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.