Introduction to �Audio and Music Engineering �

Lecture 2�

•  A few mathematical prerequisites�•  Limits and derivatives �•  Simple harmonic oscillators�•  Strings, Oscillations & Modes�

1 �

Puzzler�The Early Commuter�

A commuter is in the habit of arriving at his suburban station each evening exactly at 5 o’clock. His wife always meets the train and drives him home. One day he takes an earlier train, arriving at the station at four. The weather is pleasant so instead of calling home he starts walking along the route always taken by his wife. They meet somewhere on the way. He gets into the car and they drive home, arriving at their house ten minutes earlier than usual. Assuming that the wife always drives at a constant speed, and that on this occasion she left just in time to meet the five o’clock train, can you determine how long the husband walked before he was picked up?�

2�

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 …�

5�

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 �

6�

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?�

7�

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�

9�

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 �

10�

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�

11 �

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 �

13�

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�

14�

ω = 1

LCi�

Inductor�

C� L �

Capacitor�

Electrical Oscillator: �

“spring”� “mass”�

f = 12π

1

LC

Resonant frequency�