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JEE Main Oscillations and Waves Revision Notes

## JEE Main Oscillations and Waves Revision Notes - PDF Download

We can define the wave motion as the disorder that originated in the medium because of the frequent periodic motion of the atoms of the medium and transmitting it from atom to atom. Due to this transmission, the particles remain vibrating about themselves and their mean positions.

Wave Equation: \[\frac{d^{2}y}{dt^{2}} = v^{2} (\frac{d^{2}y}{dx^{2}})\]

## Transverse Wave Motion

Transverse wave motion is a kind of wave motion where the atoms of the medium are vibrating. Their vibration is continuous in a direction at 90 degrees to the direction of dissemination (propagation) of a wave.

i. Transverse Wave’s Velocity:

\[V_{t} = \sqrt{\frac{T}{\pi r^{2} \rho}} = \sqrt{\frac{T}{m}}\]

ii. As we mentioned earlier, the atoms’ vibrations in the medium are at 90 degrees to the direction of wave propagation.

Longitudinal Wave Motion

Longitudinal wave motion is a kind of wave motion where the vibration of the elements of the medium lies in the same direction of wave propagation.

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i. Longitudinal wave’s velocity, \[V_{l} = \sqrt{\frac{E}{\rho}}\]

ii. The particle's direction of vibration seems to be in parallel with wave propagation.

### Terms Related to Wave Motion

Wavelength (λ)

The wavelength can be distinct as the space covered by a wave by the time particles complete one vibration due to SHM.

The wavelength is the space between two successive particles performing SHM in the same phase.

Wavelength = the distance between two successive crests or troughs.

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Wave Number (n)

The definition of the wave-number of a wave can be explained as the reciprocal of the wavelength of the wave.

n = 1 / λ

Meter is the unit of wave number.

Velocity of Wave

The velocity of the wave can be illustrated as the length (λ) covered by the wave through the time (T) at the time of completion of one vibration of a particle.

(frequency) * (wavelength) = velocity of the wave

V = f * λ

Phase

The phase of a particle is its state, which expresses its position and direction of motion.

Phase Difference (ϕ)

The definition for the phase signifies that it is a difference between two particles or a difference between their instantaneous phases.

The relationship between the phase difference (ϕ) and path difference (λ), is:

ϕ = (2π / λ) × (path difference)

### Waves IIT JEE Notes PDF

Simple Harmonic Motion (SHM)

The simple harmonic motion of a wave signifies that a wave initiates from a source experiencing simple harmonic motion and is termed as the simple harmonic wave.

Mathematically,

Y = r sin (ω t)

= r sin 2πf t

= r sin 2π (v / λ) t

= r sin 2π x / λ (x = vt)

Equation of Progressive Wave

Equation of progressive waves can be explained as an association between the instantaneous displacement of a particle affecting SHM and time.

r sin 2π (ωt ± ϕ) = y

r sin [(ωt ± (2π / λ) x] = y

r sin (ωt ± kx) = y

r sin 2π (t / T ± x / λ) = y

r sin 2π / λ (vt ± x) = y

### Oscillation and Waves Revision Notes

The particle velocity (V) and wave velocity (v) relationship

(2πr / λ) v cos [(2π / λ) (vt ± x)] = V

(2πr / λ) v = Vmax

### Solved Oscillation and Waves Examples

Q1. In an SHM, a particle’s amplitude of oscillation is 2cm, and its displacement is 1cm from the mean position. Let’s say that the magnitude of its acceleration is equal to that of its velocity. Then calculate the maximum velocity, maximum acceleration of SHM, and the time of SHM.

Ans: Here, a = 6 cm,

When y = 1 cm

It is said that, the magnitude of velocity = magnitude of the acceleration

\[\omega (a^{2} - y^{2})^{\frac{1}{2}} = \omega^{2}y\]

or, \[a^{2} - y^{2} = \omega^{2}y^{2}\]

or, \[2^{2} - 1^{2} = \omega^{2}\]

or, \[\omega = \sqrt{3}\]

T = time period = \frac{2 \pi}{\omega} = \frac{2 \pi}{\sqrt{3}} = 3.63 s\]

Maximum velocity = \[a \omega = 2 \times \sqrt{3} = 6 \times 1.732 = 10.39 cm/s\]

Maximum acceleration = \[\omega^{2}a = 3 \times 6 = 18 cm/s^{2}\]

Q2. Can you find out the second’s pendulum’s time period when its length is doubled?

Ans: We know that

Time, \[T = 2\pi \sqrt{\frac{l}{g}}\]

Second’s pendulum = 2s

Then, \[T = 2\pi \sqrt{\frac{l}{g}} = 2\] . . . . .(1)

L = 2l as the length is doubled

\[T' = 2\pi \sqrt{\frac{2l}{g}} = \sqrt{2} \times 2\pi \sqrt{\frac{l}{g}} = \sqrt{2} \times 2 = 2.828\]s.

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Q1. Mention the Second Kind of Waves.

Ans: As per scientific experiments, electromagnetic waves are classified into seven basic types such as,

Radio Waves

Microwaves

Visible Light Rays

Ultraviolet Waves

Infrared Waves

Gamma Rays

X - rays

Q2. What are the Features of the Oscillation of Waves?

Ans: The oscillating waves lie perpendicular to the direction where oscillation takes place for transverse waves. The wave transfers energy, instead of moving in the direction of propagation.

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