A sonometer wire having a length of 50 cm is vibrating in the fundamental mode with a frequency of 100 Hz. Which of the following is the type of propagating wave and its peed?

1. Classification of Waves: Mechanical vs. Electromagnetic (basic)

At its most fundamental level, a wave is a disturbance that transfers energy from one point to another without the bulk transport of matter. In the UPSC syllabus, particularly within Physics and Geography (Seismology), we classify waves based on whether they require a material medium (like air, water, or rock) to travel. This gives us two broad categories: Mechanical Waves and Electromagnetic (EM) Waves.

Mechanical Waves are those that must have a medium to propagate. These waves travel through the elastic properties of the medium. For example, Sound waves travel by creating a series of compressions (high density) and rarefactions (low density) in the air Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. Because they rely on particle interaction, sound waves actually travel faster as the density and elasticity of the medium increase. Seismic waves, such as Primary (P) waves, are also mechanical; they are longitudinal disturbances that squeeze and stretch the Earth's interior as they pass Physical Geography by PMF IAS, Earths Interior, p.60.

Electromagnetic Waves, on the other hand, are "self-sufficient." They do not require any material medium and can travel through the absolute vacuum of space. They consist of oscillating electric and magnetic fields. Light is the most familiar example of a transverse electromagnetic wave Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. Interestingly, while mechanical waves speed up in denser media, EM waves like light actually slow down because a higher density increases the refractive index Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. Light reaches its maximum speed of 3×10⁸ m s⁻¹ in a vacuum.

Feature	Mechanical Waves	Electromagnetic Waves
Medium Required?	Yes (Solid, Liquid, or Gas)	No (Can travel in vacuum)
Speed in Vacuum	Zero (Cannot propagate)	Maximum (~3×10⁸ m s⁻¹)
Examples	Sound, Seismic waves, Water waves	Light, Radio waves, X-rays, Microwaves

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Physical Geography by PMF IAS, Earths Interior, p.60; Science , class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148

2. Wave Motion: Transverse vs. Longitudinal (basic)

At its core, a wave is a disturbance that transfers energy from one point to another without the bulk movement of matter. To master wave motion, we categorize waves based on how the particles of the medium vibrate relative to the direction in which the energy is traveling. There are two primary ways this happens: Longitudinal and Transverse.

Longitudinal Waves (or compressional waves) occur when the displacement of the medium is in the same direction as the wave's travel. Imagine a spring being pushed and pulled; the energy moves forward by creating regions of high pressure called compressions (squeezing) and regions of low pressure called rarefactions (stretching) Physical Geography by PMF IAS, Earths Interior, p.60. A classic example is a Primary wave (P-wave) during an earthquake. Because they transmit energy through direct pressure, P-waves are the fastest seismic waves and can travel through solids, liquids, and gases FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20.

Transverse Waves, on the other hand, occur when the particles vibrate perpendicular (at a 90° angle) to the direction of wave propagation. Think of a rope tied to a wall that you shake up and down. This motion creates crests (the highest points) and troughs (the lowest points) Physical Geography by PMF IAS, Tsunami, p.191. Secondary waves (S-waves) and light waves are perfect examples. Because they require a "shear" or sideways force to propagate, S-waves are generally slower than P-waves and cannot travel through liquids (like the Earth's outer core), as liquids do not resist changing their shape Physical Geography by PMF IAS, Earths Interior, p.61.

Feature	Longitudinal Waves	Transverse Waves
Particle Motion	Parallel to wave direction	Perpendicular to wave direction
Key Characteristics	Compressions and Rarefactions	Crests and Troughs
Seismic Example	P-waves (Primary)	S-waves (Secondary)

Sources: Physical Geography by PMF IAS, Earths Interior, p.60-61; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20; Physical Geography by PMF IAS, Tsunami, p.191

3. Core Wave Parameters: λ, f, and v (basic)

To understand how waves move—whether they are seismic waves traveling through the Earth's interior or sound waves in the air—we must master three fundamental parameters: Wavelength (λ), Frequency (f), and Speed (v). Imagine a wave as a series of hills and valleys. The highest point is the crest, and the lowest is the trough FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. Wavelength (λ) is simply the horizontal distance between two successive crests or troughs Physical Geography by PMF IAS, Tsunami, p.192. Think of it as the 'physical length' of one complete wave cycle.

While wavelength describes space, Frequency (f) describes time. It is the number of waves passing a fixed point in one second, measured in Hertz (Hz) Physical Geography by PMF IAS, Tsunami, p.192. Closely related is the Wave Period (T), which is the time it takes for one full wave to pass. If you know the frequency, you know the period (f = 1/T). Finally, Wave Speed (v) is the rate at which the wave energy moves through a medium FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. It’s important to note that wave speed isn't constant everywhere; it changes based on the density and composition of the material it travels through, which is how scientists map the Earth's interior Physical Geography by PMF IAS, Earth's Interior, p.63.

The beauty of physics lies in how these three are interconnected by a simple, elegant formula: v = f × λ (Speed = Frequency × Wavelength). This equation tells us that for a wave traveling at a constant speed, frequency and wavelength are inversely proportional Physical Geography by PMF IAS, Earth's Atmosphere, p.279. If the frequency increases (more waves per second), the wavelength must decrease (the waves get shorter) to maintain the same speed.

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, Earth's Atmosphere, p.279; Physical Geography by PMF IAS, Earth's Interior, p.63

4. Seismic Waves: A Geographical Application (intermediate)

When we look at the Earth's interior, we aren't just looking at rocks; we are looking at how energy moves through them. Seismic waves are essentially energy waves generated by the sudden release of stress during an earthquake. To master this for Geography, you must distinguish between two main categories: Body Waves (which travel through the interior) and Surface Waves (which travel along the crust). Body waves are the real "scientists" of the group because their behavior reveals exactly what the Earth is made of FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

The first type of body wave is the P-wave (Primary wave). Think of these as "Push-Pull" waves. They are longitudinal, meaning the particles of the medium vibrate parallel to the direction of the wave, creating a series of compressions and rarefactions (squeezing and stretching). Because they are the fastest, they arrive first at seismographs. Most importantly, P-waves are versatile: they can travel through solids, liquids, and gases Physical Geography by PMF IAS, Earths Interior, p.60. This makes them similar to sound waves in their physical behavior.

The second type is the S-wave (Secondary wave). These are transverse or shear waves, where the particles move perpendicular to the direction of the wave—much like a ripple on a pond or a flicked rope. S-waves are slower than P-waves (about 1.7 times slower) and arrive second Physical Geography by PMF IAS, Earths Interior, p.61. However, they have a critical limitation: S-waves can only travel through solid materials. They cannot pass through liquids because liquids do not have the "shear strength" to snap back after being distorted. This specific property is how geographers discovered that the Earth's outer core is liquid—since S-waves simply disappear when they hit it FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

Feature	P-Waves (Primary)	S-Waves (Secondary)
Nature	Longitudinal (Compressional)	Transverse (Shear/Distortional)
Medium	Solid, Liquid, and Gas	Solid only
Speed	Fastest (~1.7x faster than S)	Slower arrival

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.60; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Interior, p.62

5. Acoustics: Speed of Sound in Different Media (intermediate)

To understand why sound travels at different speeds, we must first look at the nature of the medium. Sound is a mechanical wave that moves through the compression and rarefaction of particles Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. Because it relies on particles bumping into one another, the speed of sound is dictated by two primary physical properties of the medium: Elasticity (the ability of the material to return to its original shape) and Density.

In general, sound travels fastest in solids, slower in liquids, and slowest in gases. This is because particles in solids are closely packed and have very strong interparticle interactions Science Class VIII NCERT, Particulate Nature of Matter, p.113. This allows the vibration to be passed from one particle to the next almost instantaneously. In contrast, gases are highly compressible and their particles are far apart, leading to a much slower transfer of energy. While we often think density slows sound down (since heavier particles are harder to move), in solids, the increase in elasticity usually far outweighs the increase in density. For example, sound travels faster in an iron rod than in liquid mercury, even though mercury is denser, because iron is significantly more elastic Physical Geography by PMF IAS, Earths Interior, p.61.

When we deal with waves on a specific medium, like a stretched sonometer wire, we categorize the wave as transverse (where particles move perpendicular to the wave direction). The speed (v) of such a wave is a product of its frequency (f) and its wavelength (λ), expressed as v = fλ. In a fundamental mode of vibration for a string of length L fixed at both ends, the wavelength is exactly twice the length of the string (λ = 2L). Understanding this relationship allows us to calculate the precise velocity of sound or vibration in that specific setup.

Medium State	Particle Arrangement	Relative Speed of Sound
Solids	Closely packed; fixed positions	Highest (e.g., ~5000+ m/s in Steel)
Liquids	Close but can move past each other	Intermediate (e.g., ~1500 m/s in Water)
Gases	Far apart; move randomly	Lowest (e.g., ~343 m/s in Air)

Sources: Science Class VIII NCERT, Particulate Nature of Matter, p.113; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

6. Stationary Waves and Nodes (intermediate)

Stationary waves (or standing waves) are a fascinating phenomenon that occurs when two identical waves traveling in opposite directions meet and superimpose. Unlike the seismic P-waves or S-waves that propagate through the Earth's interior to transport energy from an epicenter to a seismograph, stationary waves appear to be 'standing' still. They do not move forward; instead, the particles of the medium simply oscillate in place with different amplitudes depending on their position.

In a stretched string, such as a sonometer wire, these waves are transverse in nature. As noted in geographical studies of seismic activity, transverse waves (like S-waves) are characterized by particles vibrating perpendicular to the direction of wave propagation, creating distinct crests and troughs Physical Geography by PMF IAS, Earths Interior, p.62. When such a wave reflects off a fixed boundary, it interferes with the incoming wave to form a stationary pattern. The most critical features of this pattern are Nodes and Antinodes.

Feature	Nodes	Antinodes
Displacement	Zero (Point of permanent rest)	Maximum (Point of peak vibration)
Pressure Change	Maximum variation	Minimum variation
Location	Fixed ends of a string	Center of a vibrating segment

Interestingly, the concept of a node also appears in human geography as a 'meeting point' or a point of origin in a network FUNDAMENTALS OF HUMAN GEOGRAPHY, CLASS XII, Tertiary and Quaternary Activities, p.48. In physics, however, a node is where the 'links' (the wave displacements) cancel each other out perfectly. For a string of length L fixed at both ends, the simplest way it can vibrate is called the fundamental mode. In this mode, there are nodes at both ends and one antinode in the middle. Here, the length of the string is exactly half of a wavelength (L = λ/2). Therefore, the wavelength (λ) is twice the length of the string (2L).

Sources: Physical Geography by PMF IAS, Earths Interior, p.62; FUNDAMENTALS OF HUMAN GEOGRAPHY, CLASS XII, Tertiary and Quaternary Activities, p.48

7. Fundamental Mode of Vibration (exam-level)

When we talk about the Fundamental Mode of Vibration (also known as the first harmonic), we are describing the simplest and lowest-frequency way a physical system—like a stretched string—can vibrate. Imagine a wire fixed at both ends, such as a sonometer wire or a guitar string. When you pluck it, the wire doesn't just move randomly; it forms a specific pattern. In its fundamental mode, the wire forms a single "loop" with two nodes (points of zero displacement) at the fixed ends and one antinode (the point of maximum displacement) right in the center.

To understand this mathematically, we look at the relationship between the length of the string (L) and the wavelength (λ) of the wave produced. In this simplest mode, the entire length of the string is exactly half of one wavelength (L = λ/2). This means that to find the full wavelength, we simply double the length: λ = 2L. These waves are transverse in nature because the particles of the wire move up and down, perpendicular to the direction the wave travels along the string. This is distinct from longitudinal waves, like sound in air, but shares the same definition of wave frequency: the number of vibrations or cycles passing a point in one second FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Movements of Ocean Water, p.109.

Once we know the wavelength (from the string's length) and the frequency (the pitch of the note), we can calculate the wave speed (v) using the universal wave equation: v = fλ. For example, if a 0.5-meter wire is vibrating at a frequency of 100 Hz (similar to the 50 Hz frequency we see in household AC power Science, Class X, Magnetic Effects of Electric Current, p.206), the wavelength would be 1.0 meter (2 × 0.5m), and the wave speed would be 100 m/s. This speed is determined by the physical properties of the string, such as its tension and mass.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Movements of Ocean Water, p.109; Science, Class X, Magnetic Effects of Electric Current, p.206

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental principles of wave mechanics, this question allows you to see how those building blocks come together in a practical application. A sonometer wire is the quintessential example of a stretched string fixed at both ends. In such a setup, the particles of the medium vibrate perpendicular to the direction of wave propagation, which defines the wave as transverse. This is a critical distinction to remember for the UPSC: while the sound produced by the instrument and traveling through the air is longitudinal, the physical wave traveling along the tensioned wire itself is transverse, similar to the behavior of Secondary Waves (S-waves) described in Physical Geography by PMF IAS.

To determine the speed, we apply the logic of the fundamental mode. In this state, the string forms a single loop, meaning the length of the wire (L) is exactly half of the wavelength (λ/2). Given the length is 50 cm, we first convert this to S.I. units (0.5 m), giving us a wavelength of 1.0 m (λ = 2 × 0.5 m). By using the wave equation v = fλ, where the frequency (f) is 100 Hz, the calculation becomes 100 Hz × 1.0 m, resulting in a speed of 100 m/s. Consequently, the correct answer is (D) Transverse, 100 m/s.

UPSC often designs distractors to catch students who rush their reasoning. Options (A) and (C) are immediate traps for those who confuse the source of the sound (the transverse vibration of the wire) with the resultant sound wave in the air (which is longitudinal). Option (B) is a common calculation error; it uses a speed of 50 m/s, which happens if a student forgets the λ = 2L relationship and simply multiplies the frequency by the length. Always ensure your units are in meters and your wave geometry is correctly identified before finalizing your calculation.