The standing wave pattern along a string of length 60 cm is shown in the below diagram. If the speed of the transverse waves on this string is 300 m/ s. in which one of the following modes in the string vibrating?

1. Classification of Mechanical Waves (basic)

Welcome to your first step in mastering waves! To understand the world around us—from the music we hear to the tremors of an earthquake—we must first understand Mechanical Waves. These are disturbances that transfer energy through a material medium (solid, liquid, or gas). Unlike light, which can travel through the vacuum of space, mechanical waves like sound require a medium to exist Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. We classify these waves based on how the particles of the medium move relative to the direction the wave is traveling.

The first category is Longitudinal Waves (also known as compressional or pressure waves). In these waves, particles of the medium vibrate parallel to the direction of the wave's travel. This creates alternating regions of high pressure called compressions (squeezing) and low pressure called rarefactions (stretching) Physical Geography by PMF IAS, Earths Interior, p.60. A classic example is a P-wave (Primary wave) during an earthquake; these are the fastest seismic waves and arrive first at a seismograph because they transmit energy very efficiently through the medium Physical Geography by PMF IAS, Earths Interior, p.61.

The second category is Transverse Waves (or shear waves). Here, the particles move perpendicular (at a right angle) to the direction of wave propagation. This motion creates crests (high points) and troughs (low points) Physical Geography by PMF IAS, Earths Interior, p.62. Think of a ripple on a pond or an S-wave (Secondary wave) in an earthquake. Because it is harder to "shear" a medium than to compress it, S-waves travel slower than P-waves and cannot pass through liquids, which have no shear strength Physical Geography by PMF IAS, Earths Interior, p.61.

Feature	Longitudinal Waves	Transverse Waves
Particle Motion	Parallel to wave direction	Perpendicular to wave direction
Structure	Compressions & Rarefactions	Crests & Troughs
Seismic Example	P-waves (Fastest)	S-waves (Slower)
Common Example	Sound waves	Waves on a plucked string

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Physical Geography by PMF IAS, Earths Interior, p.60; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Interior, p.62

2. Wave Parameters: Frequency, Wavelength, and Speed (basic)

To understand waves—whether they are the ripples in a pond, the sound of a flute, or seismic tremors—we must first master the vocabulary used to describe their physical properties. Imagine a wave as a series of hills and valleys. The highest point is the crest and the lowest point is the trough FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. The vertical distance from the very bottom of a trough to the top of a crest is the wave height, while the amplitude is exactly one-half of that height, representing the maximum displacement from the rest position Physical Geography by PMF IAS, Tsunami, p.192.

Beyond height, we look at how waves repeat in space and time. Wavelength (λ) is the horizontal distance between two successive crests. In contrast, frequency (f) is a measure of how often these waves pass a fixed point in one second Physical Geography by PMF IAS, Tsunami, p.192. These two parameters share an inverse relationship: if the wavelength is very long (like radio waves), the frequency is low; if the wavelength is short, the frequency is high Physical Geography by PMF IAS, Earths Atmosphere, p.279. This is a crucial concept for UPSC aspirants, as it explains why different waves (like P-waves and S-waves) behave differently as they travel through the Earth's interior.

Finally, we consider wave speed (v), which is the rate at which the wave moves through a medium. A fundamental rule in physics is that Speed = Frequency × Wavelength (v = fλ). It is important to remember that the speed of a wave is not fixed; it changes depending on the density and nature of the material it passes through FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20. Generally, waves move faster through denser materials.

Parameter	Definition	Unit (Common)
Wavelength	Distance between two peaks	Meters (m)
Frequency	Number of cycles per second	Hertz (Hz)
Wave Period	Time for one full wave to pass	Seconds (s)

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; Physical Geography by PMF IAS, Earths Atmosphere, p.279; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20

3. Superposition Principle and Wave Reflection (intermediate)

At the heart of acoustics and wave mechanics lies the Principle of Superposition. This principle states that when two or more waves overlap in the same medium, the resulting displacement at any point is simply the algebraic sum of the displacements of the individual waves. Think of it as 'constructive' or 'destructive' addition. This is why when seismic waves interact with surface rocks, they can generate entirely new sets of waves with different properties FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20. When a wave hits a boundary—like a string tied to a wall—it undergoes Reflection. Just as light reflects off a mirror where the angle of incidence equals the angle of reflection Science, Class VIII, Light, p.158, mechanical waves reflect back, often flipping upside down (a 180° phase change) if the end is fixed.

When an incident wave and its reflected counterpart travel back and forth on a string, they interfere via superposition to create Standing Waves. Unlike the P-waves (longitudinal) or S-waves (transverse) that travel through the Earth's interior Physical Geography, Earth's Interior, p.60-62, standing waves appear to stay in one place, vibrating in segments called loops. The points that do not move at all are called Nodes, while the points of maximum vibration are Antinodes.

The pattern in which a string vibrates is defined by its Harmonics. The simplest pattern is the Fundamental Mode (1st Harmonic), consisting of exactly one loop. As we add energy, we get overtones. It is crucial to master the nomenclature here, as it is a common point of confusion in competitive exams:

Mode of Vibration	Harmonic Number (n)	Number of Loops	String Length (L) Relationship
Fundamental	1st Harmonic	1	L = 1(λ/2)
1st Overtone	2nd Harmonic	2	L = 2(λ/2) = λ
2nd Overtone	3rd Harmonic	3	L = 3(λ/2)

In any nth harmonic, there are always n antinodes and n+1 nodes (including the fixed ends). For example, a string vibrating in its 2nd overtone is actually in its 3rd harmonic, meaning it has three distinct vibrating loops.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20; Science, Class VIII, Light, p.158; Physical Geography, Earth's Interior, p.60-62

4. Acoustics: Sound Waves and Medium Properties (intermediate)

In acoustics, sound is understood as a mechanical wave that requires a material medium—such as air, water, or solid rock—to propagate. Unlike light, which travels fastest in a vacuum at approximately 3×10⁸ m s⁻¹ Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148, sound waves rely on the physical interaction of particles. When a sound wave passes through a medium, it causes a series of compressions (high-pressure regions) and rarefactions (low-pressure regions). This makes sound a longitudinal wave, similar to the P-waves (primary waves) generated during seismic events, which can travel through the Earth's interior at speeds ranging from 5 to 13.5 km/s Physical Geography by PMF IAS, Earths Interior, p.61.

The speed of sound is not constant; it is dictated by the physical properties of the medium. While we often intuitively think that density is the only factor, the speed is actually determined by the interplay between elasticity (the ability of a material to return to its original shape) and density. A common misconception is that denser materials always slow down waves. However, if a material is significantly more elastic (stiff), the wave will actually travel faster. For instance, even though mercury is denser than iron, sound travels faster in iron because iron has a much higher modulus of elasticity Physical Geography by PMF IAS, Earths Interior, p.61.

Medium State	General Speed Trend	Reasoning
Solids	Highest Speed	High elasticity/stiffness allows particles to snap back and transfer energy quickly.
Liquids	Intermediate Speed	Less stiff than solids but denser than gases.
Gases	Lowest Speed	Particles are far apart and interactions are less frequent.

When sound waves are confined—such as in a musical string or an organ pipe—they form standing waves. These waves occur at specific frequencies called harmonics. The simplest pattern is the fundamental mode (first harmonic). Higher-frequency patterns are called overtones. For a string of length (L) fixed at both ends, the relationship is defined by L = n(λ/2), where n is the harmonic number. For example, the second overtone corresponds to the third harmonic (n=3), creating three distinct vibrating loops or segments.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Physical Geography by PMF IAS, Earths Interior, p.61

5. Applications of Wave Phenomena: Doppler Effect and SONAR (exam-level)

When we apply the principles of wave motion to real-world technology, two phenomena stand out for their transformative impact: the Doppler Effect and SONAR. At its core, sound is a mechanical wave that travels through the compression and rarefaction of a medium Physical Geography by PMF IAS, Earths Magnetic Field, p.64. The speed and frequency of these waves can be manipulated or measured to tell us about the motion and position of objects.

The Doppler Effect is the change in the observed frequency of a wave when there is relative motion between the source and the observer. Imagine an ambulance speeding toward you: as it approaches, the sound waves are "bunched up," leading to a higher frequency (higher pitch). As it moves away, the waves are "stretched," leading to a lower frequency. In modern technology, this is used in speed guns to catch speeding cars and in Doppler Ultrasound to monitor blood flow. In the cosmos, it helps astronomers determine if galaxies are moving away from us (Redshift), illustrating how technology serves as an application of deep scientific knowledge Exploring Society: India and Beyond, Factors of Production, p.176.

SONAR (Sound Navigation and Ranging) utilizes the reflection of sound waves—specifically ultrasonic waves—to detect objects underwater. Ultrasonic waves are preferred because they can travel long distances in water with minimal scattering. A transmitter sends a pulse, which hits an object (like a submarine or the seabed) and reflects back to a detector. By knowing the speed of sound in water—which is affected by the density of the medium Physical Geography by PMF IAS, Earths Magnetic Field, p.64—and measuring the time interval (t), we calculate depth using the formula: d = (v × t) / 2.

Application	Core Wave Principle	Primary Use-Case
Doppler Effect	Frequency shift due to relative motion	Speed detection, Satellite tracking, Medical imaging
SONAR	Reflection of ultrasonic waves (Echo)	Seabed mapping, Underwater navigation, Fishing

Sources: Physical Geography by PMF IAS, Earths Magnetic Field, p.64; Exploring Society: India and Beyond, Factors of Production, p.176

6. Formation of Standing (Stationary) Waves (intermediate)

When we talk about Standing Waves (or stationary waves), we are describing a fascinating physical phenomenon where a wave appears to be vibrating in place rather than traveling through a medium. This occurs due to the superposition of two waves of the same frequency and amplitude traveling in opposite directions. In a practical sense, imagine plucking a string fixed at both ends: the wave you create travels to the end, reflects back, and interferes with the incoming wave. As noted in the study of seismic and ocean waves, reflection causes waves to rebound, which is the foundational step for a standing wave to form FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

The anatomy of a standing wave consists of two critical points: Nodes and Antinodes. Nodes are positions where the displacement is always zero (the string stays still), while Antinodes are positions where the displacement is at its maximum. For a string of length L fixed at both ends, the boundaries must always be nodes. This physical constraint means that only specific wavelengths can form stable standing waves. These specific patterns are called harmonics. The basic relationship is given by the formula L = n(λ/2), where n is an integer (1, 2, 3...) known as the harmonic number, and λ is the wavelength.

We classify these modes based on the number of "loops" or vibrating segments they form:

• Fundamental Mode (1st Harmonic): n = 1. The string forms a single loop. Here, the length of the string is half the wavelength (L = λ/2).
• First Overtone (2nd Harmonic): n = 2. The string forms two loops. The length equals one full wavelength (L = λ).
• Second Overtone (3rd Harmonic): n = 3. The string forms three loops. The length is 1.5 times the wavelength (L = 3λ/2).

Just as we measure the frequency of waves as the number of oscillations passing a point in one second FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109, in standing waves, higher harmonics correspond to higher frequencies and more complex vibration patterns.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Moveings of Ocean Water, p.109

7. Modes of Vibration: Harmonics and Overtones (exam-level)

When we pluck a string or blow into a pipe, the resulting sound is rarely a single, pure frequency. Instead, the object vibrates in several different patterns simultaneously, known as modes of vibration. These modes are determined by the physical boundaries of the medium—such as the ends of a 100 cm string fixed to a support Science-Class VII, Measurement of Time and Motion, p.109. In a standing wave, these boundaries force the wave to have zero displacement at the ends (nodes) and maximum displacement at specific points in between (antinodes). The most basic pattern, where the string vibrates as a single loop, is called the fundamental mode or the first harmonic.

The relationship between the length of the string (L) and the wavelength (λ) is governed by the formula L = n(λ/2), where n is an integer representing the harmonic number. As n increases, the wavelength decreases and the wave frequency—the number of waves passing a point in a second FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Movements of Ocean Water, p.109—increases. These higher frequencies are called harmonics if they are integer multiples of the fundamental frequency (f₁, 2f₁, 3f₁, etc.). While the fundamental is the 1st harmonic, the next vibration mode (2nd harmonic) is also called the first overtone. This terminology can be tricky: the "overtone" simply counts the frequencies above the fundamental, whereas the "harmonic" refers to the math behind the frequency itself.

To visualize this, consider the physical structure of these vibrations. In the second overtone (which is the 3rd harmonic, n=3), the string vibrates in three distinct segments or loops. This means there are three antinodes where the material stretches and squeezes the most, similar to how S-waves create crests and troughs perpendicular to their direction of travel FUNDAMENTALS OF PHYSICAL GEOGRAPHY, The Origin and Evolution of the Earth, p.20. Between these loops are nodes—points that remain perfectly still. For any n-th harmonic, there are n antinodes and n+1 nodes (counting the fixed ends).

Mode of Vibration	Harmonic Name	Overtone Name	Number of Loops (n)
Fundamental	1st Harmonic	-	1
1st Higher Mode	2nd Harmonic	1st Overtone	2
2nd Higher Mode	3rd Harmonic	2nd Overtone	3

Sources: Science-Class VII, Measurement of Time and Motion, p.109; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Movements of Ocean Water, p.109; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20

8. Solving the Original PYQ (exam-level)

In this problem, we apply the building blocks of wave interference and boundary conditions that you just mastered. When a string is fixed at both ends, the standing wave must have nodes at the extremities. This constraint limits the possible vibrations to specific harmonics where the length of the string (L) is an integer multiple of half-wavelengths (nλ/2). As you learned, the number of "loops" or segments visible in a standing wave diagram directly indicates the harmonic number (n). A diagram showing three distinct vibrating loops represents the third harmonic, which is the physical manifestation of the wave's frequency being exactly three times the fundamental frequency.

To arrive at the correct answer, you must navigate the specific terminology used in acoustics and wave mechanics. The first harmonic (n=1) is known as the fundamental mode. Every harmonic above that is considered an "overtone." Following this logic, the second harmonic (n=2) is the first overtone, and the third harmonic (n=3)—which corresponds to the three-loop pattern described—is the second overtone. By identifying the three segments in the standing wave pattern, we conclude that the string is vibrating in its Second overtone, making (C) the correct choice. This relationship is a cornerstone of vibrating systems, as explored in ScienceDirect.

A classic UPSC trap lies in the naming convention mismatch. Many students see three loops and instinctively jump to "Third overtone" (Option D), forgetting that the overtone count starts after the fundamental frequency. Another common pitfall is attempting to use the speed (300 m/s) and length (60 cm) to calculate frequency unnecessarily; while those values define the pitch, the mode of vibration is determined solely by the visual geometry of the wave pattern. Always remember the coach's rule: Harmonic number = Overtone number + 1.