Mechanical Waves and Wave Equation A wave is a nonlocal perturbation traveling in media or vacuum. A wave carries energy from place to place without a bulk flow of matter. A mechanical wave is a wave disturbance in the positions of particles in medium. Types of waves Electromagnetic waves (light), plasma waves, gravitational waves, … Periodic and solitary waves Parameters of periodic waves: (i) period T, cyclic frequency f, and angular frequency ω : T = 1/ f = 2π / ω ; (ii) wavelength λ and wave number k : λ = 2π / k ; v = λ/T=ω/k vgroup = dω/dk . (iii) phase velocity (wave speed) (iv) group velocity Sinusoidal (harmonic) wave traveling in +x: y ( x , t ) A cos(t kx ) A cosk (vt x ) t x x A cos 2 A cos t v T Solitons compression rarefaction Longitudinal Sound Waves Wave Equation Longitudinal (l ) waves (r ) in a 1-D lattice of identical particles: yn = xn – nL Xn-1Fn Xn Fn Xn+1 is a displacement of the n-th particle from its equilibrium position xn0 = nL. yn-1 y X nLyn (n+1)L n+1 (n-1)L Restoring forces exerted on the n-th particle: from left spring Fnx(l) = - k (xn-xn-1-L), from right spring Fnx(r) = k (xn+1-xn-L). Newton’s 2nd law: manx = Fnx(l) + Fnx(r) = k [xn+1-xn-(xn-xn-1)], anx= d2yn/dt2. Limit of a continuous medium: xn+1-xn= L∂y/∂x, xn+1-xn-(xn-xn-1)= L2∂2y/∂x2 2 2 y y 2 Wave Equation 2 v 0, 2 t x wave speed v kL2 m Transverse waves on a stretched string: y(x,t) is a transverse displacement. Restoring force exerted on the segment F is a tension force. Δx of spring: μ = Δm/Δx is a linear mass y y density (mass per unit length). Slope = -F1y/F=∂y/∂x Slope= F2y/F=∂ y/∂x Fy F2 y F1 y F x x x x x Newton’s 2nd law: μΔx ay= Fy , ay= ∂2y/∂t2 2 2 y 2 y Wave Equation 2 v 0,v 2 t x F Wave Intensity and Inverse-Square Law Power of 1D transverse 3-D waves wave on stretched string = Instantaneous rate of energy transfer along the string y y P ( x, t ) Fy v y F x t y 0 For a traveling wave y(x,t) = A cos (kx – ωt) , X F 2 vy v F 2 A 2 sin 2 (kx t ), F 2 A 2 Pmax Pav , 2 2 P ( x, t ) since vy = - v ∂y/∂x = = ωA sin (kx - ωt). Fy does work on the right part of string and transfers energy. Exam Example 33: Sound Intensity and Delay A rocket travels straight up with ay=const to a height r1 and produces a pulse of sound. A ground-based monitoring station measures a sound intensity I1. Later, at a height r2, the rocket produces the same second pulse of sound, an intensity of which measured by the monitoring station is I2. Find r2, velocities v1y and v2y of the rocket at the heights r1 and r2, respectively, as well as the time Δt elapsed between the two measurements. (See related problem 15.25.) Exam Example 34: Wave Equation and Transverse Waves on a Stretched String (problems 15.49, 15.63) Data: λ, linear mass density μ, tension force F, and length L of a string 0<x<L. Questions: (a) derive the wave equation from the Newton’s 2nd law; (b) write and plot y-x graph of a wave function y(x,t) for a sinusoidal wave traveling in –x direction with an amplitude A and wavelength λ if y(x=x0, t=t0) = A; (c) find a wave number k and a wave speed v; y A L (d) find a wave period T and an angular frequency ω; 0 X (e) find an average wave power Pav . Solution: (b) y(x,t) = A cos[2π(x-x0)/λ + 2π(t-t0)/T] where T is found in (d); (c) k = 2π / λ , v = (F/μ)1/2 as is derived in (a); (d) v = λ / T = ω/k → T = λ /v , ω = 2π / T = kv (e) P(x,t) = Fyvy = - F (∂y/∂x) (∂y/∂t) = (F/v) vy2 Pav = Fω2A2 /(2v) =(1/2)(μF)1/2ω2A2. (a) Derivation of the wave equation: y(x,t) is a transverse displacement. Restoring force exerted on the segment F is a tension force. Δx of spring: μ = Δm/Δx is a linear mass y y density (mass per unit length). Slope = -F1y/F=∂y/∂x Slope= F2y/F=∂ y/∂x Fy F2 y F1 y F x x x x x Newton’s 2nd law: μΔx ay= Fy , ay= ∂2y/∂t2 2 2 y 2 y Wave Equation 2 v 0,v 2 t x F Principle of Linear Superposition. Wave Interference and Wave Diffraction y ( x, t ) yi ( x, t ) i Energy is conserved, but redistributed in space. Constructive interference at the time of overlapping of two wave pulses. Destructive interference at the time of overlapping of two wave pulses: Energy is conserved, but redistributed in space. Diffraction is the bending of a wave around an obstacle or the edges of an opening. Direction of the first minimum: sin θ = λ / D for a single slit , sin θ = 1.22 λ / D for a circular opening. The phenomenon of beats for two overlapping waves with slightly different frequencies Reflection of Waves and Boundary Conditions Example: Transverse waves on a stretched string. Traveling and Standing Waves. Normal (Natural) Modes. Traveling waves (in ±x direction): y(x,t) = A cos (±kx - ωt) = = A cos [ k (±x - vt) ] When a guitar string is plucked (pulled into a triangular shape) and released, a superposition of normal modes results. Transverse Standing Waves. 2ASW=4A Standing wave: y(x,t) = A [cos (kx + ωt) – cos (kx - ωt)]= = 2A sin (kx) sin (ωt) Amplitude of standing wave ASW = 2A λn = 2L/n Longitudinal Standing Waves Tube open at both ends: Tube open at only one end: fn = nf1, n= 1, 2, 3, …; L=nλ1/2 fn = nf1, n= 1, 3, 5, …; L=nλ1/4 . Only odd harmonics f1, f3, f5, … exist.