Introduction to ���Audio and Music Engineering
Lecture 2
• A few mathematical prerequisites • Limits and derivatives • Simple harmonic oscillators • Strings, Oscillations & Modes
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Limits and Derivatives
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x
f(x)
x0+∆x
f(x0+∆x)
x0
f(x0)
Slope = f(x0+∆x) - f(x0)
x0+∆x – x0 f(x0+∆x) - f(x0)
∆x =
Make ∆x à 0
lim∆x→0f (x 0 + ∆x ) − f (x 0 )
∆x ≡ ddx f (x )
x 0
≡ f '(x 0 )
tangent
tangent x0 x
f(x)
x0+∆x
f(x0+∆x)
f(x0)
x0+∆x
f(x0+∆x)
f’(x0)
A few simple derivatives we will need …
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ddx x = 1
ddx ax = a
ddx xn = nxn−1
ddx f (x ) ⋅ g (x )⎡⎣ ⎤⎦ = f (x ) ⋅ g '(x ) + f '(x ) ⋅ g (x )
ddx f (g (x ))⎡⎣ ⎤⎦ = f '(g (x )) ⋅ g '(x )
ddx sin(x ) = cos(x )
ddx cos(x ) = − sin(x )
Product rule:
ddx ex = ex
Chain rule:
ddx x sin(x )⎡⎣ ⎤⎦ = x ⋅ cos(x ) + 1 ⋅ sin(x )
ddx sin(x2 )⎡⎣ ⎤⎦ = cos(x2 ) ⋅2x d
dx eax = aeax
ddx Const = 0
Question
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2x sin(2x )What is the derivative w.r.t. x of:
a) b) c) d)
2x cos(2x )
2sin(2x )+2x cos(2x )
2sin(2x )+ 4x cos(2x )
2cos(2x )+ 4x sin(2x )
What’s so special about e?
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ddx ex = ex The only exponential function (ax) where the slope of
the function equals the value of the function at every point.
Bacteria colony growth: Let’s say that there is a colony of bacteria growing in a petri dish and that the rate of increase of the number of bacteria is 2 times the number already present. How does the population grow over time?
Let y(t) = number of bacteria at time t, and y(t=0) = N0 , (initial condition)
ddt
y = 2⋅ ythen y (t )= Ae 2tsolution is …
e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 …
and since y(t=0) = N0 à A = N0
y (t )=N0e2t
0 time
# b
acte
ria
N0
sin(t) ddt sin(t ) = cos(t )
cos(t)
ddt cos(t ) = − sin(t )
- sin(t)
ddt − sin(t )( ) = − cos(t )
- cos(t) ddt − cos(t )( ) = sin(t )
Derivatives of sin, cos
vmax
-vmax t
xmax
-xmax t
Simple Harmonic Oscillator
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F
m k
A
x = xmax v = 0
B
C
x = -xmax v = 0 D
E
x(t) = xmaxsin(t) B
D
v(t) = vmaxcos(t) A
B
C
D
E
C A E
x = 0 v = vmax
x = 0 v = - vmax
x = 0 v = vmax
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m k
Simple Harmonic Oscillator
ddt x = vddt v = a
ddt
dxdt ≡ d 2x
dt2 = a
, velocity
, acceleration F = ma
Newton’s 2nd Law
F = -kx
Hooke’s Law
m d 2xdt2 = −kx
x
d 2xdt2 = − k
m x d 2xdt2 = −ω2xω2 ≡ k
mlet , so
Can we find a function that satisfies this differential equation?
x = x 0 sin ωt( ) ddt x = x 0ω cos ωt( ) d 2
dt2 x = −x 0ω2 sin ωt( )
x (t ) = x 0 sin ωt( ) ω ≡ kmwhere so
It works!
Frequency and Period
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m k
x
x (t ) = x 0 sin ωt( ) ω ≡ kmwhere
ωt = 2π
Sine repeats every 2π sin ωt( )
t
t = T = Period
T = 2πω
ωt = 0
Period (seconds per cycle)
Frequency (cycles per second) f = 1T
= ω2π
ω = 2π f
Angular frequency
(radians per second) Frequency
(cycles per second)
ωT = 2π
Question
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What is approximate frequency (in Hertz) of a simple harmonic oscillator of mass 1 kg with a spring constant of 9 Nts/m?
a) b) c) d)
2 Hz 0.5 Hz 9 Hz 1/9 Hz
Other systems that display simple harmonic oscillation
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Simple pendulum ω ≡ gl
f ≡ 12π
glg l
m
Helmholtz resonator
V (volume)
L S surface area
Spring
mass m = ρSL
air density
k = ρ S 2c 2
Vc = sound velocity
f = 12π
km
= c2π
SVL
for … c = 340 m/sec S = π x 10-4 m2 V = 100 cc = 10-4 m3 L = 3 x 10-2 m
f = 500 Hz
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ω = 1
LCi
Inductor
C L
Capacitor
Electrical Oscillator:
“spring” “mass”
f = 12π
1
LC
Resonant frequency