Waves Class 12 One shot in Telugu | EAPCET JEE Physics | EAPCET JEE 2025

Waves are a crucial topic in physics, closely linked with simple harmonic motion. This lesson focuses on understanding waves without the Doppler effect, as it has been removed from the syllabus. The session covers wave equations, progressive waves, stationary waves (including strings and pipes), resonance tubes experiments like sonometers, and prepares for beats discussion in subsequent classes.

A wave is a disturbance created in a medium, such as liquid or gas, that carries energy from one point to another without any loss of energy. The particles within the medium remain at their mean position while transmitting this energy through vibrations. This process represents the transmission of energy via waves, which can occur in two primary forms: transverse waves and longitudinal waves.

Understanding Transverse and Longitudinal Waves Transverse waves involve particle vibrations perpendicular to the wave's direction of propagation, while longitudinal waves have particles vibrating parallel to it. For example, in transverse waves like water ripples, particles move up and down as the wave travels horizontally. In contrast, longitudinal waves such as sound compress and expand along their path.

Frequency and Time Period Explained The time period is defined as the duration required for one complete oscillation or cycle of a wave. Frequency refers to how many oscillations or cycles occur per second; its SI unit is Hertz (Hz). These concepts are fundamental in understanding periodic motion within various types of waves.

The progressive wave equation is expressed as y = A sin(ωt - kx), where 'y' represents particle displacement, 'A' denotes amplitude (maximum displacement from mean position), and ω signifies angular frequency. The term 'k', known as the wave number or constant, equals 2π/λ and describes a specific property of a given wave. This equation illustrates how particles vibrate in one direction while waves propagate energy in another—particles move vertically ('y'), whereas waves travel horizontally ('x'). For transverse progressive waves like S-waves, maximum upward or downward displacements define their amplitudes.

The initial phase of a wave can be positive, negative, or zero. At time T=0 and position X=0, the equation simplifies to y = A sin(ωt - kx + φ), where φ represents the phase constant. When considering specific cases like maximum displacement (y_max), adjustments are made to include terms such as π/2 for accurate representation of oscillatory motion.

The phase difference (Δϕ) is directly related to the path difference (Δx) through the equation Δϕ = 2π/λ × Δx, where λ represents wavelength. For two points P1 at X1 and P2 at X2, their phase difference arises from subtracting wave equations: Y₁ = A sin(ωt - kX₁) and Y₂ = A sin(ωt - kX₂). This results in a relation of K(X₂-X₁), linking spatial separation with oscillatory behavior. Additionally, waves propagating along positive or negative x-axes are distinguished by opposite signs in their sinusoidal expressions.

Understanding Wave and Particle Velocities Wave velocity refers to the speed at which a wave propagates through space, while particle velocity describes how individual particles of the medium move as the wave passes. When time varies but position is constant, particle velocity can be derived using derivatives with respect to time. The maximum particle velocity equals amplitude multiplied by angular frequency (Aω). A key relationship exists: particle velocity equals negative wave velocity times slope of the wave.

Analyzing Progressive Waves and Particle Movements In transverse progressive waves, snapshots reveal positions where particles either move upward or downward based on their slopes. Positive slopes correspond to upward movement; negative slopes indicate downward motion; zero slope means no movement occurs for those points. For example, specific points like C and G remain stationary due to zero slope in such scenarios.

Understanding Wave Velocity and Repeated Question Patterns The wave velocity is calculated using the formula V = fλ, where frequency (f) and wavelength (λ) are key parameters. For a given progressive wave equation with specific values of ω=628 rad/s and k=31.4 cm⁻¹, the velocity computes to 20 cm/s. Questions on this topic often repeat across years in exams like those from 2021-2024, emphasizing its importance for students preparing for competitive tests.

Phase Difference, Wavelength Relations & Superposition Concepts For waves with phase differences or superpositions traveling along axes such as X-direction, formulas involving λ=f/v help determine properties like distance between points or maximum particle velocities. A notable example includes calculating wavelength when phase difference equals π/3 radians over a known distance of 6 meters at a frequency of 500 Hz; resulting in an answer showing consistent patterns year-on-year e.g., solving yields answers around ~18km/sec ranges etc repetitive trends seen!

The velocity of a transverse progressive wave on a string is determined by the square root of tension divided by linear mass density (v = √(T/μ)). Tension, represented as T, equals the weight force acting on the string (mg), while μ represents mass per unit length. The formula can also incorporate material properties like density and cross-sectional area to express μ. Changes in temperature affect thermal stress and consequently influence wave velocity through parameters such as Young's modulus, coefficient of thermal expansion, and temperature change.

Calculating Wave Velocity and Time The velocity of a wave is determined using the formula V = √(T/μ), where T represents tension, μ denotes mass per unit length. For a steel wire with given parameters, calculations yield specific values for speed and time taken by the wave to travel certain distances. The relationship between distance traveled, velocity, and time (time = distance/velocity) plays an essential role in these computations.

Wave Dynamics in Ropes: Average Velocity & Segment Ratios In ropes under varying tensions along their lengths due to gravity or other forces, average wave velocities are calculated as V_avg = (V_top + V_bottom)/2. Using free body diagrams helps determine tension at different points on the rope affecting local velocities like root(T/mu). Additionally, ratios of segmental velocities depend directly on square roots of respective tensions; e.g., if T2= 2*T1 then ratio becomes sqrt(T1/T2).

Understanding Wave Velocity and Its Dependencies The velocity of a longitudinal progressive wave is directly proportional to the square root of the modulus of elasticity (E) and inversely proportional to the square root of density (ρ). For solids, Young's modulus determines E, while for fluids it’s governed by bulk modulus. The speed at standard temperature and pressure ranges between 330-340 m/s. In gases, sound velocity depends on factors like molecular weight and specific heat ratio ( γ), with formulas derived from kinetic theory.

Temperature Effects on Sound Speed Sound speed varies with temperature; as temperature increases, so does its velocity following V = V₀(1 + T/330 or T/340). This relationship highlights how thermal energy influences particle motion in a medium. At STP conditions (~0°C), typical speeds are around 330 m/s but adjust proportionally based on given temperatures.

The superposition principle explains how waves combine, with resultant amplitude determined by the sum or difference of individual amplitudes. Constructive interference occurs when wave amplitudes add positively, while destructive interference happens when they subtract due to phase differences. The formula for resultant amplitude involves the square root of squared individual amplitudes plus a term accounting for their phase difference using cosine functions.

Stationary waves form when an incident wave superimposes with its reflected counterpart, creating points of maximum amplitude (antinodes) and zero displacement (nodes). The behavior differs at rigid ends versus free supports; for example, the direction of the reflected wave changes in a rigid end scenario. In composite strings, transmitted and reflected pulses exhibit varying signs depending on their interaction. Harmonics play a crucial role in musical instruments: the lowest frequency produced is called fundamental frequency or first harmonic, while higher frequencies are termed overtones.

Stationary waves in strings are explained through harmonics, where the fundamental frequency is determined by the formula f = v/2L. The first harmonic corresponds to one loop of a wave, while higher harmonics involve additional loops with frequencies proportional to 1:2:3 ratios. In pipes, stationary waves differ based on whether they are open or closed; for example, closed pipes produce only odd-numbered harmonics due to their boundary conditions. Applications include musical instruments like organ pipes and sonar systems.

In closed pipes, the fundamental frequency (first harmonic) is determined by the formula f = V / 4L. The second harmonic corresponds to three times this frequency, making it a third harmonic or first overtone. Subsequent harmonics follow an odd-numbered sequence: 1:3:5 for frequencies. For open pipes, particles are free to move at both ends. Here, the fundamental frequency follows f = V / 2L with subsequent harmonics forming a simple integer ratio of frequencies like 1:2:3.

In a closed pipe, resonance occurs with one end open and the other closed. The length of the air column (L) is calculated using formulas involving velocity (V), frequency (F), and wavelength ( ). For fundamental frequencies, L equals V divided by 4 times F; for higher harmonics like third or fifth, it involves multiplying F accordingly. Corrections are applied to account for factors such as radius adjustments during calculations.

Understanding Stationary Waves and Gas Properties Stationary waves are analyzed through their fundamental frequencies, harmonics, and the relationship between tension in strings and wave speed. The properties of gases like helium (monatomic) and hydrogen (diatomic) are explored using specific heat capacities to determine ratios such as RMS speed relative to sound velocity. Calculations involve molecular weights affecting sound speeds across different gases under standard conditions.

Practical Applications: Strings, Harmonics, Tension & Frequency String vibrations depend on linear mass density, length between supports, harmonic modes (e.g., fifth overtone), with frequency proportional to square root of tension. Problems include calculating percentage decrease in frequency due to reduced string tension or determining wavelength from given vibration data for closed pipes' harmonics. Practical examples emphasize formula applications for solving real-world physics problems efficiently.