Sine and Cosine are the y and x
- Slides: 21
Sine and Cosine are the y and x components of a point on the rim of a rotating wheel
Degree and radians on the unit circle 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 ) 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 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/ λ Angular frequency: ω = 2π/ T = 2π*f Wave solution: u(x, t) = A * sin( k*x – ω *t ) (m)
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 Three phase power: three sinusoids phase separated by 120⁰.
Phase advance/delay and Unit circle 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
Longitudinal (P) vs. Transverse (S) waves: vibration versus energy transport direction
Water and Rayleigh waves particle motions • Acoustic medium (water) • Prograde circular particle motion • Elastic medium • Rayleigh surface wave • Synchronized P-SV motions • Retrograde Circular particle motion
Two different wavelength waves added Together: 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.
- Antigentest åre
- Graph of sine and cosine functions
- How to find period from sine graph
- Practice 9-2 sine and cosine ratios
- Sine rule for obtuse angles
- Sine maze
- Soh cah toa rule
- 9-5 practice graphing trigonometric functions
- Sin rule for sides
- 4-4 graphing sine and cosine functions
- Sin^-1(-1)
- Sine rule cosine rule
- Negative cosine graph
- Sine word problems
- Tan cosine and sine
- Burj khalifa trigonometry
- Horizontal phase shift
- Site:slidetodoc.com
- Sine cosine partner
- What is amplitude in trigonometry
- Trigonometry graph
- Graph of sine and cosine functions