1. WAVE MOTION AND ITS APPLICATIONS

 Welcome to Rajasthan Polytechnic Physics (2002) Notes for 2nd Semester Students!

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1. WAVE MOTION AND ITS APPLICATIONS

Wave Motion: As mentioned earlier, wave motion is the movement of energy through a medium or space. It's important to differentiate between mechanical and electromagnetic waves:

• Mechanical Waves: Require a medium to propagate (e.g., water waves, sound waves, and seismic waves). Without a medium, these waves cannot travel.
• Electromagnetic Waves: Do not require a medium and can propagate through a vacuum (e.g., light, radio waves, and X-rays). These waves move at the speed of light.

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1.1 Transverse vs Longitudinal Waves

Transverse Waves:

• In these waves, particles of the medium move perpendicular to the direction of wave propagation.
• Example: Light waves, electromagnetic waves, and water waves.
• Important Points: In a transverse wave, you’ll observe crests (high points) and troughs (low points). The wave travels in a direction, but the medium’s particles move up and down.

Longitudinal Waves:

• In longitudinal waves, particles of the medium move parallel to the direction of wave propagation.
• Example: Sound waves, seismic P-waves (Primary waves).
• Important Points: In these waves, you will find compressions (regions where particles are close together) and rarefactions (regions where particles are spread out).

Key Difference:

• Transverse waves have oscillations perpendicular to the direction of wave travel (e.g., waves on a rope).
• Longitudinal waves have oscillations parallel to the direction of wave travel (e.g., sound waves).

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1.2 Wave Velocity, Frequency, and Wavelength

1. Wave Velocity (v):

   • This is the speed at which the wave travels through a medium. The formula is $v = \frac{d}{t}$, where $d$ is the distance traveled by the wave, and $t$ is the time taken.
   • Example: If a sound wave travels 340 meters in 1 second, its velocity is $v = 340 \, \text{m/s}$.

2. Frequency (f):

   • The frequency is the number of complete oscillations or cycles of the wave that occur per unit time (usually seconds).
   • Unit: Hertz (Hz), where 1 Hz = 1 cycle per second.
   • Example: A wave with 2 cycles every second has a frequency of 2 Hz.

3. Wavelength (λ):

   • Wavelength is the distance between two consecutive points in phase, typically two crests or two troughs.
   • Unit: Meter (m).
   • Example: In water waves, the distance between one crest and the next crest is the wavelength.

Relationship:

$$v = f \times \lambda$$

This means the wave velocity is the product of frequency and wavelength. If you increase the frequency, the wavelength decreases for a constant wave speed and vice versa.

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1.3 Numerical Example on Wave Velocity, Frequency, and Wavelength:

Problem: A sound wave travels through air with a speed of 340 m/s. If its frequency is 170 Hz, what is the wavelength of the sound wave?

Solution: Using the formula $v=f\times \mathrm{\lambda ,\; we\; can\; rearrange\; it\; to\; find\; the\; wavelength:}$

$$\lambda = \frac{v}{f} = \frac{340}{170} = 2 \, \text{m}$$

Answer: The wavelength of the sound wave is 2 meters.

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1.4 Principle of Superposition

The principle of superposition applies when two or more waves meet at a point. The resultant displacement at that point is the algebraic sum of the displacements of the individual waves.

• Constructive Interference: If the two waves meet in phase (i.e., crests meet crests, and troughs meet troughs), the resultant displacement will be greater.
• Destructive Interference: If the two waves meet out of phase (i.e., crests meet troughs), they cancel each other out, resulting in no displacement at the point of overlap.

Example: Two sinusoidal waves:

$$y_1 = A \sin(kx - \omega t)$$
$$y_2 = A \sin(kx - \omega t + \phi)$$

The resultant displacement:

$$y_{\text{total}} = y_1 + y_2 = 2A \cos\left(\frac{\phi}{2}\right) \sin\left(kx - \omega t + \frac{\phi}{2}\right)$$

If the waves are in phase ($\phi = 0$), they will constructively interfere.

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1.5 Stationary Waves and Resonance Tube

Stationary Waves: When two waves of the same frequency, amplitude, and wavelength travel in opposite directions, they interfere to form a stationary wave. A stationary wave consists of nodes (points of zero displacement) and antinodes (points of maximum displacement).

Resonance Tube: A resonance tube is a tube open at both ends. When sound waves resonate inside it, stationary waves form. The resonance condition occurs when the tube’s length corresponds to specific wavelengths of the sound.

Example:
In a resonance tube, at certain lengths of the tube, standing waves will form and produce maximum sound intensity. The frequency of the sound can be related to the length of the tube and the speed of sound.

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1.6 Simple Harmonic Motion (SHM)

SHM is the most basic form of oscillatory motion where the restoring force is proportional to the displacement and acts in the opposite direction.

The motion is governed by Hooke's Law:

$$F = -kx$$

Where:

• $F$ is the restoring force,
• $k$ is the spring constant,
• $x$ is the displacement from equilibrium.

Energy in SHM:
The total mechanical energy in SHM is constant, and it’s the sum of kinetic energy and potential energy:

$$E_{\text{total}} = KE + PE$$

The energy oscillates between kinetic and potential forms, but the total remains constant throughout the motion.

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1.7 Numerical Example on SHM:

Problem: A block of mass 0.5 kg is attached to a spring with a spring constant of 100 N/m. It is displaced by 0.1 m from the equilibrium position. Find the amplitude of the motion and the maximum velocity of the block.

Solution:

1. Amplitude: The amplitude $A$ is the maximum displacement from the equilibrium position, which is 0.1 m.
2. Maximum Velocity:
   The maximum velocity in SHM occurs when the object passes through the equilibrium point (where the displacement is zero):

$$v_{\text{max}} = A \cdot \omega$$

where $\omega$ is the angular frequency and is given by:

$$\omega = \sqrt{\frac{k}{m}}$$

Substitute the given values:

$$\omega = \sqrt{\frac{100}{0.5}} = \sqrt{200} = 14.14 \, \text{rad/s}$$

Now calculate the maximum velocity:

$$v_{\text{max}} = 0.1 \cdot 14.14 = 1.414 \, \text{m/s}$$

Answer: The amplitude is 0.1 m, and the maximum velocity is 1.414 m/s.

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Final Summary:

• Wave Velocity, Frequency, and Wavelength: $v = f \times \lambda$, with examples for calculating wavelength and frequency.
• Transverse vs Longitudinal Waves: Key differences in the direction of oscillation and types of waves (e.g., light vs sound).
• Superposition Principle: Constructive and destructive interference with mathematical derivations.
• Stationary Waves: Examples like resonance tubes and their application in sound.
• Simple Harmonic Motion (SHM): The basic oscillatory motion with energy exchange between kinetic and potential forms.

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