Q.You have learnt that a travelling wave in one dimension is represented by a function $y = f(x, t)$ where $x$ and $t$ must appear in the combination $x - vt$ or $x + vt$, i.e. $y = f(x \pm vt)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave:

The concept first

The electromagnetic spectrum is one continuous family — all of it is $\vec E$ and $\vec B$ oscillating and travelling at $c$. What changes across the spectrum is the wavelength $\lambda$ and hence the photon energy $E=h\nu=hc/\lambda$. Every practical application is chosen because $\lambda$ (or $E$) fits the physical scale of the problem. Learn that logic and you never have to memorise the table.

Step-by-step

1. X-rays $\to$ crystal structure study (R).

X-rays have $\lambda\sim10^{-10}\,\text{m}=1\ \text{Å}$, which is precisely the spacing between atomic planes in a crystal. A wave diffracts strongly only when the obstacle spacing is comparable to $\lambda$ — Bragg's law,

So X-rays (and only X-rays) give a usable diffraction pattern from a crystal lattice. This is how DNA's structure and countless mineral structures were found.

2. UV rays $\to$ finger prints in forensic labs (Q).

UV photons ($E\approx3$–$10$ eV) are energetic enough to excite electrons in the organic residues (oils, amino acids) left behind by a fingertip, and in the fluorescent powders dusted on a surface. These re-emit visible light — the print glows against its background. Visible light cannot do this; X-rays would be far too penetrating.

3. Radio waves $\to$ TV communication system (S). …

The question tests your understanding of how mechanical waves travel through different media. The crucial distinction is between transverse waves (where particles oscillate perpendicular to the wave’s direction) and longitudinal waves (where particles oscillate parallel to the wave’s direction). Each type requires a specific kind of elastic restoring force from the medium.

1. Transverse waves rely on shear stress — the medium must resist being twisted or bent sideways.

   • Solids have a fixed shape and can provide shear restoring forces, so transverse waves (like waves on a string or seismic S-waves) can travel through solids.
   • Fluids (liquids and gases) cannot sustain shear stress; they flow to eliminate sideways deformation. Therefore, transverse mechanical waves cannot propagate through liquids or gases.
   • This immediately tells us statement (C) is true: “Transverse waves can propagate through solids.”
   • It also tells us statement (A) is false: transverse waves cannot propagate through all media (they fail in fluids), but the phrasing “cannot propagate through all media” is ambiguous — it actually means “there exist media through which they cannot propagate,” which is true, but the intended meaning here is likely “they cannot propagate through any medium,” which is false. In standard MCQs, (A) is considered incorrect because transverse waves can propagate through solids.

2. Longitudinal waves rely on bulk compression — the medium must be able to be squeezed and expanded.

   • Solids, liquids, and gases all resist compression (they have bulk modulus), so longitudinal waves (sound waves, seismic P-waves) travel through all three. …

The question tests your understanding of wave fundamentals: what physical properties enable wave propagation, what defines a progressive wave, and the crucial distinction between energy and matter transport.

Longitudinal waves compress and expand the medium along the direction of propagation. This compression requires the medium to resist volume changes, which is quantified by the bulk modulus of elasticity $K$. Solids, liquids, and gases all possess bulk modulus (though gases have the smallest values), so longitudinal waves—sound being the classic example—can indeed propagate through all three states of matter. Statement A captures this correctly.

A progressive wave (or traveling wave) is one that carries a disturbance from one location to another through the medium. The wave profile moves spatially, described by equations like $y = A \sin(kx - \omega t)$, where the phase $(kx - \omega t)$ advances in space and time. Statement B is a straightforward, correct definition.

The critical conceptual point lies in statement C. Waves are mechanisms of energy transfer, not mass transfer. When a wave passes through a medium, individual particles oscillate about their equilibrium positions—they do not travel with the wave. A water wave moves across a pond, but the water molecules themselves bob up and down; they don't flow horizontally with the wave. Energy propagates, matter stays local.

Let me evaluate each statement: …

Concept & Intuition

In physics, every equation must be dimensionally consistent: the dimensions on both sides must match. A subtle but crucial rule is that transcendental functions (trigonometric, logarithmic, exponential, etc.) can only act on pure numbers — quantities with no physical dimension. If you see $\sin(m)$ where $m$ has dimensions of mass (M), that’s a red flag. The argument inside a sine, cosine, or log must be dimensionless. We’ll check each option by examining the arguments of these functions.

1. Option (A): $l = l_0 \dfrac{\sin(m)}{\sin(m_0)}$

   • Here $l$ and $l_0$ have dimension $L$, so the right side must also have dimension $L$. The ratio $\sin(m)/\sin(m_0)$ should be dimensionless.
   • But $m$ and $m_0$ have dimension $M$ (mass). The sine function requires a dimensionless argument — you cannot take the sine of a mass.
   • Conclusion: This is dimensionally inconsistent because $\sin(m)$ is meaningless.
   • Already we suspect (A) is the answer, but let’s verify the others.

2. Option (B): $l = l_0 \dfrac{\sin(m/m_0)}{\cos(m/m_0)}$

   • The arguments $m/m_0$ are ratios of masses, so they are dimensionless (M/M = 1).
   • $\sin$ and $\cos$ of a dimensionless number are fine. The ratio $\sin/\cos$ is also dimensionless.
   • Multiplying by $l_0$ (dimension L) gives L on the right, matching $l$ on the left.
   • Conclusion: Dimensionally consistent.

3. Option (C): $l = l_0 \dfrac{\log(m/m_0)}{\sin(m/m_0)}$

   • Again $m/m_0$ is dimensionless, so $\log$ and $\sin$ are valid. The ratio is dimensionless.
   • Right side has dimension L, left side L.
   • Conclusion: Dimensionally consistent. …

The Core Concept

Frequency modulation (FM) is a method of encoding information in a carrier wave by varying its instantaneous frequency according to the amplitude of the input signal. The key insight: the amplitude of the FM wave is deliberately kept fixed — this is what distinguishes FM from AM (amplitude modulation). Think of it like a singer holding a steady loudness (constant amplitude) but changing pitch (frequency) to convey a melody.

Step-by-Step Reasoning

1. What is modulated?

   In FM, the frequency of the carrier wave is varied around a central (carrier) frequency. The amount of variation is proportional to the instantaneous amplitude of the modulating signal (e.g., audio). So the frequency is not constant — it changes continuously with time.

2. What stays constant?

   The amplitude of the FM wave is kept constant. This is a deliberate design choice: by not varying the amplitude, FM is immune to amplitude noise (like static from lightning or electrical interference). The transmitted power is also constant, which simplifies amplifier design.

3. Mathematical representation

   An FM wave can be written as:

Here, $A_c$ is the constant amplitude, $f_c$ is the carrier frequency, and $m(t)$ is the modulating signal. The instantaneous frequency is $f_c + k_f m(t)$, which clearly varies with time. The amplitude $A_c$ does not depend on time.

4. Eliminating the wrong options …

The concept here is the classification of waves based on the direction of particle displacement relative to the wave's propagation. In a transverse wave, particles of the medium oscillate perpendicular to the direction the wave travels. In a longitudinal wave, particles oscillate parallel to the wave's direction. The critical physical distinction is that transverse waves need a medium with shear rigidity — the ability to resist sideways deformation — while longitudinal waves only need bulk elasticity (compressibility). This is why sound, which travels through fluids (gases and liquids) that cannot support shear, is always longitudinal.

Let’s examine each option:

1. Light waves (A) — These are electromagnetic waves. They consist of oscillating electric and magnetic fields, and the field oscillations are perpendicular to the direction of propagation. Light does not require a medium at all, but by definition it is transverse. So this is a transverse wave.

2. Sound waves (B) — Sound propagates through air, water, or solids as a series of compressions and rarefactions. The air molecules move back and forth along the direction the sound travels. In fluids, there is no shear strength, so transverse waves cannot exist. Even in solids, sound can have a longitudinal component, but the primary mode in air is longitudinal. Hence, sound is not a transverse wave. …