Ultrasonic waves of frequency 3 * 105 Hz are passed through a medium where speed of sound is 10 times that in air (Speed of sound in air is 300 m/s). The wavelength of this wave in that medium will be of the order of

1. Basics of Mechanical Waves (basic)

A mechanical wave is a disturbance that travels through a material medium — whether solid, liquid, or gas — by transferring energy from one particle to the next. Unlike electromagnetic waves (like light), mechanical waves cannot travel through a vacuum; they require the physical presence of atoms or molecules to propagate. The most common examples you will encounter in your syllabus are sound waves and seismic waves (earthquake waves).

These waves primarily move through two physical processes: compression (where particles are pushed together) and rarefaction (where particles are stretched apart). Because of this, mechanical waves like sound and Primary seismic waves (P-waves) are often called longitudinal waves, as the displacement of the medium is parallel to the direction the wave travels Physical Geography by PMF IAS, Earths Interior, p.60. A crucial principle to remember is that the velocity of these waves is not constant; it depends heavily on the medium. Generally, the denser and more elastic the material, the faster the wave travels. This is why sound travels faster in water than in air, and fastest in solids like steel FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

To mathematically describe these waves, we use the fundamental wave relationship: v = fλ. Here, v represents the velocity (speed) of the wave, f is the frequency (the number of vibrations per second), and λ (lambda) is the wavelength (the distance between two consecutive compressions). In a mechanical wave, the frequency is usually determined by the source, while the wavelength must adjust whenever the wave enters a new medium and its speed changes Physical Geography by PMF IAS, Earths Atmosphere, p.279.

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; Physical Geography by PMF IAS, Earths Atmosphere, p.279

2. Wave Parameters: Frequency and Wavelength (basic)

To understand waves, we must first visualize them as a series of repeating patterns—much like ripples on a pond. Every wave has a crest (the highest point) and a trough (the lowest point). The wavelength (represented by the Greek letter λ) is the horizontal distance between two successive crests or two successive troughs FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. It essentially tells us the physical length of one complete wave cycle.

While wavelength measures space, frequency (f) measures time and repetition. It is defined as the number of waves passing a specific fixed point during a one-second time interval Physical Geography by PMF IAS, Tsunami, p.192. Frequency is measured in Hertz (Hz), where 1 Hz equals one wave per second. Closely related to this is the wave period, which is the time it takes for one full wave cycle to pass a point. Mathematically, frequency and period are reciprocals of each other (f = 1/T).

The most critical concept for a UPSC aspirant to grasp is the inverse relationship between these two parameters. For any wave traveling at a constant speed (v), the relationship is governed by the formula: v = fλ (Speed = Frequency × Wavelength). This means that if the frequency of a wave increases, its wavelength must decrease to maintain the same speed. For instance, in the electromagnetic spectrum, high-frequency waves like X-rays have incredibly short wavelengths, while low-frequency radio waves can have wavelengths longer than a football field Physical Geography by PMF IAS, Earths Atmosphere, p.279.

Parameter	Definition	Unit
Wavelength (λ)	Horizontal distance between two consecutive crests.	Meters (m)
Frequency (f)	Number of wave cycles passing a point per second.	Hertz (Hz)

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Tsunami, p.192; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), Earths Atmosphere, p.279

3. The Fundamental Wave Equation (v = fλ) (basic)

At its heart, every wave—whether it is a sound wave, a seismic wave, or light—is a pattern of energy moving through space. To understand how these waves behave, we use the Fundamental Wave Equation: v = fλ. In this equation, v represents the velocity (speed) of the wave, f is the frequency (how many waves pass a point per second), and λ (lambda) is the wavelength (the physical distance between two consecutive crests or troughs).

Think of it this way: if each wave is a "step" of a certain length (λ), and the wave takes a certain number of steps every second (f), the total distance covered in one second is the speed (v). This relationship is vital because it shows that for a constant speed, frequency and wavelength are inversely proportional. As noted in the study of our atmosphere, if the frequency increases, the wavelength must decrease to maintain the balance Physical Geography by PMF IAS, Earth’s Atmosphere, p.279.

It is important to remember that the speed of a wave (v) is not a fixed universal constant; it depends heavily on the medium it travels through. For instance, in seismic studies, we observe that the velocity of body waves changes as they move through materials of different densities; generally, the denser the material, the higher the velocity Fundamentals of Physical Geography, NCERT Class XI, The Origin and Evolution of the Earth, p.20. This is why primary waves (P-waves) travel faster than secondary waves (S-waves) and are the first to be recorded Physical Geography by PMF IAS, Earth’s Interior, p.60.

Sources: Physical Geography by PMF IAS, Earth’s Atmosphere, p.279; Fundamentals of Physical Geography, NCERT Class XI, The Origin and Evolution of the Earth, p.20; Physical Geography by PMF IAS, Earth’s Interior, p.60

4. The Acoustic Spectrum: Infrasonic to Ultrasonic (intermediate)

To understand the acoustic spectrum, we must first look at the range of human perception. Sound is a mechanical wave that requires a medium to travel, and its frequency—measured in Hertz (Hz)—determines its pitch. The Audible Range for a healthy human ear typically spans from 20 Hz to 20,000 Hz (20 kHz). However, sound exists far beyond these limits in two fascinating realms: the Infrasonic and the Ultrasonic. These divisions are not just arbitrary numbers; they dictate how waves interact with the environment, from the deep vibrations of the Earth to high-precision medical imaging.

Infrasonic waves are low-frequency sounds below 20 Hz. Because they have very long wavelengths, they can travel vast distances and penetrate solid objects easily. In nature, these are produced by large-scale phenomena like earthquakes, volcanic eruptions, and even the movement of tectonic plates. For instance, seismic P-waves during an earthquake travel at speeds between 5 to 8 km/s through the crust Physical Geography by PMF IAS, Earths Interior, p.61. On the other end of the spectrum, Ultrasonic waves have frequencies above 20 kHz. Due to their high frequency, they possess a very short wavelength, which allows them to be focused into narrow beams for applications like SONAR or medical ultrasounds.

The relationship between speed (v), frequency (f), and wavelength (λ) is defined by the formula v = fλ. In a medium where sound travels at 3000 m/s (ten times the speed of air), an ultrasonic wave with a frequency of 3 × 10⁵ Hz would have a wavelength of exactly 1 cm (calculated as 3000 / 300,000). This short wavelength is why ultrasonic waves are so effective at detecting small flaws in metal or imaging tiny structures in the human body. Conversely, within the audible range, excessive sound intensity—known as noise pollution—can lead to severe physiological effects, including increased blood pressure, heart rate fluctuations, and even permanent hearing loss Environment, Shankar IAS Academy, Environmental Pollution, p.81.

Category	Frequency Range	Key Characteristic
Infrasonic	Below 20 Hz	Long wavelength; used by whales and elephants for long-distance communication.
Audible	20 Hz – 20 kHz	The range of human hearing; subject to physiological stress if levels are too high.
Ultrasonic	Above 20 kHz	Short wavelength; used for imaging, cleaning, and by bats for navigation.

Sources: Physical Geography by PMF IAS, Earths Interior, p.61; Environment, Shankar IAS Academy, Environmental Pollution, p.81; Environment and Ecology, Majid Hussain, Environmental Degradation and Management, p.42

5. Speed of Sound in Different Media (intermediate)

To understand why sound travels at different speeds, we must first look at what sound is: a mechanical wave that moves through a medium via the compression and rarefaction of particles. Since sound relies on particles bumping into one another to transfer energy, the physical state and arrangement of those particles dictate how fast the wave can travel. In solids, particles are closely packed and held together by strong interparticle interactions (Science, Class VIII. NCERT(Revised ed 2025), Particulate Nature of Matter, p.113). This proximity allows the vibration to pass from one particle to the next almost instantaneously.

The speed of sound in a material is primarily determined by two properties: elasticity and density. Elasticity refers to how quickly a medium returns to its original shape after being deformed. Solids are generally much more elastic than liquids or gases because of their strong internal bonds. While we often associate higher density with faster sound, it is actually the high elasticity of solids that outweighs their density, leading to significantly higher velocities (Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64). For instance, in the Earth's interior, P-waves (a type of sound wave) travel much faster through the dense, solid lower mantle than through the crust (Physical Geography by PMF IAS, Earths Interior, p.61).

Medium State	Particle Arrangement	Relative Speed	Reasoning
Solids	Closely packed, fixed positions	Highest	High elasticity; particles interact immediately.
Liquids	Close but can move past each other	Intermediate	Incompressible but less elastic than solids.
Gases	Far apart, move randomly	Lowest	Highly compressible; particles must travel to collide.

An important nuance for your UPSC preparation: density and speed don't always have a linear relationship. For example, mercury is much denser than iron, yet sound travels faster in iron because iron is far more elastic (Physical Geography by PMF IAS, Earths Interior, p.61). Furthermore, when a sound wave moves from a slower medium (like air) to a faster medium (like steel), its frequency remains constant (as it depends on the source), but its wavelength must increase to account for the higher velocity, following the relation v = fλ.

Sources: Science, Class VIII. NCERT(Revised ed 2025), 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. Calculating Wavelength in High-Speed Media (exam-level)

To master wave physics, we must first anchor ourselves in the fundamental wave equation: v = fλ. Here, v represents the velocity of the wave, f is the frequency (number of cycles per second), and λ (lambda) is the wavelength, defined as the horizontal distance between two successive crests Physical Geography by PMF IAS, Tsunami, p.192. When a wave enters a different medium, its frequency—determined by the source—remains constant. However, the speed of the wave changes based on the properties of the material it is traveling through. For mechanical waves like sound, speed typically increases with the density of the material; for example, primary (P-waves) move faster through denser rock than through air FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

When we encounter a "high-speed medium," the wavelength must adapt to the increased velocity. If the speed (v) increases while the frequency (f) stays the same, the wavelength (λ) must also increase to satisfy the equation λ = v/f. This relationship shows that wavelength is directly proportional to the speed in that specific medium and inversely proportional to the frequency Physical Geography by PMF IAS, Earths Atmosphere, p.279. For high-frequency waves, such as ultrasonic waves (above 20,000 Hz), the wavelengths often become quite small, even in high-speed media, moving from the scale of meters down to centimeters or millimeters.

Let's look at the calculation logic using a practical example. Imagine a wave with a frequency of 3 * 10⁵ Hz (300,000 Hz) entering a medium where sound travels 10 times faster than it does in air (taking air as 300 m/s). The speed in this medium would be 3,000 m/s. To find the wavelength, we rearrange our formula to λ = v / f. Dividing 3,000 by 300,000 gives us 0.01 meters. In the world of precise measurements, 0.01 meters is exactly 1 cm. Understanding this conversion is vital for identifying the physical scale of waves in different environments, from seismic activity in the Earth's crust to medical ultrasound technology.

Variable	Relationship with Wavelength (λ)	Impact in High-Speed Media
Velocity (v)	Directly Proportional (λ ∝ v)	As speed increases, wavelength increases.
Frequency (f)	Inversely Proportional (λ ∝ 1/f)	Higher frequency leads to shorter wavelength.

Sources: Physical Geography by PMF IAS, Tsunami, p.192; 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 Atmosphere, p.279

7. Solving the Original PYQ (exam-level)

This question is a perfect synthesis of the wave equation (v = fλ) and the behavior of ultrasonic waves in different media. You’ve just learned that while frequency is a characteristic of the source, the speed of sound depends entirely on the medium's properties. Here, the UPSC asks you to bridge these building blocks: first, by deriving the velocity in the new medium (300 m/s × 10 = 3000 m/s), and second, by rearranging the relationship to solve for wavelength (λ = v / f). It is a classic application of scientific principles to a specific environmental scenario.

To arrive at the correct answer, focus on the units and powers of ten. By substituting the values into the rearranged formula, we get λ = 3000 / (3 × 10^5). This simplifies to 3 × 10^3 / 3 × 10^5, which leaves us with 10^-2 meters. Think about the conversion: since 1 meter equals 100 cm, 10^-2 meters is exactly 1 cm. This makes (A) the correct choice. Understanding these wave dynamics is not just for physics; as noted in Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.) > Chapter 15: Tsunami > Basics > p. 192, these same principles explain how energy moves through the ocean and crust during seismic events.

The incorrect options are designed to catch students who make unit conversion errors or calculation slips. Option (C) 100 cm (which is 1 meter) is a common trap if you forget to divide by the full frequency power, while Option (B) 10 cm usually results from a decimal placement error during division. Option (D) 0-1 cm is an outlier meant to distract those who are unsure of their precise calculation. In the UPSC exam, maintaining accuracy with scientific notation is the difference between a correct response and a narrow miss.