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 )
θ θθ θ
θ θθ θ
° °°
° °°
= − = −− =+ −
= − = −− =− −
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Sine and Cosine - Introduction to Geophysics - Lecture Slides, Slides of Geology
In the course of introduction to geophysics, The key points are:Sine and Cosine, Rim, Rotating Wheel, Components, Degree and Radians, Unit Circle, Arclength, Sinusoidal Wave Amplitudes, Points, One Cycle Apart
Typology: Slides
2012/2013
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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 )
θ θ θ θ θ θ θ θ
° ° ° ° ° °
Degree and radians on the unit circle
Equivalence relation
r
for unit circle r
π
π
°
°
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).
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 snapshot in space and time
F(x,t) amplitude in space/time
Wavelength Wave period
Phase of
sinusoidal wave
sin( ) cos(90 )
cos( ) sin(90 )
sin( / 4
sin
sin
y
y x
y x phase
x
Three phase power: three sinusoids phase separated by 120⁰.
Phase advance/delay and Unit circle
( ) *sin( ( ))
:
f A f
initial phase
θ θ ψ
ψ
= − 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.
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?
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
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.