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

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Presentation transcript:

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.sine wavephasezero crossings

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?

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 120 ⁰.

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