Waves: Comprehensive NEET Physics Notes

1. Introduction to Waves

Waves are disturbances that transfer energy from one point to another without the actual physical transfer of matter. Waves can be broadly classified into mechanical waves, which require a medium for propagation, and electromagnetic waves, which do not require a medium. Mechanical waves include types such as sound waves, water waves, and seismic waves, while electromagnetic waves include light, radio waves, and X-rays.

2. Transverse and Longitudinal Waves

2.1 Transverse Waves

In transverse waves, the oscillations of the particles of the medium are perpendicular to the direction of wave propagation. A common example is waves on a stretched string, where the displacement of the string is perpendicular to the direction in which the wave travels.

Formula:

• Displacement relation: $y(x,t)=a\sin (kx-\omega t)$
• • $y(x,t)$: Displacement at position $x$ and time $t$
  • $a$: Amplitude of the wave
  • $k$: Angular wave number ($k=\frac{2\pi }{\lambda }$)
  • $\omega$: Angular frequency ($\omega =2\pi \nu$)

Explanation:

• $a$ represents the maximum displacement of the particles from their equilibrium position.
• $k$ relates to the wavelength $\lambda$, and $\omega$ relates to the frequency $\nu$.

Example Application: A wave traveling along a string with amplitude 0.005 m, angular wave number 80 rad/m, and angular frequency 3 rad/s can be described by the equation $y(x,t)=0.005\sin (80x-3t)$. At $x=0.3$ m and $t=20$ s, the displacement is $y(0.3,20)=0.005\sin (1.699)\approx 0.005$ m.

Common Mistake: Students often confuse the direction of the wave's motion with the direction of the particle's motion. In transverse waves, these are perpendicular.

2.2 Longitudinal Waves

In longitudinal waves, the oscillations of the particles are parallel to the direction of wave propagation. Sound waves in air are a prime example, where regions of compression and rarefaction travel through the medium.

Formula:

• Displacement relation: $s(x,t)=a\sin (kx-\omega t)$
• • $s(x,t)$: Displacement in the direction of wave propagation

Explanation: The displacement amplitude $a$ and other quantities have the same significance as in transverse waves, but the oscillations occur in the direction of the wave's travel.

Example Application: Consider a sound wave in air with amplitude 0.002 m, frequency 500 Hz, and speed 340 m/s. The wavelength $\lambda$ is calculated as $\lambda =\frac{v}{\nu }=\frac{340}{500}=0.68$ m. The wave equation is $s(x,t)=0.002\sin (9.24x-3141.6t)$.

Common Misconception: A common error is to assume that longitudinal waves cannot propagate through solids. However, they do, as seen in seismic P-waves.

3. Displacement Relation in a Progressive Wave

3.1 Amplitude and Phase

• Amplitude (a): Maximum displacement from the equilibrium position.
• Phase ($\varphi$): Describes the position within the wave cycle at a given point.

3.2 Wavelength ($\lambda$) and Angular Wave Number ($k$)

• Wavelength ($\lambda$): The distance between two successive points in phase.
• Angular Wave Number ($k$): Defined as $k=\frac{2\pi }{\lambda }$.

3.3 Period ($T$), Angular Frequency ($\omega$), and Frequency ($\nu$)

• Period ($T$): The time taken for one complete oscillation.
• Angular Frequency ($\omega$): Defined as $\omega =\frac{2\pi }{T}$.
• Frequency ($\nu$): The number of oscillations per second, $\nu =\frac{1}{T}$.

Formula Recap:

• Wave equation: $y(x,t)=a\sin (kx-\omega t+\varphi )$

4. Speed of a Travelling Wave

4.1 Speed of a Transverse Wave on a Stretched String

Formula: $v=\sqrt{\frac{T}{\mu }}$

• T: Tension in the string
• $\mu$: Linear mass density of the string

Explanation: The speed of the wave depends on the tension in the string and its mass per unit length. Higher tension or lower mass per unit length results in a faster wave.

Example Application: For a string with tension 60 N and linear mass density $6.9\times 10^{-3}$ kg/m, the wave speed is $v=\sqrt{\frac{60}{6.9\times 10^{-3}}}\approx 93$ m/s.

Common Mistake: Confusing the dependence of wave speed on frequency or wavelength, when it actually depends only on the medium properties (tension and mass density).

5. Reflection of Waves

5.1 Reflection at a Rigid Boundary

When a wave reflects off a rigid boundary, it undergoes a phase change of $\pi$ (180 degrees), which effectively inverts the wave.

Formula:

• Incident wave: $y_i(x,t)=a\sin (kx-\omega t)$
• Reflected wave: $y_r(x,t)=-a\sin (kx-\omega t)$

Explanation: The inversion occurs because the boundary cannot move, creating a node at the reflection point.

Example Application: When a wave on a string hits a fixed end, it reflects with an inverted phase. If the incident wave is $y_i=0.005\sin (80x-3t)$, the reflected wave is $y_r=-0.005\sin (80x-3t)$.

NEET Tip: Remember to consider phase change when dealing with wave reflection problems.

6. Doppler Effect

The Doppler Effect describes the change in frequency observed when there is relative motion between a wave source and an observer.

Formula:

• For a source moving towards the observer: ${\nu }^{\prime }=\nu \left(\frac{v+v_o}{v-v_s}\right)$
• • ${\nu }^{\prime }$: Observed frequency
  • $\nu$: Source frequency
  • $v$: Speed of sound in the medium
  • $v_o$: Speed of the observer
  • $v_s$: Speed of the source

Example Application: If a car approaches a stationary observer with a horn frequency of 500 Hz and speed of 30 m/s, the observed frequency is ${\nu }^{\prime }=500\times \frac{343+0}{343-30}\approx 538$ Hz.

Common Mistake: Forgetting to adjust the formula for whether the source or observer is moving towards or away from each other.

Quick Recap

• Transverse and longitudinal waves have distinct oscillation directions.
• Key wave parameters include amplitude, wavelength, frequency, and speed.
• Reflection can cause phase inversion, particularly at rigid boundaries.
• The Doppler Effect explains frequency shifts due to relative motion.

Practice Questions

1. A wave on a string is described by $y(x,t)=0.01\sin (100x-200t)$. Find the amplitude, wavelength, and speed of the wave.
2. Calculate the frequency observed by a stationary observer when a sound source moving at 40 m/s approaches them at 340 m/s.
3. Derive the relationship between wave speed, frequency, and wavelength for a transverse wave on a string.
4. Explain the phase change when a wave reflects off a fixed boundary.
5. A pipe open at both ends resonates at its second harmonic. If the length of the pipe is 0.5 m, what is the frequency of the sound wave inside?

Solutions:

1. Amplitude = 0.01 m, Wavelength = 0.0628 m, Speed = 2 m/s.
2. Frequency observed = 558.8 Hz.
3. Derivation involves using $v=\lambda \nu$.
4. The phase change is due to the boundary condition that the displacement must be zero at the rigid boundary.
5. Frequency = 680 Hz.

These notes provide a structured overview of the key formulae and concepts in the chapter on waves, aligning closely with the NEET syllabus and emphasizing problem-solving techniques.