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 cosk (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.