Mechanical Waves and Wave EquationA 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

compression

rarefaction

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 ;(iii) phase velocity (wave speed) v = λ/T=ω/k (iv) group velocity vgroup = dω/dk .

Sinusoidal (harmonic) wave traveling in +x:

v

xtA

x

T

tA

xvtkAkxtAtxy

cos2cos

)(cos)cos(),(

Solitons

Longitudinal Sound Waves

Wave Equation

m

kLvspeedwave

x

yv

t

yEquationWave

2

2

22

2

2

,0

Longitudinal waves in a 1-D lattice of identical particles: yn = xn – nL is a displacement of the n-th particle from

its equilibrium position xn0 = nL.

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

Transverse waves on a stretched string: y(x,t) is a transverse displacement. Restoring force exerted on the segment Δx of spring:

(n-1)L nL (n+1)L

Xn-1 Xn Xn+1

Xyn-1 ynyn+1

)(lnF

)(r

nF

xxx

yyy x

y

x

yFFFF 12

F is a tension force.μ = Δm/Δx is a linear massdensity (mass per unit length).

Newton’s 2nd law: μΔx ay= Fy , ay= ∂2y/∂t2

F

vx

yv

t

yEquationWave

,02

22

2

2

Slope=F2y/F=∂ y/∂xSlope = -F1y/F=∂y/∂x

Wave Intensity and Inverse-Square LawPower of 1D transverse wave on stretched string =Instantaneous rateof energy transferalong the string

t

y

x

yFvFtxP yy

),(

),(sin

),(

222

2

tkxAF

vv

FtxP y

For a traveling wavey(x,t) = A cos (kx – ωt) ,

,22

22max AFP

Pav

Fy does work on the right partof string and transfers energy.

X

y

0

3-D waves

since vy = - v ∂y/∂x = = ωA sin (kx - ωt).

Exam Example 33:Sound Intensity and Delay

A rocket travels straight upwith ay=const to a height r1

and produces a pulse of sound. A ground-based monitoringstation measures a soundintensity I1. Later, at a heightr2, the rocket produces thesame second pulse of sound,an intensity of which measuredby the monitoring station is I2.Find r2, velocities v1y and v2y ofthe rocket at the heights r1 andr2, respectively, as well as the time Δt elapsed between the two measurements.(See related problem 15.25.)

(a) Derivation of the wave equation: y(x,t) is a transverse displacement. Restoring force exerted on the segment Δx of spring:

xxx

yyy x

y

x

yFFFF 12

F is a tension force.μ = Δm/Δx is a linear massdensity (mass per unit length).

Newton’s 2nd law: μΔx ay= Fy , ay= ∂2y/∂t2

F

vx

yv

t

yEquationWave

,02

22

2

2

Slope=F2y/F=∂ y/∂xSlope = -F1y/F=∂y/∂x

Exam Example 34: Wave Equation and Transverse Waves on a Stretched String (problems 15.51 – 15.53)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 travelingin –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;(d) find a wave period T and an angular frequency ω;(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);

y

X0

LA

(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) vy

2 Pav = Fω2A2 /(2v) =(1/2)(μF)1/2ω2A2.

Principle of Linear Superposition.Wave Interference and

Wave Diffraction

i

i txytxy ),(),(

Constructive interferenceat the time of overlappingof two wave pulses.

Energy is conserved, butredistributed in space.

Energy is conserved, butredistributed in space.

Destructive interferenceat the time of overlappingof two wave pulses:

Diffraction is the bending of a wave aroundan 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 beatsfor two overlapping waves withslightly different frequencies

Reflection of Waves and Boundary Conditions

Example: Transverse waves on a stretched string.

Traveling and Standing Waves. Transverse Standing Waves.Normal (Natural) Modes.

When a guitar string is plucked (pulled into a triangular shape) and released, a superpositionof normal modes results.

Traveling waves (in ±x direction):y(x,t) = A cos (±kx - ωt) = = A cos [ k (±x - vt) ]

Standing wave: y(x,t) = A [cos (kx + ωt) – cos (kx - ωt)]= = 2A sin (kx) sin (ωt) Amplitude of standing wave ASW = 2A

2ASW=4A

λn = 2L/n

Longitudinal Standing Waves

Tube open at both ends:

fn = nf1, n= 1, 2, 3, …; L=nλ1/2

Tube open at only one end:

fn = nf1, n= 1, 3, 5, …; L=nλ1/4 .

Only odd harmonics f1, f3, f5, … exist.