Two identical piano wires have same fundamental frequency when kept under the same tension. What will happen if tension of one of the wires is slightly increased and both the wires are made to vibrate simul- taneously ?

1. Basics of Sound Waves and Propagation (basic)

Welcome to our journey into the world of acoustics! To understand how music, speech, or even noise reaches our ears, we must first understand the fundamental nature of sound. At its core, sound is a mechanical wave. Unlike light, which can travel through the emptiness of space, sound requires a physical medium—such as air, water, or solid metal—to propagate. It travels by causing the molecules of that medium to vibrate, passing energy from one particle to the next. Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

Sound waves are characterized as longitudinal waves (also known as compression waves). As the wave moves forward, it creates regions of high pressure called compressions and regions of low pressure called rarefactions. In these waves, the particles of the medium vibrate parallel to the direction in which the wave is traveling. This is distinct from transverse waves, such as light or S-waves in earthquakes, where the medium moves perpendicular to the wave direction, creating visible crests and troughs. Physical Geography by PMF IAS, Earths Interior, p.61-62

One of the most fascinating aspects of sound is how the medium dictates its speed. Generally, sound travels fastest in solids, slower in liquids, and slowest in gases. This is because solids have higher elasticity and density, allowing the mechanical energy to be transmitted more efficiently between tightly packed molecules. This is the opposite of light waves, which actually slow down as the density of a medium increases. Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64

Feature	Sound Waves	Light Waves
Type	Mechanical Wave	Electromagnetic Wave
Medium	Required (Solid, Liquid, Gas)	Not Required (Can travel in vacuum)
Motion	Longitudinal (Parallel)	Transverse (Perpendicular)

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

2. Characteristics of Sound: Pitch, Loudness, and Quality (basic)

To understand sound, we must first recognize it as a mechanical wave that travels through a medium via compression and rarefaction Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. While the speed of sound is a property of the medium (influenced by factors like elasticity and density), the sound itself is defined by three distinct characteristics that our ears and brain perceive: Loudness, Pitch, and Quality.

Loudness is the measure of the sound's intensity and is determined by the amplitude of the sound wave. If you strike a drum harder, you give it more energy, creating a larger vibration (amplitude), which we hear as a louder sound. Pitch, however, depends on the frequency of the vibration. A high-frequency sound—meaning the source vibrates many times per second—is perceived as a 'shrill' or high-pitched sound (like a bird chirping), whereas a low-frequency sound is perceived as a 'deep' or low-pitched sound (like a lion's roar). In instruments like the piano, the pitch is adjusted by changing the tension of the strings; a tighter string vibrates faster, producing a higher frequency.

Finally, Quality (or Timbre) is the characteristic that allows us to distinguish between two sounds even if they have the same loudness and the same pitch. For instance, if a flute and a guitar play the exact same musical note at the same volume, you can still tell them apart. This is because most sounds are not 'pure' tones; they are a complex mix of different frequencies called overtones or harmonics. The unique shape of the resulting waveform gives the sound its 'color' or quality.

Characteristic	Physical Property	Perception
Loudness	Amplitude	Volume (Faint vs. Loud)
Pitch	Frequency	Shrillness (High vs. Low)
Quality (Timbre)	Waveform / Harmonics	Distinction between sources

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

3. Applications of Sound: Echo and SONAR (intermediate)

To understand Echo and SONAR, we must first look at the reflection of sound. Just as a ball bounces off a wall, sound waves reflect when they hit a hard surface. An Echo is simply a reflected sound that arrives at the listener with a delay after the direct sound. For our brain to distinguish an echo from the original sound, there must be a time interval of at least 0.1 seconds (this is known as the persistence of hearing). If the speed of sound in air is roughly 344 m/s, the sound must travel a total distance of 34.4 meters to be heard as a distinct echo. This means the reflecting object must be at least 17.2 meters away from the source.

While echoes can be a natural wonder, they also have high-tech industrial and military applications through SONAR (Sound Navigation and Ranging). SONAR uses ultrasonic waves (frequencies above 20 kHz) because these high-frequency waves can penetrate deep into water without dissipating quickly. A SONAR device consists of a transmitter and a detector. The transmitter sends out a pulse of ultrasound which travels through the water, strikes an object (like a submarine or the seabed), and reflects back to the detector. By measuring the time (t) taken for the pulse to return and knowing the speed of sound in water (v), we can calculate the distance (d) using the formula: 2d = v × t.

In the context of the environment, it is important to note that while sound is a tool for navigation, excessive or irregular sound levels can be harmful. Constant exposure to high-intensity sound fluctuations can lead to physiological effects such as increased heart rate, blood pressure, and even hearing loss Environment, Shankar IAS Academy, Environmental Pollution, p.81. This is why maritime SONAR is often regulated to prevent disturbing marine life like whales and dolphins, who rely on their own natural biological sonar (echolocation).

Feature	Echo (Natural)	SONAR (Technology)
Wave Type	Audible Sound Waves	Ultrasonic Waves
Medium	Usually Air	Usually Water
Primary Use	Acoustic analysis, nature	Depth sounding, locating underwater objects

Sources: Environment, Shankar IAS Academy, Environmental Pollution, p.81

4. Doppler Effect and Sonic Boom (intermediate)

The Doppler Effect is a phenomenon where the frequency (or pitch) of a wave changes for an observer moving relative to the source of the wave. Think of a police siren: as the car speeds toward you, the sound seems higher-pitched, and as it passes you and moves away, the pitch suddenly drops. This happens because sound is a mechanical wave that travels through the compression and rarefaction of a medium Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64. When the source moves toward you, it "catches up" to the sound waves it just emitted, bunching them together and increasing the frequency. When it moves away, the waves are stretched out, decreasing the frequency.

While we often notice this with sound, the Doppler Effect is equally vital in astronomy regarding light waves. Just as sound pitch changes, the color of light shifts depending on motion. This is known as Redshift and Blueshift. If a galaxy is moving away from Earth, its light waves are stretched, shifting toward the red end of the spectrum (longer wavelengths). Conversely, if it moves closer, it shifts toward blue. Astronomer Edwin Hubble used this to show that galaxies are drifting apart, leading to the conclusion that our universe is expanding Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3. Scientists even use specific "beacons" like Type Ia supernovae to measure this cosmological redshift and determine how fast the universe is accelerating Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.13.

When an object, like a fighter jet, travels faster than the speed of sound, it outpaces the sound waves it produces. Instead of the waves spreading out ahead of the craft, they pile up behind it, forming a single, massive shock wave. This constructive interference of sound waves creates a Sonic Boom—a thunder-like noise that carries a tremendous amount of energy. The transition from subsonic to supersonic speed is often visualized by a "vapor cone" caused by the sudden drop in air pressure and temperature around the shock wave.

Scenario	Relative Motion	Observed Effect
Approaching	Source moves toward Observer	Higher Frequency (Higher Pitch / Blueshift)
Receding	Source moves away from Observer	Lower Frequency (Lower Pitch / Redshift)
Supersonic	Source moves faster than Wave	Shock Wave (Sonic Boom)

Sources: Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.3; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.13

5. Resonance and Forced Vibrations (intermediate)

To understand resonance, we must first distinguish between free and forced vibrations. Every physical object, whether it is a simple pendulum or a complex bridge, has a 'natural frequency'—the rate at which it prefers to vibrate when disturbed once and left alone. As seen in simple experiments with a bob and string, a pendulum's time period (and thus its frequency) remains remarkably consistent regardless of the number of oscillations, provided its length is unchanged Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110. These are free vibrations. However, when an external periodic force is applied to an object, it is compelled to vibrate at the frequency of that external force. This is known as forced vibration. For example, when radio waves hit electrons in the ionosphere, they force those electrons to vibrate at the wave's specific frequency Physical Geography by PMF IAS, Earths Atmosphere, p.279.

Resonance is a special, powerful case of forced vibration. It occurs when the frequency of the external 'driving' force exactly matches the natural frequency of the object. When this synchronization happens, the object absorbs energy from the driver with maximum efficiency, leading to a dramatic increase in the amplitude of vibration. Think of it like pushing a child on a swing: if you push exactly when the swing reaches its peak, the swing goes higher and higher with very little effort. In the atmosphere, such coupling and energy exchange between systems—like the ocean and the air—can lead to large-scale oscillations like El-Niño Geography of India, Majid Husain, Climate of India, p.13.

Understanding resonance is critical because it can be both useful and destructive. In communication, we 'tune' a radio to match the frequency of the incoming signal to achieve resonance and clear sound. Conversely, if soldiers march in step across a bridge at a frequency that matches the bridge's natural frequency, the resulting resonant vibrations can become so violent that the structure collapses. This is why resonance is often described as the 'sweet spot' of energy transfer in the physical world.

Type of Vibration	Frequency Determined By	Amplitude Behavior
Free	The object's own physical properties.	Decreases over time due to friction/damping.
Forced	The external driving force.	Usually small; depends on the force applied.
Resonance	Matching (Driver = Natural).	Increases significantly to a maximum point.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110; Physical Geography by PMF IAS, Earths Atmosphere, p.279; Geography of India, Majid Husain, Climate of India, p.13

6. Principle of Superposition and Interference (exam-level)

At the heart of wave mechanics lies the Principle of Superposition. It states that when two or more waves travel through the same medium simultaneously, the resultant displacement at any point is the algebraic sum of the individual displacements of each wave. Unlike solid objects that collide and bounce, waves simply pass through one another, momentarily combining their energies. This phenomenon of wave interaction is known as interference.

Interference generally manifests in two ways. Constructive interference occurs when waves meet "in phase" (crest meets crest), leading to a combined wave of greater amplitude. Conversely, destructive interference occurs when waves are "out of phase" (crest meets trough), causing them to cancel each other out. This is why sound waves, which behave as longitudinal pressure waves involving compressions and rarefactions Physical Geography by PMF IAS, Earths Interior, p.60, can sometimes result in silence or amplified noise depending on how they overlap. This behavior is fundamental to all waves, including seismic P-waves, which are also longitudinal and move through the Earth by compressing and stretching material FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

A fascinating application of this principle is the production of beats. When two sound waves of slightly different frequencies interfere, they drift in and out of phase with each other. This creates a periodic rise and fall in volume—a "wa-wa-wa" sound—that we call a beat. For example, in a piano, if two strings are under slightly different tensions, their frequencies will differ because the frequency (f) of a stretched string is proportional to the square root of its tension (T), expressed as f ∝ √T. The number of beats heard per second, or the beat frequency, is simply the absolute difference between the two original frequencies (fʙ = |f₁ - f₂|). Musicians and tuners rely on this phenomenon to bring instruments into perfect unison by adjusting tension until the beats disappear.

Type of Interference	Phase Relationship	Resultant Amplitude
Constructive	In-phase (Crest-Crest)	Increased (Loud sound)
Destructive	Out-of-phase (Crest-Trough)	Decreased (Faint/No sound)
Beats	Slightly different frequencies	Periodic variation (Pulsing)

Sources: 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

7. Vibrations in Stretched Strings (exam-level)

When we talk about vibrations in stretched strings, such as those in a piano or a guitar, we are observing a fascinating interplay of tension, mass, and wave mechanics. A string fixed at both ends, when plucked, generates standing waves. The rate at which the string moves back and forth is its wave frequency, defined as the number of vibrations passing a point in one second FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Movements of Ocean Water, p.109. The most basic mode of this vibration is called the fundamental frequency (or the first harmonic).

The speed of a wave on a string—and consequently its frequency—is heavily dependent on the tension applied to it. Just as hanging heavier objects from a spring causes a greater stretch due to the force of gravity Science, Class VIII, Exploring Forces, p.73, increasing the tension in a musical string makes it "stiffer" and quicker to return to its equilibrium position. Mathematically, the fundamental frequency (f) is directly proportional to the square root of the tension (√T). If you tighten a string, the pitch (frequency) goes up; if you loosen it, the pitch goes down. These vibrations are transverse in nature, meaning the particles of the string move perpendicular to the direction of the string itself, creating crests and troughs similar to the movement of S-waves in the Earth's crust FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20.

A critical phenomenon occurs when two strings are tuned to almost, but not exactly, the same frequency. When played together, their sound waves interfere with each other. Because their frequencies are slightly different, they periodically move in and out of phase—sometimes reinforcing each other (constructive interference) and sometimes cancelling each other out (destructive interference). This results in a periodic variation in volume known as beats. The beat frequency is simply the absolute difference between the two individual frequencies (f₂ - f₁). For example, if one wire vibrates at 440 Hz and another at 444 Hz, you will hear a distinct "throbbing" sound or "beat" 4 times per second. This is the primary tool used by piano tuners: they adjust the tension until the beats disappear, signifying that the strings are perfectly in unison.

Change in String	Effect on Frequency (Pitch)
Increase Tension (Tightening)	Increases (Pitch becomes higher)
Increase Length	Decreases (Pitch becomes lower)
Increase Thickness (Mass)	Decreases (Pitch becomes lower)

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Movements of Ocean Water, p.109; Science, Class VIII, Exploring Forces, p.73; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.20

8. The Physics of Beats (exam-level)

When we talk about waves, interference is the magic that happens when two waves meet. Beats are a specific and beautiful manifestation of this interference. Imagine two sound sources—like two piano wires—vibrating at slightly different frequencies. Because their frequencies aren't identical, the waves they emit will alternately reinforce and cancel each other out over time.

This happens through the principle of superposition. At one moment, the peaks (compressions) of both waves arrive at your ear at the same time, leading to constructive interference and a loud sound. A fraction of a second later, due to the slight frequency mismatch, the peak of one wave aligns with the trough (rarefaction) of the other, leading to destructive interference and near silence. This periodic waxing and waning of sound intensity is what musicians call a "beat." While frequency is a concept we see across nature—from the daily cycles of tides (NCERT Class XI Fundamentals of Physical Geography, Movements of Ocean Water, p.110) to the recurring patterns of earthquakes (PMF IAS Physical Geography, Earthquakes, p.182)—in acoustics, it creates a distinct physical pulse.

The Beat Frequency (the number of pulses you hear per second) is mathematically simple: it is the absolute difference between the two source frequencies. If Source A is 440 Hz and Source B is 442 Hz, you will hear 2 beats per second. This is the primary tool for instrument tuners. For a stretched string, the frequency is proportional to the square root of its tension (f ∝ √T). By slightly tightening a wire, a tuner changes its frequency, listening closely as the "beats" slow down and eventually disappear, signifying that the frequencies are perfectly matched.

Feature	Constructive Interference	Destructive Interference
Phase	In-phase (Peak meets Peak)	Out-of-phase (Peak meets Trough)
Sound Intensity	Maximum (Loud)	Minimum (Quiet)

Sources: NCERT Class XI Fundamentals of Physical Geography, Movements of Ocean Water, p.110; PMF IAS Physical Geography, Earthquakes, p.182

9. Solving the Original PYQ (exam-level)

To solve this question, you must synthesize the concepts of wave mechanics and superposition. Recall from NCERT Class 11 Physics that the fundamental frequency of a stretched string is directly proportional to the square root of its tension ($f \propto \sqrt{T}$). Initially, the two wires are in unison. By slightly increasing the tension of one wire, you introduce a marginal frequency shift. You now have two sound sources vibrating simultaneously with nearly identical, but not exactly equal, frequencies. This specific setup is the textbook requirement for the phenomenon of interference in time.

As your coach, I want you to walk through the logic: the waves from these two wires will periodically move in and out of phase. When they are in phase, they interfere constructively (increasing volume), and when out of phase, they interfere destructively (decreasing volume). This rhythmic rise and fall in sound intensity is known as Beats, making (B) the correct answer. The number of beats heard per second is simply the difference between the two frequencies.

UPSC often uses technical-sounding distractors to test your conceptual clarity. Resonance (C) is a common trap; however, it requires an external driving frequency to match the natural frequency, whereas here the frequencies are being pulled apart. Noise (A) implies a random, non-periodic collection of frequencies, which contradicts the precise nature of piano wires. Finally, Non-linear effects (D) usually refer to complex interactions at very high amplitudes or in specific media, which is an overcomplication of this basic acoustic principle.