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.sine wavephasezero crossings

6. 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/ λ Angular 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?

7. Wave snapshot in space and time

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

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 120 ⁰.

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 Elastic medium Rayleigh surface wave Synchronized P-SV motions Retrograde Circular particle motion Acoustic medium (water) Prograde 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.