In the given progressive wave $y = 5\sin(100\pi t - 0.4\pi x)$ where $y$ and $x$ are in m, $t$ is in s. What is the (a) amplitude (b) wave length (c) frequency (d) wave velocity (e) particle velocity amplitude.

> [!TLDR] > By comparing the given wave equation with the standard form of a progressive wave, we can directly identify its key characteristics. The amplitude is $5$ m, wavelength is $5$ m, frequency is $50$ Hz, wave velocity is $250$ m/s, and the particle velocity amplitude is $500\pi$ m/s. When analyzing a progressive wave, its mathematical representation contains all the information about its physical properties. The key is to understand the standard form of a wave equation and then compare it with the given equation to extract the relevant parameters. A general equation for a one-dimensional progressive harmonic wave traveling in the positive $x$-direction can be written as: $$y(x, t) = A\sin(kx - \omega t + \phi)$$ or $$y(x, t) = A\sin(\omega t - kx + \phi)$$ where: * $y$ is the displacement of a particle from its equilibrium position at a given $x$ and $t$. * $A$ is the amplitude, representing the maximum displacement of any particle from its equilibrium position. * $k$ is the angular wave number (or propagation constant), related to the wavelength $\lambda$ by $k = \frac{2\pi}{\lambda}$. * $\omega$ is the angular frequency, related to the frequency $f$ by $\omega = 2\pi f$. * $t$ is time. * $x$ is the position. * $\phi$ is the initial phase constant. The given wave equation is $y = 5\sin(100\pi t - 0.4\pi x)$. This equation matches the form $y(x, t) = A\sin(\omega t - kx)$, with the phase constant $\phi = 0$. Let's extract each parameter by direct comparison and using the relevant definitions. 1. **Identify Amplitude ($A$)** The amplitude is the coefficient of the sine function in the wave equation. Comparing $y = 5\sin(100\pi t - 0.4\pi x)$ with $y = A\sin(\omega t - kx)$, we see that $A = 5$. Since $y$ is in meters, the amplitude is in meters. 2. **Identify Angular Frequency ($\omega$) and Angular Wave Number ($k$)** By comparing the terms inside the sine function: The coefficient of $t$ is the angular frequency $\o

In the given progressive wave y = 5sin(100π t - 0. – Class 11 Physics

NCERT Exemplar · Q32

y = 5\sin(100\pi t - 0.4\pi x)

(a) amplitude

(b) wave length

(d) wave velocity

(e) particle velocity amplitude.

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Concept understanding — Wave Type Classification

Wave Type Classification: From Intuition to Precision

Imagine you're standing at the edge of a still pond. You drop a pebble. Ripples spread outward in circles. Now imagine you're holding one end of a long rope tied to a wall. You flick your wrist once — a single hump travels down the rope, hits the wall, and comes back. These are both waves, but they behave differently. Why?

The key difference lies in what is moving versus what is waving.

The Core Intuition

A wave is a disturbance that carries energy from one place to another without permanently moving the medium itself. But the direction of that disturbance relative to the wave's travel direction gives us our first major classification.

Think of a crowd at a stadium doing "the wave." People stand up and sit down (the disturbance moves up-down), but the wave itself travels around the stadium (left-right). The motion of each person is perpendicular to the wave's travel. That's one type.

Now think of a slinky stretched on a table. If you push one end toward the other, a compression travels along the slinky. Each coil moves along the same line as the wave — forward and back. That's the other type.

The Precise Classification

Waves are classified into two fundamental types based on the relationship between the direction of particle displacement (how the medium moves) and the direction of wave propagation (which way the energy travels).

\perp

Transverse Waves

In a transverse wave, the particles of the medium oscillate perpendicular to the direction the wave travels.

Example: Light waves (electromagnetic waves), waves on a string, water waves (surface component), seismic S-waves
Key feature: The wave has crests (high points) and troughs (low points)
Visual: Think of a rope shaken up and down — the rope moves vertically, the wave moves horizontally

Longitudinal Waves

In a longitudinal wave, the particles of the medium oscillate parallel to the direction the wave travels.

Example: Sound waves in air, seismic P-waves, slinky compressions
Key feature: The wave has compressions (regions of high density) and rarefactions (regions of low density)
Visual: Think of a slinky pushed and pulled — coils bunch up and spread out along the same line the wave moves

Note

Some waves are neither purely transverse nor purely longitudinal. Water waves, for instance, have particles moving in circular paths — a combination of both. These are called surface waves or Rayleigh waves in seismology.

Why This Matters

This classification isn't just academic. It determines:

What materials a wave can travel through: Transverse waves (like light) can travel through vacuum. Longitudinal waves (like sound) need a medium. But mechanical transverse waves (like rope waves) also need a medium — the distinction is about how the medium moves, not whether a medium exists. …

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By comparing the given wave equation with the standard form of a progressive wave, we can directly identify its key characteristics. The amplitude is $5$ m, wavelength is $5$ m, frequency is $50$ Hz, wave velocity is $250$ m/s, and the particle velocity amplitude is $500\pi$ m/s.

When analyzing a progressive wave, its mathematical representation contains all the information about its physical properties. The key is to understand the standard form of a wave equation and then compare it with the given equation to extract the relevant parameters.

A general equation for a one-dimensional progressive harmonic wave traveling in the positive $x$ -direction can be written as:

y(x, t) = A\sin(kx - \omega t + \phi)

y(x, t) = A\sin(\omega t - kx + \phi)

where:

$y$ is the displacement of a particle from its equilibrium position at a given $x$ and $t$ .
$A$ is the amplitude, representing the maximum displacement of any particle from its equilibrium position.
$k$ is the angular wave number (or propagation constant), related to the wavelength $\lambda$ by $k = \frac{2\pi}{\lambda}$ .
$\omega$ is the angular frequency, related to the frequency $f$ by $\omega = 2\pi f$ .
$t$ is time.
$x$ is the position.
$\phi$ is the initial phase constant.

The given wave equation is $y = 5\sin(100\pi t - 0.4\pi x)$ . This equation matches the form $y(x, t) = A\sin(\omega t - kx)$ , with the phase constant $\phi = 0$ .

Let's extract each parameter by direct comparison and using the relevant definitions.

Identify Amplitude ( $A$ )

The amplitude is the coefficient of the sine function in the wave equation.

Comparing $y = 5\sin(100\pi t - 0.4\pi x)$ with $y = A\sin(\omega t - kx)$ , we see that $A = 5$ .

Since $y$ is in meters, the amplitude is in meters.
Identify Angular Frequency ( $\omega$ ) and Angular Wave Number ( $k$ )

By comparing the terms inside the sine function:

The coefficient of $t$ is the angular frequency $\omega$ .

So, $\omega = 100\pi$ rad/s.

The coefficient of $x$ is the angular wave number $k$ .

So, $k = 0.4\pi$ rad/m.
Calculate Wavelength ( $\lambda$ )

The angular wave number $k$ is defined as $k = \frac{2\pi}{\lambda}$ . We can rearrange this to find the wavelength $\lambda$ .

\lambda = \frac{2\pi}{k}

Substitute the value of $k$:

\lambda = \frac{2\pi}{0.4\pi} = \frac{2}{0.4} = 5 \text{ m}

Calculate Frequency ( $f$ ) The angular frequency $\omega$ is defined as $\omega = 2\pi f$ . We can rearrange this to find the frequency $f$ .

f = \frac{\omega}{2\pi}

Substitute the value of $\omega$:

f = \frac{100\pi}{2\pi} = 50 \text{ Hz}

Calculate Wave Velocity ( $v$ ) The wave velocity (or phase velocity) $v$ is the speed at which the wave propagates through the medium. It can be calculated using the relationship $v = \frac{\omega}{k}$ or $v = f\lambda$ . Using $v = \frac{\omega}{k}$ :

v = \frac{100\pi \text{ rad/s}}{0.4\pi \text{ rad/m}} = \frac{100}{0.4} = \frac{1000}{4} = 250 \text{ m/s}

Alternatively, using $v = f\lambda$:

v = (50 \text{ Hz}) \times (5 \text{ m}) = 250 \text{ m/s}

Calculate Particle Velocity Amplitude ( $v_{p,max}$ ) …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 19 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. Why gases cannot support the propagation of transverse wave?$
(1) Gases are incompressible
(2) Gases have no shearing stress
(3) Density of gas is less
(4) Gases can flow
›Reveal solutionSolution

A transverse wave needs the medium to resist a sideways (shearing) deformation; gases cannot offer this, so they cannot carry transverse waves.

In a transverse wave, particles of the medium oscillate perpendicular to the direction of wave travel. For layers of the medium to pull each other sideways and pass this disturbance on, the medium must be able to sustain a shearing (tangential) stress — a restoring force that resists relative sliding of adjacent layers. Solids can do this because of rigidity (shear modulus). Gases (and liquids, to first approximation) have zero shear modulus — their molecules can slide past one another freely with n …

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$Q. The waves in which the particles of the medium vibrate perpendicular to the direction of wave propagation are called -$
(a) Transverse waves
(b) Progressive waves
(c) Longitudinal waves
(d) Stationary waves
›Reveal solutionSolution

A wave in which particles of the medium oscillate perpendicular to the direction of wave propagation is called a transverse wave (e.g. waves on a stretched string, electromagnetic waves).

Waves are broadly classified by how the medium's particles move relative to the wave's direction of travel. In a transverse wave, particles vibrate at right angles (perpendicular) to the direction of propagation — producing crests and troughs, as seen in waves on a string or water surface waves. This is different from a longitudinal wave (particles vibrate parallel to the direction of propagation, forming compressions and rarefactions, e.g. sound waves), a progressive wa …

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$Q. Assertion (A) : Sound waves cannot travel in Vacuum but light can travel in Vacuum. Reason (R) : Sound waves are longitudinal waves and cannot be polarised but electromagnetic waves are transverse and they can be polarised.$
(a) Both A and R are true, and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
›Reveal solutionSolution

Both statements are individually true, but the reason about polarisation does not correctly explain why sound cannot travel in vacuum while light can.

Checking Assertion (A): Sound is a mechanical wave — it needs a material medium (solid, liquid or gas) whose particles vibrate to carry the disturbance forward. In vacuum there are no particles to vibrate, so sound cannot travel through it. Light is an electromagnetic wave, made of oscillating electric and magnetic fields, and needs no material medium — it travels through vacuum (this is how sunlight reaches Earth). So A is TRUE.

Checking Reason (R): Sound waves are longitudinal (particle vibration is along the direction of propagation) and longitudinal waves cannot be polarised, since there is no transverse direction to restrict. Light (an electromagnetic wave) is transverse, and transverse waves can be polarised. So R is also TRUE as a standalone fact.

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. The path difference between the two waves y1 = a1 sin(ωt - 2πx/λ) and y2 = a2 cos(ωt - 2πx/λ + φ) is$
(a) (2π/λ)φ
(b) (λ/2π)φ
(c) (2π/λ)(φ + π/2)
(d) (λ/2π)(φ + π/2)
›Reveal solutionSolution

Convert cos to sin; extra phase = phi + pi/2; path difference = (lambda/2 pi)(phi + pi/2). Answer (D).

Given:

y1 = a1 sin(omega t - 2 pi x/lambda)

y2 = a2 cos(omega t - 2 pi x/lambda + phi).

Using cos(theta) = sin(theta + pi/2), rewrite y2:

y2 = a2 sin(omega t - 2 pi x/lambda + phi + pi/2).

Comparing the arguments of the two sine waves, the phase difference is:

delta_phi = (phi + pi/2).

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q.When sound wave is refracted from air to water, which of the following will remain unchanged ?$
(a) Wavelength
(b) Velocity
(c) Frequency
(d) Angular wave number
›Reveal solutionSolution

On refraction the frequency stays the same; speed and wavelength change. Answer (C).

The frequency of a wave is determined by the source producing it. When the wave crosses from one medium to another, the number of wavefronts arriving per second (frequency) cannot change, otherwise energy would accumulate at the boundary.

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Sound waves in air cannot be polarized because$
(a) their speed is less
(b) they require medium
(c) these are longitudinal
(d) their speed is temperature dependent
›Reveal solutionSolution

Polarization is a property exclusive to transverse waves, where oscillation direction (perpendicular to propagation) can be confined to a plane; sound waves in air are longitudinal (oscillation along the direction of propagation), so there is no perpendicular plane to restrict -- hence they cannot be polarized.

Polarization refers to restricting the oscillations of a transverse wave to a single plane containing the direction of propagation. This is possible only when the oscillation direction is perpendicular to the direction the wave travels (as with light or a wave on a string), since there are multiple perpendicular directions to choose from/restrict.

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Which of the following is an example of longitudinal wave?$
(a) Sound wave in air
(b) Water waves
(c) Light waves
(d) Electromagnetic waves
›Reveal solutionSolution

Sound waves in air are longitudinal waves.

In a longitudinal wave, particles of the medium oscillate parallel to the direction of wave propagation, producing alternating regions of compression (particles close together) and rarefaction (particles spread apart). Sound waves in air are the classic example of this — as a sound wave passes through air, air molecules vibrate back and forth along the same line the sound travels.

…

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$Q. Fill in the blank: The distance travelled by a wave in one time period is called .........$
›Reveal solutionSolution

The distance a wave travels in one time period is its wavelength.

A travelling wave repeats its pattern in space after a distance called the wavelength (lambda), and it repeats its pattern in time after an interval called the time period (T). Since the wave moves with speed v, and it advances exactly one wavelength during one time period, the two are related by v = lambda/T, or lambda = vT. Thus the dist …

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CBSE 2024Set ANNUAL1 markMCQ
$Q. Waves produced in open organ pipe is (A) longitudinal (B) transverse (C) both longitudinal and transverse (D) none of these$
›Reveal solutionSolution

Waves in an organ pipe (sound in air) are longitudinal.

In an organ pipe, the vibrating air column produces sound by alternating compressions and rarefactions of air, with the air particles oscillating back and forth parallel to the pipe's length (the direction of wave propagation) — the defining property …

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CBSE 2024Set ANNUAL1 markMCQ
$Q. What type of waves are produced in a vibrating string or organ pipe?$
(a) Transverse waves
(b) Longitudinal waves
(c) Electromagnetic waves
(d) Surface waves
›Reveal solutionSolution

A vibrating stretched string produces transverse waves; an organ pipe's air column actually produces longitudinal waves — these are two different cases, and since the question groups them together with only one answer choice allowed, this needs an honest caveat rather than a guess.

A vibrating string (as in a guitar or veena) has its particles oscillating perpendicular to the string's length, while the wave itself travels along the string — this is a transverse wave.

An organ pipe, however, produces sound through a vibrating AIR COLUMN, where air particles oscillate back and forth ALONG the direction the wave travels — this is a longitudinal wave, not transverse.

…

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$Q. What are progressive waves?$
›Reveal solutionSolution

Progressive waves are waves that travel continuously through a medium, transferring energy from one point to another, unlike stationary waves which stay fixed in place.

In a progressive wave, disturbance (and the energy associated with it) actually propagates through the medium with a definite wave speed v. Every particle of the medium undergoes the SAME kind of periodic (often simple harmonic) oscillation with the same amplitude and frequency, but a particle farther along the direction of travel lags in phase behind particles closer to the source — this progressive phase lag is what makes the wave pattern appear to move.

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. Water waves are$
(a) longitudinal
(b) transverse
(c) both longitudinal & transverse
(d) neither longitudinal nor transverse
›Reveal solutionSolution

Water-surface waves make particles trace elliptical paths, combining both longitudinal and transverse motion at once.

In a purely longitudinal wave, particles oscillate parallel to the direction of wave propagation (e.g. sound); in a purely transverse wave, they oscillate perpendicular to it (e.g. a wave on a string). Water waves are neither purely one nor the other: a water particle at the surface moves in a roughly circular or elliptical path, which has both a component along the direc …

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Q.In the given progressive wave y=5sin⁡(100πt−0.4πx)y = 5\sin(100\pi t - 0.4\pi x)y=5sin(100πt−0.4πx) where yyy and xxx are in m, ttt is in s. What is the

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Q.In the given progressive wave $y = 5\sin(100\pi t - 0.4\pi x)$ where $y$ and $x$ are in m, $t$ is in s. What is the

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