Basic properties of waves
One-dimensional wave
f(t, x)  Acos( x   t)
t
t=0
phase ( x   t) measures the position of a wave feature, such as a wave peak or trough
wavelength 
distance between repeating units of a propagating wave
wave-number 1
  2
circular wave-number
period T
duration of one cycle in a repeating event
frequency f  1
T
angular frequency
phase speed
number of occurrences of a repeating event per unit time
  2 f

c  T  
Euler formula
Ae i( x  t) , e i( x  t)  cos( x   t)  i sin(  x   t)
f(t, x)  Re(Ae i( x  t) )
i: imaginary unit
Harmonic wave : one amplitude and one frequency
f1 (t, x)  Acos( x   t)
Right moving wave
f 2 (t, x)  Acos( x   t)
Left moving wave
Waves superimpose together
Example 1: standing wave
f  f1  f 2  Acos( x   t)  Acos( x   t)  2cos( x)cos( t)


c1  1 , c 2  1 ;
1
1
Example 2: Non-dispersive and dispersive wave
f1 (t, x)  A1cos(1 x  1 t)
f1
f 2 (t, x)  A 2 cos( 2 x   2 t)
f2
1  2
, ;
1  2
f1
c1 , c 2
Specific case 1
f2
f1  f 2
f1 (t, x)  sin( x  t)
f 2 (t, x)  sin( 1.2x  1.2t)
f(t, x)  sin( x  t)  sin( 1.2x  1.2t)
c1  c 2 Non-dispersive wave-packet
Specific case 2
f1 (t, x)  sin( x  t)
f 2 (t, x)  sin( 1.2x  1.1t)
f(t, x)  sin( x  t)  sin( 1.2x  1.1t)
c1  c 2 Dispersive wave-packet
Non-dispersive wave: wave phase speed dos not depend on wave number.
Dispersive wave: wave phase speed depends on wave number.
Specific case 3
Specific case 4
Because of the reinforcement or cancellation of wave amplitude, the
energy of wave group will be concentrated in regions where the wave
amplitude is large, and those regions with small wave amplitude
contain less energy. Thus, wave energy does not propagate at the
phase speed of individual waves, but at the speed of wave envelope,
called group velocity.
Group velocity
f1 (t, x)  sin( 1 x  1 t)
f 2 (t, x)  sin(  2 x   2 t)
sin   sin   2sin
 
 
cos
2
2
f  f1  f 2  2sin( 0 x   0 t)cos( x   t),
where  0 
1  2
,
2
high frequency carrier wave
sin( 0 x   0 t)
slow moving envelope wave
cos( x   t)
group velocity


c g    
0 
1  2
,
2
   2  1,    2  1
Longitudinal wave: parcel's oscillation is in the same direction
as the direction of wave propagation.
Transverse wave: parcel's oscillation is perpendicular to the
direction of wave propagation.
Orbital wave: involving components of both longitudinal
and transverse wave.
X direction acoustic wave
d
dt
Mass conservation equation
  ux  0
Energy conservation equation
d  0
dt
  T(1000/p)
R d / Cp
Momentum equation
cu
p
 u  R d T

du  1 p
dt  x
p  R d T
0
Non-dispersive wave
Adiabatic sound speed
c a  R d T  287 1.4  300  347 ms -1
Sound frequency    c   (u  R d T )
Doppler effect
Stationary sound source
Sound source moving at a speed slower than sound spee
Sound source is moving at the same speed as the sound spe
This pressure wall is
called shock wave
Wave front
sound speed
barrier
supersonic
sound source moving faster than sound speed
so-called Mach cone
2.45 c