Sine and Cosine are the y and x components of a point on the rim of a rotating wheel.
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
sin( ) cos(90 ) cos( ( 90 )) cos( 90 )cos( ) sin(90 ) sin( ( 90 )) sin( 90 )
Degree and radians on the unit circle
360 2 *
for unit circle r
s (m) = r (m) * θ (radians) arclength = radius * radians
Sinusoidal wave Amplitudes
Wavelength (meters)Wavelength defined between any two points on wave that are one cycle apart (2*pi radians).
•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 )
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
y x phase
Three phase power: three sinusoids phase separated by 120⁰.
Phase advance/delay and Unit circle
( ) *sin( ( ))
f A f
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( ),
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
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.