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

3

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

12

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