A sound wave travelling at a speed of 330 m/s produces 20 crests and 20 troughs in 0.1 second. The wavelength of the sound wave is :

1. Nature of Waves: Longitudinal vs. Transverse (basic)

At its simplest level, a wave is a disturbance that transfers energy from one point to another without transferring matter. Imagine a crowd doing 'the wave' in a stadium: the people stay in their seats, but the energy of the wave travels across the entire stand. In physics, we categorize waves based on how the particles of the medium move relative to the direction the energy is traveling. The two primary types are Longitudinal and Transverse waves.

Longitudinal waves are those in which the particles of the medium vibrate parallel to the direction of the wave's travel. This creates alternating regions of high pressure called compressions (where particles are squeezed together) and low pressure called rarefactions (where they are stretched apart). A classic example is a Sound wave, which travels by compressing and rarefying the air Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. In geophysics, the Primary waves (P-waves) released during an earthquake are also longitudinal; they are the fastest seismic waves because they push and pull the material they pass through Physical Geography by PMF IAS, Earths Interior, p.60.

In contrast, Transverse waves involve particles moving perpendicular (at a right angle) to the direction of energy flow. If you wiggle a rope up and down, the wave moves forward, but the rope itself only moves up and down. These waves are characterized by crests (the highest points) and troughs (the lowest points). Light is a prime example of a transverse electromagnetic wave Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. While longitudinal waves like sound require a medium to travel, electromagnetic transverse waves like light can travel through a vacuum.

Feature	Longitudinal Waves	Transverse Waves
Particle Motion	Parallel to wave direction	Perpendicular to wave direction
Structure	Compressions & Rarefactions	Crests & Troughs
Examples	Sound waves, P-seismic waves	Light waves, S-seismic waves, water ripples

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Physical Geography by PMF IAS, Earths Interior, p.60; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20

2. Anatomy of a Wave: Understanding Crests and Troughs (basic)

When we visualize a wave, whether it is a ripple in a pond or a sound vibration traveling through the air, we are looking at energy moving through a medium. To understand this movement, we look at the wave's anatomy. The crest is the highest point or the peak of a wave, while the trough is the lowest point or the valley. These two components represent the maximum displacement of the medium from its resting position in opposite directions. Physical Geography by PMF IAS, Tsunami, p.191

To measure the size of a wave, we look at its vertical dimensions. Wave height is defined as the total vertical distance from the bottom of a trough to the top of a crest. However, in physics and geography, we often focus on the amplitude, which is exactly one-half of the wave height. You can think of amplitude as the distance from the "still water level" to either the crest or the trough. FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109

Beyond height, we must consider how waves are spaced out and how fast they move. The horizontal distance between two successive crests (or two successive troughs) is known as the wavelength (represented by the Greek letter λ). Crucially, for calculations, remember that one complete wave cycle is formed by exactly one crest and one trough. The time it takes for one full cycle to pass a fixed point is the wave period, while the number of such cycles passing per second is the frequency. Physical Geography by PMF IAS, Tsunami, p.192

Feature	Crest	Trough
Definition	The highest point of a wave.	The lowest point of a wave.
Displacement	Maximum upward/positive displacement.	Maximum downward/negative displacement.

Sources: Physical Geography by PMF IAS, Tsunami, p.191; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Tsunami, p.192

3. Key Characteristics: Frequency, Amplitude, and Time Period (intermediate)

To understand waves, we must first look at their 'anatomy.' Imagine a wave moving across the ocean: the highest point is the crest, and the lowest point is the trough. The distance from the calm water level (the rest position) to the top of a crest is the Amplitude. Essentially, amplitude measures the 'strength' or displacement of the wave; in sound, higher amplitude means a louder noise, and in ocean waves, it represents greater energy Physical Geography by PMF IAS, Tsunami, p.192. It is important to note that the total vertical distance from the bottom of a trough to the top of a crest is called the wave height, meaning amplitude is exactly half of the wave height. Moving from physical shape to timing, we encounter Frequency and Time Period. These two are two sides of the same coin. The Time Period (T) is the time interval required for one complete wave cycle (one crest and one trough) to pass a specific point. Conversely, Frequency (f) is the number of these full cycles that pass a given point in exactly one second FUNDAMENTALS OF PHYSICAL GEOGRAPHY (NCERT 2025 ed.), Movements of Ocean Water, p.109. Frequency is measured in Hertz (Hz). If a wave has a high frequency, it means many crests are passing by every second, which often correlates with higher destructive power in seismic events like S-waves Physical Geography by PMF IAS, Earths Interior, p.62. Finally, the physical 'stretch' of the wave in space is the Wavelength (λ). This is the horizontal distance between two successive identical points, such as from one crest to the next Physical Geography by PMF IAS, Tsunami, p.192. These characteristics are linked by a fundamental equation: Speed (v) = Frequency (f) × Wavelength (λ). This means that if a wave travels at a constant speed, an increase in frequency must result in a shorter wavelength.

Characteristic	Definition	Measurement Units
Amplitude	Maximum displacement from the rest position (half of wave height).	Meters (m)
Frequency	Number of wave cycles passing a point per second.	Hertz (Hz)
Time Period	Time taken for one complete wave cycle to pass.	Seconds (s)
Wavelength	Horizontal distance between two consecutive crests/troughs.	Meters (m)

Sources: Physical Geography by PMF IAS, Tsunami, p.192; FUNDAMENTALS OF PHYSICAL GEOGRAPHY (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Earths Interior, p.62

4. Speed of Sound in Different Media (intermediate)

To understand why sound travels at different speeds, we must first look at the particulate nature of matter. Sound is a mechanical wave, meaning it requires a medium to travel. It moves by vibrating particles which then collide with their neighbors, passing the energy along. Because this process depends on particle-to-particle interaction, the physical state of the medium—whether it is a solid, liquid, or gas—is the primary factor in determining speed.

In solids, particles are held together by very strong interparticle forces and have minimum interparticle space Science, Class VIII NCERT, Particulate Nature of Matter, p.113. Because the particles are so tightly packed and rigidly connected, they respond almost instantly to vibrations, allowing sound to travel the fastest. In liquids, the interparticle attraction is slightly weaker and the spacing is larger Science, Class VIII NCERT, Particulate Nature of Matter, p.113, which slows down the transfer of energy compared to solids. Finally, in gases, particles are far apart with negligible attraction Science, Class VIII NCERT, Particulate Nature of Matter, p.113; the sound wave must wait for these widely spaced particles to collide, making the speed of sound the slowest in this medium.

Medium State	Interparticle Space	Relative Speed	Example
Solid	Minimum / Tightest	Highest (~5000+ m/s)	Steel, Iron, Granite
Liquid	Intermediate	Moderate (~1500 m/s)	Water, Sea water
Gas	Maximum / Loose	Lowest (~340 m/s)	Air, Helium

Beyond the state of matter, environmental conditions like temperature and humidity play a crucial role. As temperature increases, particles gain kinetic energy and move more rapidly, allowing them to transmit sound vibrations faster. Interestingly, humidity also increases the speed of sound. Moist air is actually less dense than dry air because water vapor molecules (H₂O) are lighter than the nitrogen (N₂) and oxygen (O₂) molecules they replace. Since sound travels faster through less dense gases, a humid day in the tropics will carry sound faster than a dry day in the desert Physical Geography by PMF IAS, Hydrological Cycle, p.326.

Sources: Science, Class VIII NCERT, Particulate Nature of Matter, p.113; Physical Geography by PMF IAS, Hydrological Cycle, p.326

5. Applied Acoustics: SONAR, Echo, and Ultrasound (exam-level)

Applied acoustics is the practical application of sound wave properties to solve real-world problems, primarily through the manipulation of reflection and frequency. At the heart of these applications is the Echo—a phenomenon where sound reflects off a surface and returns to the listener. To perceive an echo as a distinct sound, the reflected wave must reach the ear at least 0.1 seconds after the original sound, as the human brain retains sound for that duration. Given that sound travels at approximately 340-344 m/s in air, the reflecting surface must be at least 17 meters away to satisfy this time gap. Sound waves, much like seismic P-waves, are longitudinal and their velocity depends heavily on the density of the medium; they travel faster in denser materials and can refract or reflect when hitting a boundary between different densities FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

Ultrasound refers to sound waves with frequencies higher than the upper limit of human hearing (above 20,000 Hz). These waves are highly energetic and have short wavelengths, allowing them to penetrate deep into media or reflect off very small objects without spreading out. This makes them ideal for SONAR (Sound Navigation and Ranging). In a SONAR system, a transmitter emits ultrasonic pulses toward the seabed. These waves travel through the water, strike an object or the ocean floor, and reflect back to a detector. By measuring the time interval (t) between transmission and reception, and knowing the speed of sound in water (v), we calculate the depth (d) using the formula: 2d = v × t. This principle of echo-ranging is essential for mapping underwater topography and detecting submerged vessels.

To analyze these waves mathematically, we use the Wave Equation: v = fλ, where v is velocity, f is frequency, and λ (lambda) is wavelength. Frequency is defined as the number of complete wave cycles (one crest and one trough) passing a point per second. For example, if a source produces 50 crests and 50 troughs in 0.5 seconds, it has completed 50 cycles in half a second, resulting in a frequency of 100 Hz. Understanding this relationship is critical because high-frequency waves (like ultrasound) possess higher destructive power and different propagation characteristics compared to low-frequency waves, a principle also observed in high-frequency S-waves during seismic events Physical Geography by PMF IAS, Earths Interior, p.62.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20; Physical Geography by PMF IAS, Earths Interior, p.62

6. The Universal Wave Equation: v = fλ (intermediate)

To understand how waves move, we use the Universal Wave Equation: v = fλ. This elegant formula links three fundamental properties of a wave: its speed (v), its frequency (f), and its wavelength (λ). Think of a wave as a series of pulses traveling through a medium. The wavelength is the horizontal distance between two successive peaks, known as crests FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. The frequency is simply the count of how many of these crests pass a fixed point every second, measured in Hertz (Hz) Physical Geography by PMF IAS, Tsunami, p.192.

The logic behind the equation is straightforward: if you know the length of one wave (λ) and you know how many waves pass by every second (f), multiplying them gives you the total distance the wave covers in one second—which is the definition of wave speed. As waves travel through different environments, their speed can change. For example, the speed of light varies depending on the medium it enters, such as moving from air into a glass slab Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159. However, the frequency is usually determined by the source of the wave and stays constant; therefore, if the speed changes, the wavelength must adjust accordingly.

One of the most critical takeaways for competitive exams is the inverse relationship between frequency and wavelength when the speed is constant. If the frequency of a wave increases, its wavelength must decrease to maintain the same speed. This is why high-frequency radio waves have much shorter wavelengths compared to low-frequency ones Physical Geography by PMF IAS, Earths Atmosphere, p.279.

Variable	Definition	Standard Unit
v (Speed)	The rate at which the wave moves through a medium.	m/s (meters per second)
f (Frequency)	Number of wave cycles passing a point per second.	Hz (Hertz)
λ (Wavelength)	The distance between two consecutive crests or troughs.	m (meters)

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Tsunami, p.192; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.159; Physical Geography by PMF IAS, Earths Atmosphere, p.279

7. Solving the Original PYQ (exam-level)

This question is a brilliant test of how you integrate the fundamental building blocks of wave mechanics. You've recently learned that a wave cycle is not just a single peak, but the combination of one crest and one trough. The problem provides the raw data—20 crests and 20 troughs—which you must first translate into 20 complete cycles. By applying the definition of frequency (cycles per second), you find that 20 cycles occurring in 0.1 second equals 200 Hz. As highlighted in Physical Geography by PMF IAS, mastering these basic parameters is the first step toward understanding complex wave phenomena like Tsunami propagation or sound travel.

To arrive at the solution, you must connect these properties using the universal wave equation: v = fλ. Here, your role as a candidate is to rearrange the formula to solve for the unknown wavelength (λ = v / f). By substituting the speed of 330 m/s and the frequency of 200 Hz, the calculation 330 / 200 brings you to the correct answer: (C) 1.65 m. This logical progression—from physical description to frequency calculation, and finally to the wave equation—is exactly the systematic approach required for UPSC Prelims science questions.

UPSC frequently includes distractors to catch students making conceptual or arithmetic slips. Option (B) 3.3 m is a classic decimal placement trap, which might occur if a student incorrectly calculates the frequency as 100 Hz instead of 200 Hz. Option (D) 2.2 m often results from miscounting cycles—for instance, if a student fails to properly divide by the 0.1-second time interval. Avoiding these pitfalls requires rigorous attention to units and a clear understanding that frequency and wavelength are inversely proportional for a constant speed.