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EE599-020
Audio Signals and Systems
Wave Basics
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Vibrating String
o
y2
y1
y
1
o
2
x
Given an elastic string is displaced in the y-direction, let o
be the mass per unit length, assume tension o is constant
and independent of position, angles (1 and 2) are small,
motion is limited to x-y plane, and element x of the string
is only displaced in the y-direction. Derive the equation
governing the motion of the string.
Vibrating String
Differential forces in x and y direction from tension o
are given by Fx   o (cos( 2 )  cos(1 )) Fy   o (sin( 2 )  sin(1 ))
For small , Fx is negligible and sin()=  = dy/dx .
Therefore Newton’s second law for y-direction results in:
o x
o
2 y
t 2
y 
 y
o 2  1 
x 
 x
t 2
2 y
1  y 2 y1 



x 
x 0 x  x
  o lim
t 2
 o 2 y

o x 2
2 y
1 2 y
2 y
t
2

c 2 x 2
D’Alembert’s Solution
The general solution of the partial differential equation:
 2 y  1 2 y


0
 x 2  c 2 t 2


is:
y( x, t ) Af ( x  ct )  Bf ( x  ct )
where f(x-ct) represents a wave traveling in the forward (or
positive) x-direction, and f(x+ct) represents a wave
traveling in the negative x-direction.
Standing Waves
Consider the first harmonic component of f(x,t) given by:
 2

f1 ( x, t )  A1 cos 
( x  ct )   A1 cos x  t 
 

where  is called the angular wave number related to space
and  is referred to angular frequency related to time.
Consider a sinusoidal forward and backward wave in
complex notation:
f1 ( x, t )  .5 A1 exp j (x  t )
f1 ( x, t )  .5 A1 exp j (x  t )
determine the real part of the summation of the forward
and backward wave. Sketch and discuss the properties of
the resulting (standing) wave and its higher harmonics.
Acoustic Tubes
For the transverse waves of an elastic string the motion
of the string was governed by the restoring force and
mass/momentum. For gasses or fluids in a tube, the
pressure p(x,t) and volume displacement u(x,t) are
directly considered (analogous to voltage and current).
The resulting equations for these quantities are:
 2 p  2 p

c
 x 2  t 2


2 
 2u   2u

c
 x 2  t 2


2 
where
c
1
(density compressibility )
Acoustic Tubes
The solution is the same as that of the transverse wave on a string:
u( x, t ) Af ( x  ct )  Bf ( x  ct )
p( x, t ) Z o Af ( x  ct )  Bf ( x  ct )
where Zo is the acoustic impedance of the gas given by:
Zo 
c
A
with  being the density, c being the propagation speed, and A
being the cross-sectional area of the tube. Newton’s second law for
volume mass results in:
p   u 
x

 
A  t 
Tube Resonance
The general solution can be expressed:

u( x, t )
 An cos( nt )  Bn sin( nt )Cn cos( n x)  Dn sin( n x)
n 1
Initial conditions determine time amplitude and phase of standing time
wave oscillation. Boundary conditions will determine the resonance
frequency.
If a tube is closed at both ends, the displacement must be zero:
u(0, t ) 0  C n cos( n 0)  Dn sin( n 0)
u( L, t )  0  C n cos( n L )  Dn sin( n L )
If a tube is closed at one end and open at the other, the velocity must be
zero:
u(0, t ) 0  Cn cos( n 0)  Dn sin( n 0)
u( x, t )
 0  Cn n sin( n L)  Dn n cos( n L)
x x  L
Tube Resonance
The boundary conditions for a tube closed at both ends, reduces solution to:

 nπ 
 ncπ 
u ( x, t )   An Dn cos(nt )  Bn Dn sin( nt ) sin  n x  where κ n    , ωn  

 L 
 L 
n 1
The boundary conditions for a tube is closed at one end and open at the
other, reduces solution to:

u ( x, t )   An Dn cos(nt )  Bn Dn sin( nt ) sin  n x 
n 1
wher e
 (2n  1)π 
 (2n  1)cπ 
κn  
,
ω




n
2
L
2
L




Sketch standing waves in the tubes for the above cases.
Homework (1)
a) Electronically generate a harmonic for a recorded
guitar note, where a node is placed at the center
of the string.
b) Repeat part (a) for when the node is placed at one
third the length of the string.
Extra credit (2 points), record your voice saying (or
singing) an “a” sound (the “a” sound in the word
about) for 2 seconds sampled at 8000 Hz. Try to
perform an analogous harmonic operation as in
part (a). Email me the script/function and the
original data file.