Transverse and longitudinal waves- Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

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Transverse and Longitudinal Waves:

  • Transverse Waves: In transverse waves, the oscillation or displacement of the medium is perpendicular to the direction of the wave’s propagation. Examples include waves on a string and electromagnetic waves. The crests and troughs are characteristic of these waves. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
  • Longitudinal Waves: In longitudinal waves, the oscillation or displacement of the medium is parallel to the direction of wave propagation. Examples include sound waves and compression waves in a spring. The regions of compression and rarefaction are characteristic of these waves. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

2. Displacement Relation for a Progressive Wave:

For a progressive wave traveling in one dimension, the displacement yyy at position xxx and time ttt can be described by a sinusoidal function: Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

y(x,t)=Asin⁡(kx−ωt+ϕ)y(x, t) = A \sin(kx – \omega t + \phi)y(x,t)=Asin(kx−ωt+ϕ)

where:

  • AAA is the amplitude of the wave.
  • kkk is the wave number (k=2πλk = \frac{2\pi}{\lambda}k=λ2π​, where λ\lambdaλ is the wavelength).
  • ω\omegaω is the angular frequency (ω=2πf\omega = 2\pi fω=2πf, where fff is the frequency).
  • ϕ\phiϕ is the phase constant.

3. Principle of Superposition of Waves:

The principle of superposition states that when two or more waves overlap in space, the resultant displacement at any point is the algebraic sum of the displacements due to each wave individually. Mathematically: Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

yresultant(x,t)=y1(x,t)+y2(x,t)+⋯y_{\text{resultant}}(x, t) = y_1(x, t) + y_2(x, t) + \cdotsyresultant​(x,t)=y1​(x,t)+y2​(x,t)+⋯

This principle is used to explain phenomena such as interference and diffraction. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

4. Reflection of Waves:

When a wave encounters a boundary or a change in medium, it can be reflected. The behavior of the reflected wave depends on the type of boundary: Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

  • Fixed Boundary: The wave reflects back with an inverted phase. For instance, if a wave on a string hits a fixed end, it reflects back upside down. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
  • Free Boundary: The wave reflects back without inversion. For instance, if a wave on a string hits a free end, it reflects back right-side up. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

5. Standing Waves:

Standing waves are formed by the interference of two traveling waves moving in opposite directions with the same frequency and amplitude. They appear to be stationary and are characterized by nodes (points of no displacement) and antinodes (points of maximum displacement). The general form of a standing wave can be expressed as: Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

y(x,t)=2Asin⁡(kx)cos⁡(ωt)y(x, t) = 2A \sin(kx) \cos(\omega t)y(x,t)=2Asin(kx)cos(ωt)

where:

  • 2A2A2A is the amplitude of the standing wave.
  • sin⁡(kx)\sin(kx)sin(kx) describes the spatial variation.
  • cos⁡(ωt)\cos(\omega t)cos(ωt) describes the temporal variation.

Standing waves are commonly observed in vibrating strings, air columns, and other resonant systems. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.

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