Sine and Cosine are the y and x components of a point on the rim of a rotating wheel.
-
Upload
matteo-herder -
Category
Documents
-
view
215 -
download
0
Embed Size (px)
Transcript of Sine and Cosine are the y and x components of a point on the rim of a rotating wheel.

Sine and Cosine are the y and x components of a point on the rim of a rotating wheel
sine-wave-cosine-wave.mov
sin( ) cos(90 ) cos( ( 90 )) cos( 90 )cos( ) sin(90 ) sin( ( 90 )) sin( 90 )

Degree and radians on the unit circle
360 2 *
1
360 2
Equivalence relation
r
for unit circle r
s (m) = r (m) * θ (radians) arclength = radius * radians

Periodic Function

Sinusoidal wave Amplitudes

Wavelength (meters)Wavelength defined between any two points on wave that are one cycle apart (2*pi radians).
e.g.,
•Peaks•Zeros crossing•Troughs•Sin(θ) where θ is an point.
Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.

Wave Period T (s) and Linear Frequency 1/T (s-1 )
Wave parameters
T: wave period (s) λ: wave length (m) f=1/T : linear frequency 1 (2π /s-1 or cycles/s)
Wave Velocity or Speed: v (m/s) = λ/T = λ * f
Angular wave number: k = 2π/ λAngular frequency: ω = 2π/ T = 2π*f
Wave solution: u(x,t) = A * sin( k*x – ω *t ) (m)
The period of a wave is the time interval for the wave to complete one cycle (2*pi radians). What is this waves period?

Wave snapshot in space and time

F(x,t) amplitude in space/time
Wavelength Wave period

Translation (space or time) of Sinusoidal wave
Horizontal axis units are radians/2*pi.
• if f(θ=w*t) = sin( w*t ) = sin( 2π*(t/T) ) >> t=T >> sin(2 π) • if f(θ=k*x) = sin( k*x ) = sin( 2π*(x/λ) ) >> x= λ >> sin(2 π)

Phase of sinusoidal wave
sin( ) cos(90 )cos( ) sin(90 )
sin( / 4sin
(
()
)
)
sin
yy x
y x phase
x
Three phase power: three sinusoids phase separated by 120⁰.

Phase advance/delay and Unit circle
( ) *sin( ( ))
:
f A f
initial phase
Note minus sign in phase argument.
The red sine phase is behind (negative) the blue line phase; hence, red sin function leads the blue sin function.

Wavefront: where and what is it ?

Pulse wave versus Sinusoidal wave
A pulse is a compact disturbance in space/time.
A sinusoidal wave is NOT compact, it is everywhere in space/time.
A pulse can be ‘built’ up mathematically as a sum of sinusoidal waves.

Superposition of wave pulses
Which is the space (x) axis and which the time (t) axis?

Waves move KE/PE energy (not mass) in time
( , ) *sin( )
( , ) *sin( )
( ) *sin( ),
( ) *sin( ),
x
r
x
r
f x t A k x t
f r t A k r t
f A k x
f A k r

Longitudinal (P) vs. Transverse (S) waves: vibration versus energy transport direction

Water and Rayleigh waves particle motions
•Elastic medium•Rayleigh surface wave
•Synchronized P-SV motions
•Retrograde Circular particle motion
•Acoustic medium (water)
•Prograde circular particle motion

Two different wavelength waves addedTogether: beating phenomena
Two 1-dimensional wave pulse traveling And superimposing their amplitudes

Huygen’s wavelets: secondary wavefronts propagated to interfere constructively and destructively to make new time advanced
wavefront

Standing waves on a string.Fixed endpoint don’t move; wave is trapped.

Harmonic motion: two forces out of phase
A mechanical wave propagates a pulse/sinusoid of KE+PE energy because the inertial forces load the springs by pushing and pulling on the springs which permits the wave energy to propagated in time.