Sine and Cosine are the y and x components of a point on the rim of a rotating wheel

1. Sine and Cosine are the y and x components of a point on the rim of a rotating wheel

2. Degree and radians on the unit circle s (m)  =  r (m) * θ (radians) arclength = radius * radians

3. Periodic Function

4. Sinusoidal wave Amplitudes

5. 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.

6. 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)

7. Wave snapshot in space and time

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

9. 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)

10. Phase of sinusoidal wave Three phase power: three sinusoids phase separated by 120⁰.

11. 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.

12. Wavefront: where and what is it ?

13. 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.

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

15. Waves move KE/PE energy (not mass) in time

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

17. 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

18. Two different wavelength waves added Together: beating phenomena Two 1-dimensional wave pulse traveling And superimposing their amplitudes

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

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

21. 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.