A man standing between two parallel hills fires a gun and hears two echoes, one 2.5 s and the other 3.5 s after the firing. If the velocity of sound is 330 ms-1, how long will it take him to hear the third echo?

1. Nature of Sound Waves and Propagation (basic)

Welcome! Let’s start at the very beginning of our journey into acoustics. To understand sound, we must first understand its nature as a mechanical wave. Unlike light, which can travel through the empty vacuum of space, sound requires a medium (solid, liquid, or gas) to travel. When an object vibrates, it disturbs the particles of the medium around it, setting off a chain reaction of energy transfer.

In gases like air, sound travels as a longitudinal wave. This means the individual particles of the medium move back and forth in a direction parallel to the direction in which the wave travels. Imagine a Slinky being pushed and pulled: you see regions where the coils are bunched up and regions where they are spread out. In scientific terms, these are called:

• Compressions: Regions of high pressure and high density where particles are squeezed together.
• Rarefactions: Regions of low pressure and low density where particles are spread apart.

This mechanism is identical to how P-waves (Primary waves) behave during an earthquake; they are pressure waves that create density differences by stretching and squeezing the material Physical Geography by PMF IAS, Earths Interior, p.60. This is distinct from transverse waves (like S-waves or ripples in a pond), where the medium moves perpendicular to the wave direction, creating peaks and valleys known as crests and troughs Physical Geography by PMF IAS, Earths Interior, p.62.

Feature	Longitudinal Waves (Sound in air)	Transverse Waves (Water ripples/S-waves)
Particle Motion	Parallel to wave propagation	Perpendicular to wave propagation
Key Characteristics	Compressions and Rarefactions	Crests and Troughs

Interestingly, the nature of the material dictates how well sound is produced. For instance, metals are described as sonorous because they produce a distinct ringing sound when struck, a property utilized in school bells and musical instruments Science-Class VII . NCERT, The World of Metals and Non-metals, p.46.

Sources: Physical Geography by PMF IAS, Earths Interior, p.60; Physical Geography by PMF IAS, Earths Interior, p.62; Science-Class VII . NCERT, The World of Metals and Non-metals, p.46

2. Speed of Sound and Environmental Variables (basic)

Welcome back! Now that we understand the basics of waves, let’s look at why sound doesn't always travel at the same speed. In a vacuum, sound cannot travel at all, but in a medium like air, water, or steel, its speed is dictated by the physical properties of that environment. The most critical rule to remember is that the speed of sound is directly proportional to the temperature of the medium. As the temperature rises, the molecules in the air gain more kinetic energy and vibrate faster, allowing the sound wave to propagate more quickly through the atmosphere Physical Geography by PMF IAS, Earths Atmosphere, p.274.

To put this into a real-world context, imagine the vast temperature differences across India. On a scorching May afternoon in Rajasthan, where temperatures can soar to 48°C, sound travels significantly faster than it would on a freezing winter morning in Leh, where temperatures might drop to -8.5°C INDIA PHYSICAL ENVIRONMENT, Geography Class XI, Climate, p.34 CONTEMPORARY INDIA-I, Geography Class IX, Climate, p.37. While pressure changes don't significantly affect the speed of sound in an ideal gas, humidity does. Moist air is actually less dense than dry air (because water vapor molecules are lighter than nitrogen and oxygen molecules), which allows sound to travel slightly faster on a humid day.

Beyond gases, the nature of the material matters immensely. We generally find that sound travels fastest in solids, slower in liquids, and slowest in gases. This is because solids are highly elastic—they snap back into shape quickly after being compressed. While we often associate density with speed, elasticity is the true hero here. For example, even though mercury is denser than iron, sound travels faster in iron because iron is much more elastic Physical Geography by PMF IAS, Earths Interior, p.61.

Variable	Change in Variable	Effect on Speed of Sound
Temperature	Increase (↑)	Increases (↑)
Humidity	Increase (↑)	Increases (↑)
Density (in gases)	Increase (↑)	Decreases (↓)

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.274; INDIA PHYSICAL ENVIRONMENT, Geography Class XI (NCERT 2025 ed.), Climate, p.34; CONTEMPORARY INDIA-I, Geography Class IX (NCERT Revised ed 2025), Climate, p.37; Physical Geography by PMF IAS, Earths Interior, p.61

3. Reflection of Sound: Echoes vs Reverberation (intermediate)

When sound waves encounter an obstacle, they behave much like light waves hitting a mirror—they reflect. Just as we use the mirror formula to understand the positioning of images in optics Science class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143, in acoustics, we study how reflected sound waves return to our ears. This reflection manifests in two distinct ways: Echoes and Reverberation.

An echo is a distinct, separate repetition of sound heard after the original sound has ceased. This happens because of a phenomenon called the persistence of hearing. The human brain retains a sound for approximately 0.1 seconds. To hear a distinct echo, the reflected sound must reach your ears after this 0.1s window. If the speed of sound is roughly 344 m/s, the sound must travel at least 34.4 meters (total round trip) to be heard as an echo. This means the reflecting surface must be at least 17.2 meters away. If the distance is shorter, the reflections overlap with the original sound, creating a blur.

Reverberation, on the other hand, is the persistence of sound due to multiple reflections in an enclosed space. Instead of a distinct repetition, you hear a prolonged "roll" of sound that gradually fades away. While some reverberation is desirable in a concert hall to add depth to music, excessive reverberation can lead to annoyance and a-periodic sound fluctuations that cause displeasure to hearing Environment, Shankar IAS Academy (ed 10th), Environmental Pollution, p.81. In urban planning, controlling these reflections is vital to maintaining ambient noise limits, which are strictly regulated for residential and commercial zones Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), Environmental Degradation and Management, p.42.

Feature	Echo	Reverberation
Definition	A distinct, separate repetition of sound.	The blurring or prolongation of sound.
Time Gap	Reflected sound arrives > 0.1s after original.	Reflected sound arrives < 0.1s after original.
Requirement	Large distance (min ~17.2m) and a single large surface.	Enclosed spaces with multiple reflecting surfaces.

Sources: Science class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143; Environment, Shankar IAS Academy (ed 10th), Environmental Pollution, p.81; Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), Environmental Degradation and Management, p.42

4. SONAR and RADAR Technology (intermediate)

At the heart of both SONAR (Sound Navigation and Ranging) and RADAR (Radio Detection and Ranging) lies a fundamental principle of physics: the Pulse-Echo method. These technologies operate by emitting a burst of energy (a pulse) and measuring the time it takes for that energy to bounce off an object and return to the source. This is a direct application of wave reflection. While the concept is the same, the medium and the type of wave used are what distinguish them. While SONAR uses ultrasonic sound waves (longitudinal mechanical waves), RADAR utilizes radio waves or microwaves (transverse electromagnetic waves).

SONAR is the gold standard for underwater exploration because electromagnetic waves (like radio) are quickly absorbed by water, whereas sound waves can travel vast distances in the ocean. This principle is not just human-made; it is deeply rooted in nature. For instance, the Gangetic Dolphin and other freshwater species rely on biological SONAR (echolocation) to navigate and hunt in murky river waters Environment, Shankar IAS Academy, Conservation Efforts, p.245. The effectiveness of SONAR depends heavily on the medium's properties; sound travels faster in denser and more elastic materials. For example, sound waves move significantly faster through the Earth's crust or iron than through air or water because of the high elasticity of solids Physical Geography by PMF IAS, Earths Interior, p.61.

RADAR, on the other hand, is designed for atmospheric use. Since radio waves travel at the speed of light (approximately 300,000 km/s), RADAR provides near-instantaneous detection of high-speed objects like aircraft or missiles. Unlike sound, radio waves do not require a medium to travel, but they are reflected by metal surfaces and water droplets, making them essential for weather forecasting and air traffic control. In both systems, the distance (d) to an object is calculated using the formula d = (v × t) / 2, where 'v' is the speed of the wave and 't' is the total round-trip time. We divide by two because the wave has to travel to the target and back.

Feature	SONAR	RADAR
Wave Type	Sound (Ultrasonic)	Electromagnetic (Radio/Microwave)
Medium	Primarily Water/Solids	Air/Vacuum
Primary Use	Submarines, Seafloor mapping	Aviation, Weather, Defense

Sources: Environment, Shankar IAS Academy, Conservation Efforts, p.245; Physical Geography by PMF IAS, Earths Interior, p.61

5. The Doppler Effect and Frequency Shift (exam-level)

The Doppler Effect is one of the most intuitive yet profound phenomena in physics. It refers to the apparent change in the frequency of a wave when there is relative motion between the source of the wave and the observer. Imagine an ambulance speeding toward you: as it approaches, the siren sounds high-pitched; as it passes and moves away, the pitch suddenly drops. This happens because the source is 'catching up' to the sound waves it emits, effectively compressing the wave crests in front of it and stretching them out behind it. Since sound is a mechanical wave that travels through the compression and rarefaction of a medium, its speed is influenced by the properties of that medium Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64.

Mathematically, we represent the observed frequency (f') using the formula: f' = f [(v ± vₒ) / (v ∓ vₛ)], where f is the actual frequency, v is the speed of sound in the medium, vₒ is the velocity of the observer, and vₛ is the velocity of the source. While sound is a longitudinal wave, the Doppler effect also applies to transverse electromagnetic waves like light Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. In astronomy, this is known as Redshift (when a star moves away, stretching light waves toward the red end of the spectrum) and Blueshift (when it moves toward us).

The behavior of these waves is summarized in the table below:

Relative Motion	Wave Compression/Stretch	Frequency Shift	Observed Effect (Sound)
Source moving toward Observer	Crests are compressed	Frequency Increases	Higher Pitch
Source moving away from Observer	Crests are stretched	Frequency Decreases	Lower Pitch
Observer moving toward Source	Observer encounters more crests/sec	Frequency Increases	Higher Pitch

It is important to remember that the actual frequency emitted by the source does not change; only the perceived frequency changes due to the relative motion. This principle is vital in modern technology, from RADAR used by the military to Echocardiograms in medicine, where frequency shifts are measured to determine the velocity of blood flow within the heart.

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

6. Infrasound and Ultrasound Applications (basic)

To master acoustics, we must first understand that the human ear is only sensitive to a narrow band of frequencies, typically between 20 Hz and 20,000 Hz. Sounds that fall below this range are known as infrasound, while those above it are called ultrasound. Even though we cannot hear them, these waves are used extensively in modern science, medicine, and industry due to their unique physical properties.

Infrasound consists of very low-frequency waves that can travel vast distances with little energy loss. Large animals like elephants and whales use infrasound to communicate over many kilometers. In nature, events like earthquakes and volcanic eruptions generate infrasound before the audible shockwaves arrive, which is why some animals appear to sense a disaster before it happens. Because these waves are so powerful, high-intensity sound in general can lead to physiological issues such as anxiety, nervousness, and increased blood pressure Environment and Ecology, Environmental Degradation and Management, p.42.

Ultrasound, on the other hand, involves high frequencies that allow the waves to be focused into narrow beams. They do not bend easily around corners, making them perfect for precision tasks. In medicine, ultrasound is used for diagnostic imaging (sonography) to visualize internal organs without the radiation risks of X-rays. Interestingly, India has become a global hub for medical tourism and the outsourcing of data interpretation, where world-class hospitals interpret ultrasound tests and radiology images for patients across the globe Fundamentals of Human Geography, Tertiary and Quaternary Activities, p.50-51.

Feature	Infrasound	Ultrasound
Frequency	Below 20 Hz	Above 20,000 Hz (20 kHz)
Key Property	Long-distance travel; passes through obstacles.	High precision; reflects off small boundaries.
Application	Seismic monitoring, animal communication.	Echocardiograms, cleaning jewelry, detecting metal cracks.

Sources: Environment and Ecology, Environmental Degradation and Management, p.42; Fundamentals of Human Geography, Tertiary and Quaternary Activities, p.50-51

7. Physics of Multiple Echoes between Parallel Barriers (exam-level)

When a sound is produced between two parallel reflecting surfaces—such as two steep mountain cliffs or parallel walls in a large hall—the sound waves do not simply reflect once and disappear. Instead, they bounce back and forth repeatedly, creating a series of multiple echoes. To understand the physics here, we must look at the time intervals between these reflections. A standard echo is heard when the sound travels to a barrier and back to the observer (a distance of 2d). However, between parallel barriers, the sound undergoes successive reflections, where the wave reflected from one wall becomes the incident wave for the opposite wall.

Imagine a person standing at distances d₁ and d₂ from two parallel hills. The first two echoes heard are the "primary echoes" from each hill independently. The time for the first echo (t₁) is 2d₁/v and the second (t₂) is 2d₂/v, where v is the speed of sound. In modern scientific measurements, recording these tiny fractions of a second is crucial for accuracy Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.112. As the sound continues to bounce, the third echo is typically heard when the sound has reflected off one surface, passed the observer, reflected off the second surface, and then returned. Mathematically, the time for this combined path is the sum of the individual primary echo times: t₃ = t₁ + t₂. This represents the sound completing one full round trip between the two barriers.

Echo Sequence	Physical Path	Time Interval
1st Echo	To the closer wall and back	t₁ = 2d₁ / v
2nd Echo	To the farther wall and back	t₂ = 2d₂ / v
3rd Echo	Reflects off Wall 1, then Wall 2, then back to observer	t₃ = t₁ + t₂

In nature, these parallel barriers are often found in coastal regions where wave-cut cliffs form steep, vertical faces through erosion FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Landforms and their Evolution, p.58. If these cliffs face each other across a narrow inlet or cove, the acoustic environment becomes a natural echo chamber. While fascinating, persistent multiple echoes in urban environments can contribute to noise pollution, leading to physiological annoyance or hearing fatigue if the sound levels remain high Environment, Shankar IAS Academy .(ed 10th), Environmental Pollution, p.81.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.112; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Landforms and their Evolution, p.58; Environment, Shankar IAS Academy .(ed 10th), Environmental Pollution, p.81

8. Solving the Original PYQ (exam-level)

To solve this, we must synthesize two core concepts you've just mastered: the Reflection of Sound and the geometry of Multiple Echoes. In a single-wall scenario, an echo is simply the sound traveling to the barrier and back ($2d = v \times t$). However, when a man stands between parallel hills, the sound doesn't stop after the first bounce. The first echo (2.5 s) is the reflection from the closer hill, and the second (3.5 s) is from the farther hill. The third echo occurs when a sound wave reflects off one hill, travels past the man to the opposite hill, and reflects back to his position. This creates a cumulative path that represents a full round-trip between the two barriers.

Think of it as a relay race: the sound that produced the first echo continues past the man, hits the second hill, and returns. Mathematically, the time for the third echo is simply the sum of the individual echo times ($t_1 + t_2$). By adding 2.5 s and 3.5 s, we arrive at (C) 6 s. Notice how the velocity of sound (330 ms⁻¹) was provided as a distractor; while you could calculate the specific distances ($d_1$ and $d_2$) using the velocity, the question is designed to test your conceptual understanding of time intervals in multiple reflections rather than your ability to perform tedious multiplication.

UPSC often includes options like (A) 4 s or (D) 8 s to catch students who try to find a linear pattern or mistakenly double the wrong interval. Option (B) 5 s is a common trap for those who might simply double the first echo time ($2.5 \times 2$), assuming the third echo is just a repeat reflection from the nearest hill. To avoid these traps, always visualize the physical path the sound wave must travel: for the third echo to reach the observer, it must have completed one reflection off each of the two parallel surfaces.