Difference between revisions of "Trigonometry"
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When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance. | When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance. | ||
| − | In general, the function may also have: | + | In general, the function may also have: |
| + | |||
a spatial variable x that represents the position on the dimension on which the wave propagates, and a characteristic parameter k called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν; | a spatial variable x that represents the position on the dimension on which the wave propagates, and a characteristic parameter k called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν; | ||
a non-zero center amplitude, D | a non-zero center amplitude, D | ||
which is | which is | ||
| + | |||
y ( x , t ) = A sin( k x − ω t + φ ) + D, if the wave is moving to the right | y ( x , t ) = A sin( k x − ω t + φ ) + D, if the wave is moving to the right | ||
| + | |||
y ( x , t ) = A sin ( k x + ω t + φ ) + D, if the wave is moving to the left. | y ( x , t ) = A sin ( k x + ω t + φ ) + D, if the wave is moving to the left. | ||
| + | |||
The wavenumber is related to the angular frequency by:. | The wavenumber is related to the angular frequency by:. | ||
| + | |||
k = ω v = 2 π f v = 2 π λ {\displaystyle k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }} | k = ω v = 2 π f v = 2 π λ {\displaystyle k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }} | ||
Revision as of 17:51, 23 March 2020
While most people associate trigonometry with triangles, it is more useful in technical fields to consider the trigonometric functions as related to circles.
Sine
function of time = Asin(2πft + φ) = Asin(ωt + φ)
where:
A, amplitude, the peak deviation of the function from zero.
t, time
f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second
φ , phase, specifies (in radians) where in its cycle the oscillation is at t = 0.
When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.
In general, the function may also have:
a spatial variable x that represents the position on the dimension on which the wave propagates, and a characteristic parameter k called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν; a non-zero center amplitude, D which is
y ( x , t ) = A sin( k x − ω t + φ ) + D, if the wave is moving to the right
y ( x , t ) = A sin ( k x + ω t + φ ) + D, if the wave is moving to the left.
The wavenumber is related to the angular frequency by:.
k = ω v = 2 π f v = 2 π λ {\displaystyle k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }}
