Waves Lecture 3 Sinusoidal Progressive Waves & Phase Class 11 Physics CBSE & JEE скачать в хорошем качестве

Waves Lecture 3 Sinusoidal Progressive Waves & Phase Class 11 Physics CBSE & JEE

In this lecture we focus on *sinusoidal progressive waves* for Class 11 Physics (CBSE) and JEE. We go from the physical picture (a string driven in SHM) to the standard mathematical form of a travelling sine wave: y(x,t) = A sin(ω t ∓ k x + φ) and learn how to read and use all wave parameters: A, λ, f, v, k, ω, and phase φ. What we cover in this video (Waves Lecture 3): • Definition of a progressive (travelling) wave: – Wave that travels in a definite direction with constant amplitude (no attenuation). – Distinguishing motion of the wave pattern from motion of individual particles. • Travelling sinusoidal wave model: – Sinusoidal wave as y = sin θ curve in space. – Rope/string driven in simple harmonic motion at one end. – Snapshot view: waveform moving along x, while each element does SHM in y. • Sine wave travelling on a string: – Source motion at x = 0: y(0,t) = A sin ωt. – Right-going disturbance with time delay x/v: y(x,t) = A sin[ω (t – x/v)]. • Introducing wave number and standard wave form: – Wave number k = 2π/λ. – Relations between parameters: v = λ/T = fλ, ω = 2π f, ω = k v. – Standard equation of a right-going sinusoidal wave: y(x,t) = A sin(ω t – k x). – Left-going wave: y(x,t) = A sin(ω t + k x). – Including initial phase φ₀: y(x,t) = A sin(ω t ∓ k x + φ₀) = A sin[2π(t/T ∓ x/λ) + φ₀]. • Speed of a sinusoidal wave: – Definition from snapshots: v = Δx / Δt. – Phase φ = ω t – k x + φ₀; keeping φ constant for a given crest: d(ω t – k x)/dt = 0 ⇒ v = dx/dt = ω/k = fλ. • Initial phase φ₀ and graph shift: – y(0,0) = A sin φ₀ determines starting displacement. – Positive φ₀: curve appears shifted to the left; negative φ₀: to the right. • Phase change with time (fixed position): – For right-going wave: y = A sin[2π(t/T – x/λ) + φ]. – At fixed x: Δφ = (2π / T) Δt. – In one period Δt = T, phase changes by 2π (same phase again). • Phase change with distance (fixed time): – At fixed t, between x₁ and x₂: Δφ = (2π / λ) Δx. – Path difference ↔ phase difference: – Separation Δx = n λ ⇒ Δφ = 2nπ (in phase). – Separation Δx = (odd) λ/2 ⇒ Δφ = (odd) π (out of phase). – These relations are fundamental later in interference, diffraction and also in *Gas Dynamics* wave problems. Important examples (Q9–Q15): • Q9: Three equally spaced particles A, B, C on x-axis: – A & B have same speed, A & C same velocity. – Using SHM phase reasoning to show: – Minimum AB distance = λ/2 (same speed, opposite directions → phase difference π). – Minimum AC distance = λ (same velocity → in phase). • Q10: Plane wave y = 4 sin[(π/2) (2t + x/8)] (y, x in cm, t in s): – Extract amplitude A, wavelength λ and wave speed v. – (a) Phase difference for the same particle for a time gap of 0.4 s. – (b) Phase difference at an instant between two particles 12 cm apart. – Uses Δφ = ω Δt and Δφ = (2π/λ) Δx. • Q11: Given A = 1.0 cm, λ = 3.0 m, f = 8 Hz: – At some instant, particle at x = 5.0 m has y = +0.5 cm and is moving upward. – Find the nearest particle with x gt. 5.0 m that is at its positive extreme. – Uses phase of SHM, direction of motion, and phase–path difference relation. • Q12: Wave y = A sin(kx – ωt + φ): – At x = 0, t = 0: y = A/2 and y is increasing. – Find initial phase φ by using both y(0,0) = A/2 (sin φ = 1/2) and sign of ∂y/∂t (cos φ lt. 0). • Q13: Wave y = 2 mm sin(3t – 6x + π/4), x in cm: – At t = 0, find phases at x₁ = π/3 cm and x₂ = π/2 cm. – Show that phase difference is π (out of phase). • Q14: Two waves on a string: – y₁ = 2.00 cm sin(20.0x – 32.0t), – y₂ = 2.00 cm sin(25.0x – 40.0t), x in cm, t in s. (a) Phase difference at x = 5.00 cm, t = 2.00 s. (b) Smallest positive x (at t = 2.00 s) where their phase difference is ±π (destructive interference). – Uses Δφ(x,t) = φ₁ – φ₂ = –5x + 8t, and Δφ = π + 2nπ. • Q15: Wave y = (2.00 cm) sin(kx – ωt) with k = π rad/m and ω = 2π rad/s: – Directly read off amplitude, wavelength λ = 2 m, frequency f = 1 Hz, speed v = 2 m/s by comparing with standard form. Channel: CBSE & JEE Physics | Dr Kedar Pathak Exam‑tight physics lectures in Hinglish with clean slides – no hype, only clear concepts, graphs, derivations and problem‑solving. These wave tools also prepare you for later chapters like sound, optics, and even Gas Dynamics.