANATOMY OF A SOUND WAVE: DISPLACEMENT VS DISTANCE
Anatomy of a periodic transverse/graphical wave disturbance illustrating Amplitude ($a$), Crest, Trough, and Wavelength ($\lambda$).
Graphic Generation Specification
Target: Wave Anatomy & Particle VibrationAspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Educational textbook scientific illustration in clean 2D vector style. A sinusoidal sound wave displacement graph plotted on coordinate axes with pure white #ffffff background. Horizontal axis labeled "Distance (x)", vertical axis labeled "Particle Displacement (y)". Clear vibrant blue sine wave showing two complete cycles. Red double-headed vertical arrows indicating Amplitude (a) from zero-line to crest and trough. High-contrast green horizontal dimension line above indicating Wavelength (λ) between two consecutive crests. Sharp, crisp labels with professional textbook typography. No cluttered background, no 3D distortion.
Classification of Mechanical (Elastic) Waves
Sound waves require a material medium and propagate through the back-and-forth oscillations of medium particles about their mean equilibrium positions. During these oscillations, kinetic energy continuously converts into potential energy and vice-versa, which is why sound waves are termed elastic or mechanical waves.
Mechanical waves are divided into two fundamental classes:
Longitudinal Waves:
The particles of the medium vibrate along (parallel to) the direction of propagation of the wave.
The wave advances via alternating regions of high particle density called Compressions ($C$) and low particle density called Rarefactions ($R$).
Because longitudinal waves involve volume changes and all states of matter possess volume elasticity (bulk modulus), longitudinal waves can travel through solids, liquids, and gases.
Examples: Sound waves travelling through air, sound waves inside water or rock strata.
Transverse Waves:
The particles of the medium vibrate perpendicular (normal) to the direction of propagation of the wave.
The wave advances in the form of alternating elevations called Crests and depressions called Troughs.
Transverse waves require elasticity of shape (shear elasticity / rigidity) to propagate. Since liquids and gases lack rigidity, transverse mechanical waves can travel ONLY through solids and along the free surface of liquids (due to surface tension), never inside fluids or gases.
Examples: Transverse waves along a plucked violin or guitar string, ripples on the surface of water.
SPEED OF SOUND IN GASES & STRINGS
1. Speed of Longitudinal (Sound) Wave in a Gaseous Medium (Laplace's Formula):
$$ V = \sqrt{\frac{\gamma P}{d}} $$
Where $P$ is the pressure of the gas, $d$ is the density of the gas, and $\gamma$ is the ratio of principal specific heat capacities ($\gamma = C_p / C_v = 1.4$ for air).
Effect of Temperature: As temperature increases, gas expands and its density $d$ decreases. Since $V \propto 1/\sqrt{d}$, the speed of sound increases with temperature. In air, speed increases by approximately $0.61\text{ m s}^{-1}$ for every $1^\circ\text{C}$ rise in temperature ($V_t = V_0 + 0.61 t$).
Effect of Humidity (Moisture): Water vapour has a lower molecular density than dry air. As humidity increases, the effective density of moist air decreases, causing the speed of sound to increase. Hence, sound travels faster on a humid day than on a dry day!
Effect of Pressure: When pressure changes at constant temperature, the gas density $d$ changes in the exact same proportion ($P/d = \text{constant}$ by Boyle's law). Hence, change in pressure has NO effect on the speed of sound in a gas!
2. Speed of Transverse Wave in a Stretched String:
$$ V = \sqrt{\frac{T}{m}} $$
Where $T$ is the tension in the string in newtons ($\text{N}$), and $m$ is the mass per unit length (linear mass density) of the string ($\text{kg m}^{-1}$).
Refraction of Sound Waves (Crossing Medium Boundary)
When a sound wave travelling in one medium enters another medium of different density or elasticity (refraction):
The speed ($V$) of the wave changes according to the medium properties.
The wavelength ($\lambda$) changes in direct proportion to velocity because $\lambda = V/f$.
The intensity ($I$) decreases because part of the incident wave energy is reflected back at the interface boundary.
The direction of propagation bends (refracts), except for normal incidence ($\angle i = 0^\circ$).
The frequency ($f$) of the sound wave DOES NOT CHANGE because frequency is governed exclusively by the original vibrating source!
Distinction Between Light Waves and Sound Waves
Feature / Property
Light Waves
Sound Waves
1. Wave Nature
These are electromagnetic waves formed by oscillating electric and magnetic fields.
These are mechanical (elastic) waves formed by physical particle oscillations.
2. Medium Requirement
They can travel through vacuum (free space) as well as transparent media.
They strictly require a material medium for propagation (cannot travel in vacuum).
3. Speed in Air
Extremely high: $c = 3 \times 10^8\text{ m s}^{-1}$ in air/vacuum.
Very low: $\approx 330 - 340\text{ m s}^{-1}$ in air at room temperature.
4. Wavelength Range
Very small: visible light is of the order of $10^{-6}\text{ m}$ ($400 - 700\text{ nm}$).
Macroscopic: ranges from $10^{-2}\text{ m}$ to $10\text{ m}$ ($1\text{ cm}$ to $10\text{ m}$).
5. Polarization & Direction
Always transverse waves in all media. Can be polarized.
In gases and liquids, strictly longitudinal waves. (Transverse only in solids/liquid surfaces). Cannot be polarized.
6. Energy Transfer Mode
Transfers energy in discrete packets of electromagnetic quanta called photons.
Transfers energy via kinetic-potential energy handoff across oscillating particles.
7.2 Reflection of Sound Waves
Like light, sound waves return back into the same medium when they strike a hard boundary or obstacle. The return of a sound wave on striking a surface (such as a brick wall, rock cliff, hillside, metal sheet, or wooden board) back into the original medium is called the reflection of sound.
LAWS OF REFLECTION OF SOUND
The angle of incidence ($\angle i$) is strictly equal to the angle of reflection ($\angle r$).
The incident sound wave ray, the reflected sound wave ray, and the normal to the reflecting surface at the point of incidence all lie in the same plane.
Crucial Difference with Reflection of Light: Light waves have extremely short wavelengths ($\approx 10^{-6}\text{ m}$), so they undergo regular reflection only from highly polished, smooth surfaces like mirrors. In contrast, sound waves have large wavelengths ($10^{-2}\text{ m}$ to $10\text{ m}$), so they reflect efficiently from any large, rigid obstacle whether smooth or rough (unpolished brick walls, hilltops, metal plates, plywood). The only requirement is that the linear size of the reflecting surface must be larger than or comparable to the wavelength of the sound wave.
Practical Devices Utilizing Reflection of Sound
Megaphone (Speaking Tube): A horn-shaped hollow conical device. Sound waves produced at the narrow end undergo successive multiple reflections along the diverging interior walls. This prevents the sound energy from spreading out in all directions and confines it into a powerful, focused beam travelling forward.
Sound Board: A large curved concave wooden board placed vertically behind a stage speaker in an auditorium. Sound emitted by the speaker strikes the sound board and, following the laws of reflection, reflects forward as a parallel beam towards the entire audience, ensuring uniform audibility across large halls.
Ear Trumpet: A funnel-shaped listening tube with a wide conical mouth and a narrow tip inserted into the ear. Sound waves entering the broad aperture undergo repeated reflections along the tapering tube, concentrating their energy and delivering amplified sound pressure to the listener's eardrum.
7.3 Echo and Conditions for Hearing an Echo Distinctly
If a person stands at some distance from a tall cliff, building, or hillside and produces a sharp loud sound (such as clapping or firing a starting gun), they hear two distinct sounds:
The original (direct) sound, heard almost instantaneously.
The reflected sound coming back from the obstacle after a perceptible pause, called the echo.
FORMAL DEFINITION OF ECHO
Echo: The sound heard after reflection from a distant obstacle (such as a cliff, a hillside, a tall wall of a building, or the edge of a forest) after the original direct sound has ceased, is called an echo.
Note: If the reflected sound arrives while the original sound is still ringing, it simply overlaps and mixes with the direct sound; this is not an echo. Only when the reflected sound arrives distinctly separate from the direct sound is an echo formed.
GEOMETRY OF ECHO FORMATION: TOTAL DISTANCE = 2d
Echo formation mechanism: Sound traverses a round-trip distance of $2d$ in time interval $t$ at medium velocity $V$.
Graphic Generation Specification
Target: Echo Formation & Minimum Distance DerivationAspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Clean academic physics textbook diagram in landscape format on a pure solid white #ffffff background. On the left side, a stylized student figure producing a sound impulse. On the right side, a tall vertical rocky cliff face. A solid blue curved wavefront travelling from student to cliff labeled "Direct Sound, travels distance d". A dashed magenta curved wavefront returning from cliff to student's ear labeled "Reflected Echo, travels distance d back". A horizontal dimension line between student and cliff labeled "Distance d = Vt/2 = 17 m". Clearly indicate "Total distance = 2d" and "Time delay t ≥ 0.1 s (persistence of hearing)". Crisp, modern flat vector styling with high pedagogical clarity.
⭐ MATHEMATICAL DERIVATION: MINIMUM DISTANCE FOR ECHO FORMATION
1. Persistence of Hearing:
The sensation of any sound persists in the human brain for approximately $0.1\text{ second}$ ($\frac{1}{10}\text{ th}$ of a second) after the exciting source ceases. Therefore, to hear an echo distinctly separated from the original sound, the reflected sound wave must reach the listener's ear at least $0.1\text{ second}$ after the direct sound has ended ($t \ge 0.1\text{ s}$).
2. Round-Trip Distance Formula:
If $d$ is the distance between the listener/source and the reflecting barrier, and $V$ is the velocity of sound in the medium, then the total distance covered by the sound wave going to the obstacle and returning is:
$$ \text{Total Distance} = d + d = 2d $$
$$ \text{Time taken } t = \frac{\text{Total Distance}}{\text{Speed of Sound}} = \frac{2d}{V} \implies d = \frac{V \times t}{2} $$
3. Minimum Distance in Air at Room Temperature:
Substituting $t = 0.1\text{ s}$ and taking the speed of sound in air at ordinary temperature ($20^\circ\text{C}$) as $V = 340\text{ m s}^{-1}$:
(Note: At freezing temperature $0^\circ\text{C}$ where $V = 330\text{ m s}^{-1}$, $d_{\text{min}} = \frac{330 \times 0.1}{2} = 16.5\text{ m}$; at $22^\circ\text{C}$ where $V = 344\text{ m s}^{-1}$, $d_{\text{min}} = 17.2\text{ m}$).
4. Minimum Distance in Sea Water:
In sea water, sound travels much faster ($V \approx 1400\text{ m s}^{-1}$). Hence, the minimum distance to hear a distinct echo under sea water is:
Conclusion: An underwater obstacle must be at least $70\text{ metres}$ away from the source for an echo to be heard distinctly!
Important Consequences: Proximity & Reverberation
If Obstacle Distance is Less Than $17\text{ m}$: The reflected sound returns to the ear in less than $0.1\text{ s}$. The original sound and reflected sound blend into one another, so no distinct echo is heard; the sound is merely prolonged or blurred.
Reverberation: If sound undergoes repeated multiple reflections from the parallel walls, floors, and curved ceiling of an enclosed auditorium or historical monument, the sensation of sound is prolonged for a noticeable time after the source has stopped. This phenomenon is known as reverberation (famously experienced in historical vaulted tombs like the Taj Mahal at Agra and Sikandra).
THREE ESSENTIAL CONDITIONS TO HEAR A DISTINCT ECHO
Minimum Distance: The distance between the sound source and the reflecting obstacle must be at least $17\text{ m}$ in air ($70\text{ m}$ in sea water) so that the time delay $t \ge 0.1\text{ s}$.
Sufficient Size of Reflector: The linear dimensions (area) of the reflecting surface must be large compared to the wavelength ($\lambda$) of the sound wave.
Audible Intensity: The sound produced must have sufficient initial intensity so that after travelling a round trip of $2d$ through the damping medium and undergoing reflection, the returning wave possesses enough energy to produce an audible sensation in the ear.
7.4 Determination of Speed of Sound by the Method of Echo
The echo principle provides an experimental method to measure the speed of sound in air in an open field:
EXPERIMENTAL PROCEDURE & FORMULA
An observer stands at a accurately measured distance $d$ (which should be at least $50\text{ m}$ to provide a clear, comfortably measurable time delay) from a flat vertical wall or rock cliff.
The observer produces a sharp sound (such as firing a pistol or clapping two wooden blocks together) and simultaneously starts a sensitive stopwatch having a least count of $0.01\text{ s}$.
The instant the echo is heard, the stopwatch is stopped, and the elapsed time interval $t$ is recorded.
The speed of sound $V$ in air is calculated using the relation:
$$ V = \frac{\text{Total Distance Travelled}}{\text{Time Interval}} = \frac{2d}{t}\text{ m s}^{-1} $$
Minimizing Human Reaction Error: The experiment is repeated multiple times (often timing 10 or 20 rhythmic claps matched to echoes), and the average value of $V$ is calculated.
7.5 Applications of Echoes: Sound Ranging & Echo Depth Sounding
Echoes find extensive practical use in sound ranging (locating distance and position of obstacles) and echo depth sounding (determining ocean floor depth). These technologies utilize ultrasonic waves ($f > 20\text{ kHz}$).
Three Reasons Why Ultrasonic Waves Are Used Instead of Audible Sound
High Directionality & Undeviated Propagation: Because of their very high frequency and correspondingly short wavelength ($\lambda = V/f$), ultrasonic waves experience negligible diffraction spreading and can travel long distances in a medium without significant deviation.
Narrow Beam Confinement: They can be concentrated and beamed in a sharp, narrow, well-defined direction like a searchlight.
Low Absorption: Ultrasonic waves penetrate dense media (sea water, soft bodily tissues) with minimal absorption and energy dissipation.
Vital Fact: Audible sound waves spread out in all directions due to diffraction and are rapidly absorbed. However, in any given medium, ultrasonic waves travel with the same velocity as audible sound.
1. Use of Echoes by Bats, Dolphins and Fishermen
Bats (Echolocation / Sound Ranging): Bats are nearly blind in darkness, yet they fly at high speeds through forests and capture tiny flying insects with uncanny precision. They emit high-frequency ultrasonic squeaks (up to $100\text{ kHz}$). When these ultrasonic waves strike an obstacle or prey, they reflect back as echoes. By analyzing the time delay, intensity, and frequency shift of the returning echo, bats determine the exact distance, direction, and speed of the target. This process is called sound ranging.
Dolphins: Dolphins navigate murky waters and detect enemies or schools of fish by emitting clicking ultrasonic pulses and interpreting the reflected echoes.
Fishermen & Trawlers (Fish Finders): Modern fishing vessels use ultrasonic echo sounders. A transducer mounted on the hull transmits high-frequency ultrasonic pulses into the sea. When the pulses encounter a shoal of fish, partial reflections return to the receiver. The depth $d$ of the fish shoal is computed using $d = \frac{Vt}{2}$ ($V \approx 1400\text{ m s}^{-1}$).
2. Use of Echoes by 'SONAR'
SONAR stands for SOund Navigation And Ranging. It is an acoustic apparatus installed on ships and submarines to locate submerged obstacles (enemy submarines, icebergs, shipwrecks) and to measure the depth of the sea floor (a process termed echo depth sounding or bathymetry).
Book Figure Placeholder: Fig. 7.1
Fig. 7.1: Principle of SONAR
Illustrates a ship floating on sea water equipped with a Transmitter ($T$) and a Receiver ($R$) mounted close to each other on the keel. Ultrasonic pulses travel downwards over depth $d$, strike an underwater obstacle or sea bed, and reflect upwards to the receiver.
Book Figure 7.1 Slot — Ready for direct image placement using <img class="img-in-note">
WORKING PRINCIPLE OF SONAR: TRANSMITTER, RECEIVER & SEA BED
SONAR operational setup: Ultrasonic pulses are emitted by Transmitter ($T$), traverse depth $d$, reflect off the seabed, and return to Receiver ($R$).
Graphic Generation Specification
Target: SONAR Echo Depth SoundingAspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Textbook-quality physics cross-section illustration in landscape orientation on a pure white #ffffff background. A modern marine exploration ship on the surface of calm light-blue seawater. Below the ship, deep clear ocean water leading down to a rocky seabed and a sunken submarine obstacle. A bright red ultrasonic transmitter labeled "Transmitter (T)" under the ship emitting sharp concentrated pink sound waves downward. A bright green detector labeled "Receiver (R)" receiving reflected dashed green echo beams. A clear vertical measurement arrow indicating ocean depth "d" with formula "d = (V × t) / 2". Highly detailed, clean scientific vector style with elegant labels.
SONAR DEPTH FORMULA & RADAR COMPARISON
If $V$ is the speed of ultrasonic waves in sea water ($\approx 1400\text{ m s}^{-1}$) and $t$ is the round-trip transit time:
$$ \text{Total Distance Covered} = 2d = V \times t \implies d = \frac{V \times t}{2} $$
Note 1 — Distinction Between SONAR and RADAR:
SONAR (Sound Navigation And Ranging): Uses ultrasonic mechanical sound waves to detect objects underwater, because radio waves attenuate rapidly in sea water.
RADAR (Radio Detection And Ranging): Uses electromagnetic waves (radio waves or microwaves) travelling at $c = 3 \times 10^8\text{ m s}^{-1}$ in air or space to detect aircraft and distant objects.
Note 2 — Placement of Apparatus: In actual SONAR and RADAR installations, the transmitter and receiver are installed immediately adjacent to each other on the vessel hull. They are shown separated in schematics solely for diagrammatic clarity.
3. Use of Echoes in the Medical Field
In medical diagnostics, ultrasonic echoes provide non-invasive, radiation-free imaging of internal tissues:
Ultrasonography (USG): High-frequency ultrasonic beams are directed into the human body. As the waves encounter boundaries between different soft tissues and organs (such as the liver, gall bladder, kidneys, uterus, and growing fetus), partial reflections (echoes) occur. These reflected signals are converted by a digital processor into high-resolution, real-time images on a monitor.
Echocardiography (ECHO): Ultrasonic pulses are directed at the heart to obtain dynamic moving images of cardiac chambers, valves, and blood flow velocity.
Why Ultrasound is Superior to X-Rays: Unlike X-rays, ultrasonic waves are completely non-ionizing; they cause zero biological tissue damage and are 100% safe for examining pregnant mothers and unborn babies.
Graphic Generation Specification
Target: Medical Ultrasonography & EchocardiographyAspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Medical physics educational illustration in landscape format on a pure white #ffffff background. Shows an ultrasound probe transducer placed against human skin emitting high-frequency ultrasonic pulse beams into internal tissue. Ultrasonic pulses partially reflect off the boundary of an internal organ (e.g., heart or liver), generating returning echo signals. The transducer converts these echoes into electrical pulses displayed as a clear cross-sectional anatomical image on an adjacent digital monitor. Clean line art, hospital diagnostic theme, clear annotations: "Transducer Probe", "Incident Ultrasound Pulses", "Tissue Boundary", "Reflected Echoes", and "Digital Image Reconstruction". Pure white background, no clutter.
(B) NATURAL, DAMPED AND FORCED VIBRATIONS; RESONANCE
7.6 Natural (or Free) Vibrations
A body clamped or pivoted at one point, when disturbed slightly from its rest (equilibrium) position and released, starts vibrating back and forth. The vibrations so produced are called the natural or free vibrations of the body.
DEFINITION: NATURAL (FREE) VIBRATIONS
Natural (or Free) Vibrations: The periodic vibrations of a body in the complete absence of any external force or friction acting on it, are called the natural (or free) vibrations.
Key properties of natural vibrations:
Natural Period & Natural Frequency ($f_0$): The fixed time period of the body is called its natural period and the corresponding number of vibrations per second is called its natural frequency ($f_0$). The natural frequency depends solely on the shape, size, mass, and elastic structure of the body.
Constant Amplitude & Frequency: Each body capable of vibrating executes oscillations with a constant natural frequency and its amplitude of vibration remains strictly constant indefinitely.
Ideal Occurrence in Vacuum: Pure natural vibrations can occur only in vacuum. In any material medium (such as air or liquid), the surrounding particles offer frictional resistance (damping), which dissipates energy and continuously reduces the amplitude until motion ceases.
Detailed Examples of Natural (Free) Vibrations
Simple Pendulum:
When the bob of a simple pendulum is displaced slightly from its mean position and released, it oscillates with its natural frequency governed by its effective length $l$ and the local acceleration due to gravity $g$:
$$ f = \frac{1}{2\pi}\sqrt{\frac{g}{l}} $$
Different pendulums having different lengths vibrate with different natural frequencies. For example, a simple pendulum of length $l = 1.0\text{ m}$ on Earth's surface ($g = 9.8\text{ m s}^{-2}$) has a natural frequency of:
$$ f = \frac{1}{2\pi}\sqrt{\frac{9.8}{1.0}} \approx 0.5\text{ Hz} \quad (\text{Time period } T = 2.0\text{ s}) $$
Load Suspended from a Helical Spring:
When a mass $m$ attached to a spring of force constant (hardness) $K$ is pulled down and released, it vibrates with natural frequency:
$$ f = \frac{1}{2\pi}\sqrt{\frac{K}{m}} $$
(The force constant $K$ is the restoring force produced per unit elongation, measured in $\text{N m}^{-1}$). The frequency increases with a stiffer spring (higher $K$) and decreases with a heavier load (higher $m$).
Tuning Fork:
When one prong of a tuning fork is struck against a hard rubber pad, its prongs vibrate back and forth at a fixed natural frequency determined by their length, thickness, and elasticity. The sound emitted consists of a single frequency (called a pure note or monotone).
Piano Strings:
When a piano key is struck, a felt hammer hits a steel string of specific length, tension, and thickness, exciting it to vibrate at its unique natural frequency.
Air Column in Flutes and Organ Pipes:
When an air column inside a pipe is set into vibration, it vibrates with a natural frequency that is inversely proportional to the effective length ($l$) of the air column ($f \propto 1/l$). In a flute, musical notes of different frequencies are produced by opening or closing finger holes, which changes the effective length of the vibrating air column.
In an organ pipe open at both ends: The frequencies of different natural modes are in the integer ratio:
$$ f_1 : f_2 : f_3 : \dots = 1 : 2 : 3 : \dots $$
In an organ pipe closed at one end: The frequencies of different natural modes are in the odd integer ratio:
$$ f_1 : f_2 : f_3 : \dots = 1 : 3 : 5 : \dots $$
Stretched Strings in Musical Instruments (Sitar, Guitar, Violin):
When a stretched wire clamped at both ends is plucked, it executes transverse vibrations of a definite natural frequency. The natural frequency $f$ of the fundamental note depends on:
The vibrating length $l$ of the string: $f \propto \frac{1}{l}$
The radius (thickness) $r$ of the string: $f \propto \frac{1}{r}$
The tension $T$ in the string: $f \propto \sqrt{T}$
LAW OF STRETCHED STRINGS
$$ f = \frac{1}{2l}\sqrt{\frac{T}{m}} = \frac{1}{2l}\sqrt{\frac{T}{\pi r^2 d}} $$
Where $T$ is the tension ($\text{N}$), $m$ is the mass per unit length ($\text{kg m}^{-1}$), $r$ is the radius of the wire ($\text{m}$), and $d$ is the density of the wire material ($\text{kg m}^{-3}$).
To increase the pitch (frequency) of a string: (a) Decrease the vibrating length $l$, (b) Decrease the radius $r$ (use a thinner string), or (c) Increase the tension $T$ (tighten the tuning peg).
Modes of Vibration in a Stretched String:
A string of length $l$ stretched under tension can vibrate in different stationary modes depending on where it is plucked:
Principal Note (Fundamental): Plucked in the middle ($l/2$) $\implies$ vibrates in one loop. Frequency $= f$, Wavelength $\lambda_1 = 2l$.
First Subsidiary Note (First Overtone / 2nd Harmonic): Plucked at one-fourth length ($l/4$) from one end $\implies$ vibrates in two loops. Frequency $= 2f$, Wavelength $\lambda_2 = \frac{2l}{2} = l$.
Second Subsidiary Note (Second Overtone / 3rd Harmonic): Plucked at one-sixth length ($l/6$) from one end $\implies$ vibrates in three loops. Frequency $= 3f$, Wavelength $\lambda_3 = \frac{2l}{3}$.
Harmonic Ratios of Stretched String
Ratio of Frequencies: $f_1 : f_2 : f_3 = 1 : 2 : 3$
Ratio of Wavelengths: $\lambda_1 : \lambda_2 : \lambda_3 = 2l : l : \frac{2l}{3} = 3 : 2 : 1$
Book Figure Placeholder: Fig. 7.5
Fig. 7.5: Different Modes of Vibrations in a Stretched String
Illustrates the three modes of vibration of a stretched string: (a) Single loop (Principal note, Frequency $f$, $\lambda = 2l$), (b) Two loops (First subsidiary, Frequency $2f$, $\lambda = l$), and (c) Three loops (Second subsidiary, Frequency $3f$, $\lambda = 2l/3$).
Book Figure 7.5 Slot — Ready for direct image placement using <img class="img-in-note">
HARMONIC MODES OF A STRETCHED STRING (LOOPS, FREQUENCIES & WAVELENGTHS)
Modes of transverse vibrations in a stretched string clamped at both ends showing nodal points, antinodes, and harmonic ratios.
Graphic Generation Specification
Target: Modes of Vibration in a Stretched String (Fig. 7.5)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics textbook vector diagram in landscape orientation on a pure white #ffffff background. Three horizontal panels showing a clamped stretched string: (a) top panel showing one full standing loop with stationary nodes at clamps and maximum antinode displacement in the middle, labeled "Fundamental Note / Principal Note: Frequency f, Wavelength λ = 2l"; (b) middle panel showing two symmetrical loops with a central node, labeled "First Subsidiary Note: Frequency 2f, Wavelength λ = l"; (c) bottom panel showing three equal loops with two internal nodes, labeled "Second Subsidiary Note: Frequency 3f, Wavelength λ = 2l/3". High-contrast vibrant blue string lines with dashed counter-oscillating envelopes, clean dark clamp supports at ends, crisp typography.
Nature of Natural Vibrations & Displacement-Time Graph
Natural vibrations are simple harmonic vibrations executed under the action of the internal restoring force of the body alone. The restoring force is directly proportional to displacement from the mean position ($F = -kx$). Once set into vibration in vacuum, the body continues to oscillate with the same amplitude and same frequency forever.
Book Figure Placeholder: Fig. 7.6
Fig. 7.6: Displacement-Time Graph for Natural or Free Vibration (In Vacuum)
Depicts a constant-amplitude sinusoidal wave oscillating symmetrically between $+a$ and $-a$ over time $t$, demonstrating zero energy dissipation in the absence of external resistive forces.
Book Figure 7.6 Slot — Ready for direct image placement using <img class="img-in-note">
DISPLACEMENT-TIME GRAPH: NATURAL VIBRATIONS IN VACUUM (CONSTANT AMPLITUDE)
Fig. 7.6: Displacement-time curve for natural vibrations in vacuum showing constant amplitude $a$ and constant frequency $f_0$.
Graphic Generation Specification
Target: Natural Vibration Displacement-Time Graph (Fig. 7.6)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics textbook graph illustration in landscape format with pure white #ffffff background. Coordinate axes: horizontal axis labeled "Time (t)" with arrow pointing right, vertical axis labeled "Displacement" with arrow pointing up, origin labeled "O". A regular sinusoidal wave in vivid blue oscillating continuously with constant peak amplitude. Horizontal red dashed boundary lines at positive maximum labeled "+a" and negative maximum labeled "-a". Wave peaks and troughs touch the red dashed lines perfectly across all cycles. Clean, high-contrast flat 2D vector style.
7.7 Damped Vibrations
In real-world conditions, any body vibrating in a material medium (air, water, oil) continuously encounters resistive forces (fluid friction, viscosity). As a consequence, the amplitude of vibration gradually diminishes over time and the body eventually comes to rest.
DEFINITION: DAMPED VIBRATIONS
Damped Vibrations: The periodic vibrations of a body of continuously decreasing amplitude in the presence of a resistive (frictional) force are called damped vibrations.
Two Forces Acting on a Body Under Damped Vibrations:
Restoring Force: An internal elastic force directed toward the mean equilibrium position ($F_{\text{restoring}} = -kx$). It tends to keep the body oscillating.
Frictional (Damping) Force: A resistive force exerted by the surrounding medium that opposes the motion at every instant. The magnitude of this frictional force is directly proportional to the instantaneous velocity of the vibrating body ($F_{\text{friction}} \propto v$).
Energy Dissipation: In each oscillation, the body does mechanical work against the frictional force of the medium. Consequently, mechanical kinetic and potential energy is continuously dissipated as heat energy into the surrounding medium. The rate of energy loss (and the rate of decay of amplitude) depends upon:
The nature of the surrounding medium (its viscosity and density).
The shape, surface area, and linear dimensions of the vibrating body.
Book Figure Placeholder: Fig. 7.7
Fig. 7.7: Displacement-Time Graph for Damped Vibrations
Shows an oscillatory wave whose amplitude decays exponentially towards zero over time within an envelope bounded by symmetrical curves converging asymptotically on the time axis.
Book Figure 7.7 Slot — Ready for direct image placement using <img class="img-in-note">
Fig. 7.7: Displacement-time curve for damped vibrations in air/medium showing amplitude decaying exponentially to zero due to energy dissipation.
Graphic Generation Specification
Target: Damped Vibration Displacement-Time Graph (Fig. 7.7)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics textbook graph illustration in landscape format with pure white #ffffff background. Coordinate axes: horizontal axis labeled "Time (t)", vertical axis labeled "Displacement", origin labeled "O". An oscillatory wave drawn in vibrant magenta/crimson whose amplitude starts at high peak (+a and -a) and decays exponentially with each successive cycle, tapering smoothly down towards zero. Dashed orange upper and lower boundary envelope curves converging toward the central axis at the far right. Crystal-clear vector line work, elegant educational labeling.
Everyday Examples of Damped Vibrations
Branch of a Tree: When a slim branch is pulled down and released, it oscillates with decaying amplitude and quickly comes to rest.
Tuning Fork in Air: When struck and held in air, the audible sound grows progressively fainter until it becomes inaudible and the prongs stop vibrating.
Simple Pendulum in Air: An oscillating pendulum bob gradually loses mechanical energy to air resistance and stops at its equilibrium position.
Loaded Spring in Air / Liquid: A vibrating mass on a spring stops rapidly in water or viscous oil due to high viscous drag.
Distinction Between Natural (Free) and Damped Vibrations
Feature / Parameter
Natural (or Free) Vibrations
Damped Vibrations
1. Amplitude Behaviour
The amplitude of vibration remains strictly constant and oscillations continue indefinitely.
The amplitude gradually decreases with time and vibrations ultimately cease altogether.
2. Energy Status
There is zero loss of energy from the vibrating system. Mechanical energy is conserved.
In each vibration, mechanical energy is dissipated as heat into the surrounding medium.
3. Forces Acting
No external or resistive force acts on the body; vibrations occur under the restoring force alone.
In addition to restoring force, a frictional (damping) resistive force opposes motion.
4. Frequency of Vibration
The frequency depends solely on the shape and size of the body and remains constant indefinitely ($f_0$).
The frequency of vibration is slightly smaller than the natural frequency of the body.
7.8 Forced Vibrations
In a resistive medium, vibrations naturally die out due to damping. However, if a continuous external periodic force is applied to the body such that it continuously feeds energy into the system to compensate for frictional losses, the body can be maintained in vibration. Such vibrations are called forced vibrations.
DEFINITION: FORCED VIBRATIONS
Forced Vibrations: The vibrations of a body which take place under the influence of an external periodic force acting on it, are called the forced vibrations.
Three Forces Acting on a Body in Forced Vibrations:
The internal elastic restoring force ($F = -kx$).
The frictional (resistive) force of the medium ($F_{\text{friction}} \propto v$).
The external periodic force (also termed the driving force): $F_{\text{ext}} = F_0 \sin(\omega t)$.
Frequency & Amplitude of Forced Vibrations
Acquired Frequency: The body no longer vibrates with its own natural frequency ($f_0$); it is forced to vibrate with the frequency of the applied external periodic force ($f$)!
Amplitude Magnitude: The amplitude of forced vibrations is governed by the difference between the driving frequency ($f$) and the natural frequency ($f_0$):
If the driving frequency is far different from the natural frequency ($f \ne f_0$), the amplitude of oscillations is very small.
If the driving frequency is close to or matches the natural frequency ($f = f_0$), the amplitude becomes exceptionally large (leading to resonance).
Amplitude Stability: Once established, the amplitude of forced vibration does not decay with time, as the driving force continuously supplies the required energy.
Everyday Examples of Forced Vibrations
Tuning Fork Stem on a Wooden Table Top: When the stem of a vibrating tuning fork is pressed firmly against a table top, the vibrating fork forces the wooden surface to oscillate at the fork's own frequency. Because the table top has a much larger surface area than the small prongs of the tuning fork, its forced vibrations displace a much larger volume of air, radiating significantly more sound energy and producing a much louder sound!
Microphone Diaphragm: The light metallic diaphragm of a microphone undergoes forced vibrations at the exact varied frequencies of the sound waves spoken into it.
Artist Plucking Strings on a Guitar: An artist repeatedly plucks strings, continuously supplying periodic force to maintain sound output.
Hollow Sound Box of Stringed Instruments: Instruments like guitars, violins, and sitars are provided with a hollow wooden box containing air. When a string is plucked, its vibrations force the large body of air inside the sound box to execute forced vibrations. The large vibrating surface area sends forth substantial acoustic energy, amplifying the loudness.
Distinction Between Natural (Free) and Forced Vibrations
Feature / Parameter
Natural (or Free) Vibrations
Forced Vibrations
1. External Driver
Vibrations taking place in the absence of any resistive or external driving force.
Vibrations taking place in a medium under the continuous influence of an external periodic force.
2. Frequency Dependence
The frequency depends solely on the shape, size, and elastic properties of the body.
The frequency is equal to the frequency of the applied periodic force (independent of the body's natural frequency).
3. Frequency Invariance
The frequency of vibration remains strictly constant.
The frequency changes whenever the frequency of the applied driving force is altered.
4. Amplitude Characteristics
The amplitude remains constant indefinitely and depends on the initial displacement.
The amplitude depends on the difference between driving and natural frequencies (small when $f \ne f_0$, huge when $f = f_0$).
7.9 Resonance (A Special Case of Forced Vibrations)
When a body is driven by an external periodic force whose frequency happens to coincide exactly with the body's natural frequency, a dramatic physical phenomenon occurs: the body absorbs energy at a maximal rate and oscillates with an extraordinary surge in amplitude. This special condition is called resonance.
DEFINITION & PRINCIPLE OF RESONANCE
Resonance: Resonance is a special case of forced vibrations. When the frequency of the externally applied periodic force on a body is exactly equal to the natural frequency of the body ($f_{\text{ext}} = f_0$), the body readily begins to vibrate with a very large (maximum) amplitude. This phenomenon is known as resonance.
The large-amplitude vibrations executed by the body are termed resonant vibrations.
Physical Mechanism: Why Loud Sound Occurs at Resonance
Suppose a body has a natural frequency $f$. Let a periodic driving force of frequency $n$ act on it:
Case 1 ($n = f$ — Resonance): In each cycle, the external force acts in perfect phase with the natural motion of the body. Energy supplied by the driver adds cumulatively to the system. The amplitude builds up to a large peak value until energy supplied per cycle equals energy dissipated by friction. Because the body vibrates with huge amplitude, it sets a large mass of air into vibration, sending forth substantial energy and producing a very loud sound!
Case 2 ($n \ne f$ — No Resonance): The external force falls out of phase with the body's natural oscillations, alternately aiding and opposing the motion. Very little energy is absorbed, resulting in forced vibrations of small amplitude and weak sound.
RESONANCE CURVE: AMPLITUDE PEAK AT f = f₀
Resonance curve showing dramatic surge in vibration amplitude when driving frequency $f$ matches natural frequency $f_0$.
Graphic Generation Specification
Target: Resonance Curve & Amplitude PeakAspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Educational physics textbook diagram in landscape format on a pure white #ffffff background. A 2D coordinate system with horizontal axis labeled "Driving Frequency (f)" and vertical axis labeled "Amplitude of Vibration (a)". A bell-shaped resonance curve drawn in rich royal purple, starting low at left, rising sharply into a towering narrow peak at the center, and descending symmetrically to a low baseline on the right. A vertical dashed red line drops from the peak to the horizontal axis at point labeled "f = f₀ (Natural Frequency)". An annotation box with arrow pointing to the summit labeled "Resonant Peak: Maximum Energy Transfer & Amplitude". Sharp vector style, clean academic typography.
Experimental Demonstrations of Resonance
EXPERIMENT 1: RESONANCE WITH TUNING FORKS
Setup: Mount two identical tuning forks A and B having the exact same natural frequency on two separate hollow wooden sound boxes whose open apertures face each other at a distance of about $1\text{ metre}$.
Book Figure Placeholder: Fig. 7.8
Fig. 7.8: Resonance with Tuning Forks
Shows two identical tuning forks A and B of identical frequency $n$ mounted vertically on separate hollow wooden sound boxes with their open ends facing each other, coupled by acoustic air waves.
Book Figure 7.8 Slot — Ready for direct image placement using <img class="img-in-note">
ACOUSTIC RESONANCE BETWEEN IDENTICAL TUNING FORKS A & B
Fig. 7.8: Acoustic resonance: Striking fork A sets the air column of box A into vibration; transmitted sound waves resonate with identical fork B.
Graphic Generation Specification
Target: Tuning Fork Acoustic Resonance Experiment (Fig. 7.8)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Scientific laboratory physics diagram in landscape orientation on a pure white #ffffff background. Two identical steel tuning forks A and B mounted vertically on two hollow polished wooden resonance sound boxes sitting on a laboratory table. The open mouth apertures of the two wooden boxes face each other. Tuning fork A is shown vibrating with blue motion arcs around its prongs. Curved periodic sound wave arcs travel through the air gap between the open mouths of the boxes. Tuning fork B on the right begins vibrating sympathetically with bold red motion blur arcs, demonstrating resonance. Clean labels: "Tuning Fork A (Frequency n)", "Sound Box A", "Sound Waves in Air", "Sound Box B", "Tuning Fork B (Resonating)". Crisp vector graphics.
Physical Mechanism: When prong A is struck and placed on sound box A, it sets the air column inside box A into forced vibrations. These vibrations radiate out through the open mouth into the surrounding air and enter the open mouth of box B, forcing the air column in box B to vibrate at frequency $n$. Because fork B has the exact same natural frequency ($n$), it readily absorbs energy from the vibrating air column through resonance and begins to vibrate vigorously. If prong A is touched and stopped, fork B is heard singing loudly!
EXPERIMENT 2: FORCED & RESONANT VIBRATIONS OF PENDULUMS
Setup: Four simple pendulums A, B, C, and D are suspended from a common tightly stretched horizontal elastic cord or rubber string PQ.
Pendulums A and B have the exact same length ($l_A = l_B$).
Pendulum C is shorter ($l_C < l_A$), so its natural frequency is higher.
Pendulum D is longer ($l_D > l_A$), so its natural frequency is lower.
Book Figure Placeholder: Fig. 7.9
Fig. 7.9: Forced and Resonant Vibrations of Pendulums
Four pendulums A, B, C, D suspended from a common horizontal stretched string PQ. Pendulums A and B have equal length; C is shorter; D is longer. Setting A in motion drives B into violent resonance.
Book Figure 7.9 Slot — Ready for direct image placement using <img class="img-in-note">
BARTON'S PENDULUMS: RESONANCE IN EQUAL-LENGTH PENDULUMS A & B
Fig. 7.9: Barton's Pendulums demonstration: Setting pendulum A oscillating drives string PQ. Pendulum B (equal length $l$) oscillates with maximum amplitude due to resonance, while C and D have small amplitudes.
Graphic Generation Specification
Target: Barton's Pendulum Resonance Setup (Fig. 7.9)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Physics laboratory experiment diagram in landscape format on a pure white #ffffff background. A horizontal elastic cord PQ clamped firmly between two vertical rigid end supports. Four simple pendulums labeled A, C, B, D hanging from the cord: Pendulum A on the left has standard length l; Pendulum C is short; Pendulum B has the exact same length l as A; Pendulum D is long. Pendulum A is shown oscillating into the page. Pendulum B is swinging with very large, dramatic red amplitude arcs (indicating resonance). Pendulums C and D have very small, barely noticeable oscillations. Clear vector style with explicit labels: "Stretched Cord PQ", "Driver Pendulum A (Length l)", "Resonant Pendulum B (Length l - Maximum Amplitude)", "Non-resonant C and D (Small Amplitude)".
Observation: Initially, pendulum B starts vibrating with a small amplitude, and gradually it acquires the exact same amplitude that pendulum A initially had. When the amplitude of pendulum B reaches its maximum, the amplitude of pendulum A drops to a minimum (due to continuous exchange and sharing of energy between the two coupled resonators). Over time, the amplitude of B decreases while that of A increases again. Furthermore, the vibrations of pendulum B are strictly in phase with those of pendulum A (i.e., both reach their extreme positions on the same side simultaneously). In contrast, pendulums C and D vibrate with very small, negligible amplitudes.
Explanation: The periodic vibrations produced in pendulum A are communicated as a periodic driving force to pendulums B, C, and D through the stretched rubber string PQ. Pendulums C and D remain in a state of ordinary forced vibrations. Pendulum B, however, enters a state of resonance because its natural frequency is exactly equal to that of A ($l_B = l_A \implies f_B = f_A$). Energy is exchanged rhythmically between pendulums A and B, so pendulum B vibrates with maximum amplitude at the frequency of A and in phase with A.
EXPERIMENT 3: RESONANCE IN AN AIR COLUMN
Apparatus: A resonance tube apparatus consists of a long vertical graduated glass tube A (held fixed) and a metallic cylindrical reservoir vessel B containing water. Tubes A and B are connected at their lower ends by a flexible rubber tube. The movable vessel B can be raised or lowered and clamped at any height on a laboratory stand. Water from vessel B enters tube A, where the water surface forms a rigid, reflecting closed boundary. The column of air between the water meniscus and the open top lip of tube A acts as a closed-end organ pipe.
Book Figure Placeholder: Fig. 7.10
Fig. 7.10: Resonance in Air Column Apparatus
Shows vertical graduated tube A connected via flexible rubber tubing to movable water reservoir B, with a vibrating tuning fork positioned horizontally just above the open mouth of tube A.
Book Figure 7.10 Slot — Ready for direct image placement using <img class="img-in-note">
RESONANCE TUBE EXPERIMENTAL APPARATUS (WATER LEVEL & AIR COLUMN LENGTH)
Fig. 7.10: Experimental resonance tube apparatus. Raising or lowering reservoir B adjusts the air column length $l$ to match tuning fork frequency.
Graphic Generation Specification
Target: Resonance Tube Air Column Apparatus (Fig. 7.10)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Laboratory physics illustration in landscape orientation on a pure white #ffffff background. A vertical clear glass tube A held on a metal stand, connected at the bottom via flexible black rubber tubing to a metal water reservoir vessel B mounted on a second stand. Light blue water partially fills both vessels. A vibrating metal tuning fork is held horizontally just above the top mouth of glass tube A. Red sound wave arcs enter the tube. An explicit vertical dimension line next to tube A labeled "Air Column Length (l)". Clearly annotated labels: "Tuning Fork", "Air Column (Resonator)", "Water Surface (Reflector)", "Graduated Scale", "Flexible Tube", and "Movable Water Reservoir B". Modern 2D educational textbook styling.
Procedure & Observations:
Tube A is initially filled with water almost to the top. A vibrating tuning fork of known frequency $f$ is held horizontally just above its open mouth.
Reservoir B is slowly lowered. As water drains into B, the water level in tube A falls, steadily increasing the length $l$ of the air column.
First Resonance ($l_1$): At a specific water level, a very loud sound is heard! At this point, the natural frequency of the air column ($f_{\text{air}} \propto 1/l$) becomes equal to the frequency of the tuning fork, establishing resonance.
Subsequent Water Lowering: As the water level is lowered further, the frequency condition is broken and the loud sound completely ceases.
Second Resonance ($l_2 = 3l_1$): When the air column is prolonged until its length becomes three times the first resonance length ($l_2 = 3l_1$), a loud resonant sound is heard once again! This corresponds to the next natural harmonic mode of a closed air pipe (where frequencies follow the odd integer ratio $1 : 3 : 5 : \dots$).
Distinction Between Forced Vibrations and Resonant Vibrations
Feature / Condition
Forced Vibrations
Resonant Vibrations
1. Driving Frequency Condition
The frequency of the external periodic force is different from the natural frequency of the body ($f_{\text{ext}} \ne f_0$).
The frequency of the external periodic force is exactly equal to the natural frequency of the body ($f_{\text{ext}} = f_0$).
2. Amplitude of Vibration
The amplitude of vibration is usually very small.
The amplitude of vibration is exceptionally large (maximum).
3. Phase Relationship
The vibrations of the body are not in phase with the external periodic driving force.
The vibrations of the body are strictly in phase with the external periodic driving force.
4. Persistence After Cessation
Vibrations cease almost immediately after the external force stops acting.
Vibrations persist for a noticeably long time after the external periodic force has ceased.
7.10 Comprehensive Real-World Examples & Applications of Resonance
Resonant Vibrations of Barton's Pendulums: When two pendulums of equal length suspended from the same support are coupled, the driver transfers energy exclusively into the equal-length pendulum, making it swing with enormous amplitude while unequal pendulums barely move.
Resonance in Machine Parts & Vehicle Rattling: When a vehicle runs, its engine pistons reciprocate at a frequency directly determined by the vehicle's speed (RPM). These mechanical vibrations are transmitted throughout the chassis. At a certain critical speed, the piston frequency matches the natural frequency of a loose body panel, frame member, or window pane. That part enters violent resonance and produces an intense rattling sound. To eliminate the rattle, the driver simply alters the vehicle speed, immediately breaking the resonance condition.
Resonance in Musical Instrument Sound Boxes & Sonometer: A vibrating string by itself displaces very little air and produces an extremely faint sound that cannot be heard at a distance. Therefore, stringed instruments (guitar, violin, sitar, cello) and laboratory sonometers are built with a hollow wooden sound box. The enclosed air cavity has natural modes matching string frequencies. When the string vibrates, the air column inside the sound box vibrates in resonance. Its large surface area displaces huge volumes of air, emitting a rich, powerful, and amplified musical sound.
Resonance in Air Column and Tuning Fork: As demonstrated in the resonance tube experiment, when the acoustic frequency of a vibrating air column matches the frequency of a tuning fork vibrating at its open mouth, maximum amplitude standing waves are formed and a loud sound is produced.
Resonance in a Suspension Bridge (Soldiers Breaking Step): When a company of troops marches over a suspension bridge, they are strictly ordered to break step. If they march in lockstep rhythm, the frequency of their collective footsteps could match the natural frequency of the suspension bridge ($f_{\text{steps}} = f_{\text{bridge}}$). This would induce resonance, causing catastrophic structural oscillation that could snap bridge cables and cause fatal collapse (as historically occurred at the Broughton Suspension Bridge in 1831).
Resonance in Radio and TV Receivers: Radio tuning circuits use an inductor-capacitor ($LC$) resonant tank circuit. Broadcast radio waves of myriad different frequencies impinge upon the antenna. When you turn the radio dial, you adjust the capacitance ($C$) of a variable capacitor, altering the natural electrical frequency of the receiver circuit ($f = \frac{1}{2\pi\sqrt{LC}}$). When the circuit frequency matches the broadcast frequency of the chosen station, electrical resonance occurs; current for that station surges to maximum amplitude, enabling clear amplification and playback while filtering out all other transmissions.
(C) CHARACTERISTICS OF SOUND & THEIR SUBJECTIVE AND OBJECTIVE NATURE
7.11 Three Fundamental Characteristics of Sound
Two musical sounds can be differentiated from one another by three distinctive perceptual characteristics:
Loudness: Differentiates a loud sound from a faint sound. Governed by amplitude ($a$).
Pitch (or Shrillness): Differentiates a sharp/shrill sound from a grave/flat sound. Governed by frequency ($f$).
Quality (or Timbre): Differentiates sounds of the same loudness and pitch emitted by different instruments. Governed by waveform.
Experimental Investigation: Cathode Ray Oscilloscope (C.R.O.) Setup
The wave pattern of a sound wave is studied experimentally using a microphone connected to the Y-input terminals of a Cathode Ray Oscilloscope (C.R.O.). The microphone converts sound pressure variations into proportional electrical voltage signals, which are displayed on the C.R.O. fluorescent screen as a real-time displacement-time graph.
Book Figure Placeholder: Fig. 7.17
Fig. 7.17: Set-up to Study the Sound Pattern (C.R.O. & Microphone)
Shows a vibrating tuning fork positioned in front of a sensitive microphone. The microphone leads connect to the Y-input terminals of a Cathode Ray Oscilloscope (C.R.O.), which displays the displacement-time wave pattern on its circular screen.
Book Figure 7.17 Slot — Ready for direct image placement using <img class="img-in-note">
EXPERIMENTAL SETUP: CATHODE RAY OSCILLOSCOPE (C.R.O.) & SOUND TRANSDUCER
Fig. 7.17: Experimental arrangement to study sound waveforms: Acoustic vibrations are converted by a microphone into electrical signals and displayed on a C.R.O. screen.
Graphic Generation Specification
Target: C.R.O. & Microphone Acoustic Analysis Setup (Fig. 7.17)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Scientific laboratory physics illustration in landscape orientation on a pure white #ffffff background. On the left side, a modern Cathode Ray Oscilloscope (C.R.O.) laboratory device with a circular display screen showing a sharp glowing green sine wave pattern, and rotary control knobs on its panel. Connected to the Y-input terminals are red and black coaxial cables leading to a desktop microphone on a metal stand in the center. On the right, a shiny steel tuning fork vibrating vigorously with concentric sound waves radiating into the microphone. Clean, crisp typography: "Cathode Ray Oscilloscope (C.R.O.)", "Y-Input", "Microphone", "Vibrating Tuning Fork". Flat 2D vector style.
1. Loudness and Intensity of Sound
DEFINITION: LOUDNESS
Loudness: Loudness is the characteristic property of sound by virtue of which a loud sound can be distinguished from a faint (feeble) one, both having the same pitch and the same quality.
Experimental Verification with Tuning Fork & C.R.O.:
Strike a tuning fork softly on a rubber pad and hold it before the microphone $\implies$ a feeble (soft) sound is heard. The C.R.O. screen traces a sine wave of small amplitude.
Strike the same tuning fork hard $\implies$ a loud sound is heard. The C.R.O. traces a sine wave of large amplitude with the exact same frequency and wavelength.
Book Figure Placeholder: Fig. 7.18
Fig. 7.18: Soft and Loud Notes (Displacement vs Time on C.R.O.)
Shows two displacement-time graphs having the same frequency and same sine waveform: (a) Soft Note with small amplitude, and (b) Loud Note with large amplitude.
Book Figure 7.18 Slot — Ready for direct image placement using <img class="img-in-note">
WAVEFORM COMPARISON: SOFT NOTE VS LOUD NOTE (SAME FREQUENCY & FORM)
Fig. 7.18: Two notes having the exact same frequency and sinusoidal waveform, but differing significantly in amplitude: larger amplitude corresponds to greater loudness.
Graphic Generation Specification
Target: Soft vs Loud Note Waveform Comparison (Fig. 7.18)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics textbook graph comparison in landscape orientation on a pure white #ffffff background. Two vertically stacked sub-graphs: (a) top graph showing a blue sine wave with identical wavelength but small peak amplitude, labeled "Soft Note (Small Amplitude)"; (b) bottom graph showing a magenta sine wave with identical wavelength and frequency but much taller peak amplitude, labeled "Loud Note (Large Amplitude)". Both graphs share matching horizontal time axes labeled "Time (t)" and vertical axes labeled "Displacement". Distinct red double-ended arrows marking amplitude height. Clear, elegant vector styling.
INTENSITY OF SOUND WAVES
Intensity ($I$): The intensity of a sound wave at any point in a medium is defined as the amount of sound energy passing per second normally through unit area placed at that point.
$$ \text{Intensity } I = \frac{\text{Energy}}{\text{Area} \times \text{Time}} = \frac{\text{Power}}{\text{Area}} \quad \left[\text{SI Unit: watt per metre}^2 \ (\text{W m}^{-2})\right] $$
Intensity of ordinary conversation: $\approx 10^{-6}\text{ W m}^{-2}$
Threshold of Audibility ($I_0$): The minimum intensity of sound audible to the normal human ear at frequency $1\text{ kHz}$ is $I_0 = 10^{-12}\text{ W m}^{-2}$.
The physical intensity ($I$) of a sound wave in air is proportional to:
The square of the amplitude of vibration ($I \propto a^2$).
The square of the frequency of vibration ($I \propto f^2$).
The density of the medium ($I \propto \rho$).
⭐ SUBJECTIVE NATURE OF LOUDNESS VS OBJECTIVE NATURE OF INTENSITY
Intensity is an OBJECTIVE physical quantity: It is a precisely measurable physical parameter ($\text{W m}^{-2}$) determined exclusively by the wave energy. It does not depend on the presence or auditory health of a human listener.
Loudness is a SUBJECTIVE auditory sensation: It depends not only on the physical intensity of the wave, but also on the biological sensitivity of the listener's ear! A sound of a given intensity may appear loud to a young person but faint to an elderly person.
Furthermore, the sensitivity of the human ear varies with frequency: normal ears possess maximum sensitivity at a frequency of approximately $1\text{ kHz}$ ($1000\text{ Hz}$). Two sounds having identical physical intensity ($I$) but different frequencies will NOT produce equal sensations of loudness in the human ear.
FIVE FACTORS GOVERNING LOUDNESS OF SOUND
Square of the Amplitude ($L \propto a^2$): When a body vibrates with a larger amplitude, it imparts greater mechanical energy to the surrounding air particles. Greater energy reaches the eardrum, producing a louder sound.
Inverse Square of Distance from Source ($L \propto \frac{1}{r^2}$): Sound energy spreads spherically as it radiates. As distance $r$ from the source increases, energy per unit area falls inversely as $r^2$. Hence, closer sounds are louder; distant sounds grow feeble and eventually inaudible.
Surface Area of the Vibrating Body: A larger vibrating surface area displaces a vastly larger volume of air, radiating significantly more sound power. For this reason:
School and temple bells are cast with large, heavy bronze rims.
A vibrating tuning fork held in air is faint, but pressed against a large table top sounds loud.
Pianos and harps possess large wooden sounding boards.
Density of the Medium: Sound energy propagation increases with medium density. Sound is louder in dense carbon dioxide or water than in light air or hydrogen gas.
Presence of Resonant Bodies: When resonant cavities or sympathetic resonators are placed in the vicinity of a vibrating source, they absorb energy and vibrate with large amplitude, significantly boosting the overall loudness of sound.
WEBER-FECHNER LAW & THE DECIBEL SCALE
Experimental studies by Weber and Fechner proved that the human ear responds logarithmically to physical sound intensity. The subjective sensation of loudness ($L$) is related to objective physical intensity ($I$) by the Weber-Fechner Law:
$$ L = K \log_{10} I \quad \dots(7.8) $$
Where $K$ is a constant of proportionality. Hence, if intensity increases by a factor of 100, loudness increases only by $\log_{10}(100) = 2$ units!
Sound Level ($L$): The difference in loudness between a sound of intensity $I$ and the threshold of audibility ($I_0 = 10^{-12}\text{ W m}^{-2}$ at $1\text{ kHz}$) defines the sound level:
$$ L = \log_{10}\left(\frac{I}{I_0}\right)\text{ bel} $$
Since the bel is a very large unit, sound level is universally expressed in decibels ($\text{dB}$), where $1\text{ decibel} = \frac{1}{10}\text{ bel}$:
$$ L = 10 \log_{10}\left(\frac{I}{I_0}\right)\text{ dB} \quad \dots(7.10) $$
(Named in honour of Alexander Graham Bell, the inventor of the telephone).
Board Question: Exact Definition of 1 Decibel (1 dB)
Noise Pollution: The disturbance produced in the environment due to undesirable, loud, and discordant sound of level exceeding $120\text{ dB}$ from sources like loudspeakers, sirens, heavy machinery, and automobiles is called noise pollution.
Safe Auditory Limit: The safe sound level for continuous human hearing is $0\text{ dB}$ to $80\text{ dB}$.
Soothing Sensation: Sounds of level between $10\text{ dB}$ and $30\text{ dB}$ produce a relaxing, pleasant, and soothing mental state.
Auditory Hazards: Constant, prolonged exposure to sounds above $120\text{ dB}$ causes severe headaches, mental agitation, hypertension, tinnitus, and irreversible permanent damage to the delicate hair cells of the cochlea, resulting in noise-induced deafness.
Crucial Distinction — Limit of Hearing vs Limit of Audibility:
Limit of Audibility: Refers to the frequency range of sound ($20\text{ Hz}$ to $20,000\text{ Hz}$) that normal human ears can perceive.
Limit of Hearing: Refers to the lowest loudness level ($0\text{ dB}$) at which sound is just audible.
2. Pitch (or Shrillness) and Frequency
DEFINITION: PITCH
Pitch: Pitch is that characteristic of sound by virtue of which an acute (shrill) note can be distinguished from a grave (flat or deep) note, both having the same loudness and same quality.
Dependence: Pitch refers strictly to musical sounds. The pitch of a sound depends directly on its frequency ($f$):
$$\text{Higher Frequency } (f) \implies \text{Higher Pitch (Shrill Sound)}$$
$$\text{Lower Frequency } (f) \implies \text{Lower Pitch (Grave / Flat Sound)}$$
In audio amplifiers and TV sets:Bass (Woofer) corresponds to low-frequency grave sound (e.g. tabla, dholak, bass drum), whereas Treble corresponds to high-frequency shrill sound (e.g. flute, violin, ghunghroo/ankle bells).
Book Figure Placeholder: Fig. 7.19
Fig. 7.19: Waves of Different Pitch (C.R.O. Screen Traces)
Shows two waves having the exact same amplitude and sinusoidal form: (a) Low Pitch Note from tuning fork A ($256\text{ Hz}$, frequency $f$), and (b) High Pitch Note from tuning fork B ($512\text{ Hz}$, frequency $2f$).
Book Figure 7.19 Slot — Ready for direct image placement using <img class="img-in-note">
PITCH VS FREQUENCY ON C.R.O.: LOW PITCH (256 Hz) VS HIGH PITCH (512 Hz)
Fig. 7.19: Demonstration that pitch depends on frequency: Both waves have equal amplitude $a$, but wave (b) with double frequency produces a shrill, high-pitched note.
Graphic Generation Specification
Target: Low Pitch vs High Pitch Comparison (Fig. 7.19)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Educational physics graph illustration in landscape format on a pure white #ffffff background. Two vertically stacked sub-plots: (a) top graph showing a blue sine wave oscillating at frequency f = 256 Hz with 2 full cycles across the window, labeled "Low Pitch Note (Grave Sound)"; (b) bottom graph showing a magenta sine wave with the exact same amplitude a, but twice as many cycles (4 full cycles, frequency 2f = 512 Hz) across the same time interval, labeled "High Pitch Note (Shrill Sound)". Clear coordinate axes labeled "Time (t)" and "Displacement". Clean vector graphics with high clarity.
Real-World Examples of Change in Pitch
Stringed Musical Instruments (Guitar, Violin, Piano): These instruments use wires of different lengths, thicknesses, and tensions ($f \propto \frac{\sqrt{T}}{l \cdot r}$). A note of higher pitch is obtained by (a) tightening the tuning peg (increasing tension $T$), (b) pressing the string against a fret (shortening vibrating length $l$), or (c) using a thinner wire (decreasing radius $r$).
Wind Instruments (Flute): In a flute, musical pitch is varied by opening or closing finger holes. Closing more holes increases the effective length of the vibrating air column, lowering its natural frequency and producing a deeper (lower pitch) note. Opening holes shortens the air column, yielding a shrill (high pitch) note ($f \propto 1/l$).
Filling a Water Pitcher under a Tap: As water fills a bucket or pitcher placed under a running tap, the rising water level steadily decreases the length of the vibrating air column inside the pitcher. Because $f \propto 1/l$, the frequency of sound increases continuously. The sound becomes progressively shriller and shriller. A listener standing in another room can tell when the pitcher is about to overflow simply by listening to the rising pitch of the sound!
Human Voice (Female vs Male): The vocal cords in adult females are shorter and tighter than those in males. Consequently, a woman's voice vibrates at a higher frequency, resulting in a distinctly shriller, higher-pitched voice compared to a man's deeper, grave voice.
Subjective Pitch vs Objective Frequency
Frequency is an objective physical quantity: It is the exact number of vibrations per second executed by the source ($f = 1/T$), accurately measurable with oscilloscopes and frequency counters, completely independent of any listener.
Pitch is a subjective auditory sensation: It is the perception of frequency registered by the listener's brain. Although pitch correlates monotonically with frequency, perceptual pitch can vary slightly depending on loudness and individual auditory physiology.
3. Quality (or Timbre) and Wave Form
DEFINITION: QUALITY (TIMBRE)
Quality (or Timbre): Quality is that characteristic of sound by virtue of which we can distinguish between two sounds of the same loudness and same pitch, emitted by two different musical instruments or different people.
Dependence: The quality of a musical sound depends entirely on its waveform (shape of the wave)!
Book Figure Placeholder: Fig. 7.20
Fig. 7.20: Two Sounds of Same Amplitude and Frequency but Different Waveforms
Shows two notes having identical amplitude $a$ (same loudness) and identical time period $T$ (same pitch), but differing in wave shape: (a) smooth sinusoidal wave, and (b) triangular wave.
Book Figure 7.20 Slot — Ready for direct image placement using <img class="img-in-note">
WAVEFORM DEPENDENCE OF QUALITY: SINE WAVE VS TRIANGULAR WAVE
Fig. 7.20: Two waves of identical amplitude $a$ and identical frequency $f$, yet producing entirely different auditory sensations due to differing wave shapes.
Principal Note and Subsidiary Overtones
A musical instrument rarely emits a pure single-frequency sine wave. Instead, when an instrument is played, it produces a complex mixture of vibrations:
Principal (Fundamental) Note: The vibration having the lowest frequency and the maximum amplitude. It determines the perceived musical pitch of the sound.
Subsidiary (Secondary) Vibrations (Overtones / Harmonics): Accompanying vibrations whose frequencies are integral multiples ($2f, 3f, 4f, \dots$) of the fundamental frequency, having smaller relative amplitudes.
Complex Resultant Waveform: By the principle of superposition, the fundamental and overtones combine into a complex resulting waveform. The quality of a musical sound depends on the number and relative amplitudes of the subsidiary notes present along with the principal note!
Book Figure Placeholder: Fig. 7.21
Fig. 7.21: Pure Note vs Musical Note (C.R.O. Waveforms)
Compares: (a) Pure sine waveform of a tuning fork ($256\text{ Hz}$, single frequency), and (b) Complex waveform of a piano note ($256\text{ Hz}$, fundamental combined with multiple subsidiary overtones).
Book Figure 7.21 Slot — Ready for direct image placement using <img class="img-in-note">
C.R.O. TRACES: PURE MONOTONE (TUNING FORK) VS COMPLEX MUSICAL NOTE (PIANO)
Fig. 7.21: Comparison of pure monotone and musical notes: A tuning fork yields a simple sine wave, whereas a piano produces a rich, complex composite waveform.
Graphic Generation Specification
Target: Pure Note vs Complex Musical Note Waveform (Fig. 7.21)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics textbook graph illustration in landscape format on a pure white #ffffff background. Two horizontal oscilloscope display panels: top panel showing the smooth, clean, perfectly regular blue sine wave of a 256 Hz tuning fork, labeled "Waveform of Tuning Fork (Pure Monotone)"; bottom panel showing the intricate, multi-peaked composite magenta wave of a piano striking the exact same 256 Hz fundamental pitch, labeled "Waveform of Piano (Complex Musical Waveform with Harmonics)". Clearly indicate matching overall amplitude "a" and matching fundamental period "T". Crisp, high-contrast flat 2D vector style.
Identification of Sound Sources & Human Voices by Quality
Every vibrating source has its own unique overtone signature and therefore a characteristic waveform:
Voice Recognition over Telephone: You can recognize a friend speaking on the phone without seeing them because the vocal cords of each human being possess unique physical dimensions and tissue elasticity, producing a personalized harmonic waveform.
Instrument Differentiation: A middle C note played on a flute has few subsidiary overtones (producing a smooth, hollow sound), while the same note played on a piano or violin contains dozens of rich overtones, allowing instant auditory differentiation.
Characteristic of Sound
Subjective / Sensory Perception
Governing Objective Physical Factor
1. Loudness
Loud vs Faint sound
Amplitude ($a$) [Energy $\propto a^2$]
2. Pitch (Shrillness)
Shrill vs Grave (flat) sound
Frequency ($f$) [Number of oscillations $\text{s}^{-1}$]
3. Quality (Timbre)
Distinguishes instrument / voice source
Waveform [Mix of subsidiary overtones]
7.12 Music and Noise
All audible sounds can be broadly classified into two categories: Music and Noise.
Music: A pleasant, continuous, and uniform sound produced by regular, periodic vibrations without sudden alterations in amplitude or wavelength. Examples: sounds emitted by a flute, violin, piano, or sitar. Its sound level typically lies between $10\text{ dB}$ and $30\text{ dB}$.
Noise: An unpleasant, harsh, and discordant sound produced by an irregular succession of non-periodic disturbances. Examples: a stone thrown against a tin roof, traffic clatter, or jet roar. Sounds having sound levels above $120\text{ dB}$ are classified as noise.
Book Figure Placeholder: Fig. 7.22
Fig. 7.22: Wave Forms of Music and Noise
Contrasts the wave patterns of: (a) Music, showing a continuous, smooth, regular periodic wave; and (b) Noise, showing an irregular, jagged, discontinuous, non-periodic waveform.
Book Figure 7.22 Slot — Ready for direct image placement using <img class="img-in-note">
WAVEFORM COMPARISON: MUSIC (REGULAR PERIODIC) VS NOISE (IRREGULAR DISCORDANT)
Target: Music vs Noise Waveform Comparison (Fig. 7.22)Aspect Ratio: Landscape (16:9)Background: Pure White #ffffff
Academic physics waveform diagram in landscape orientation on a pure white #ffffff background. Two horizontal waveform traces: (a) top trace in green showing a smooth, harmonically periodic, repetitive musical waveform, labeled "(a) Music (Regular and Periodic Vibrations)"; (b) bottom trace in sharp red showing an irregular, chaotic, spiked, and jagged noise pattern, labeled "(b) Noise (Irregular Succession of Disturbances)". Clear horizontal coordinate axes labeled "Time" and vertical axes labeled "Displacement". Clean, high-contrast vector aesthetic.
Comparison Between the Music and Noise (Official Textbook Table)
S.No. / Parameter
Music
Noise
1. Sensation
It is regular, smooth and pleasant to the ears.
It is harsh, discordant and unpleasant to the ear.
2. Nature of Vibrations
It is produced by the vibrations which are periodic.
It is produced by an irregular succession of disturbances.
3. Component Waves
All the component waves are similar without any sudden change in their wavelength and amplitude.
The component waves change their character suddenly and they are of short duration.
4. Sound Level
The sound level is low (between $10\text{ dB}$ to $30\text{ dB}$).
The sound level is high (above $120\text{ dB}$).
5. Wave Form
The wave form is regular.
The wave form is irregular.
Example
The sound produced by the musical instruments (e.g., flute, violin, piano, sitar).
The sound produced by an aeroplane, road roller, industrial machines, etc.
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