function module

Mathematical functions.

Date

2019-10-27

cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False)

A cosine wave is said to be sinusoidal, because, \(\cos(x) = \sin(x + \pi/2)\), which is also a sine wave with a phase-shift of π/2 radians. Because of this head start, it is often said that the cosine function leads the sine function or the sine lags the cosine.

Parameters

• A (float or int) – amplitude

• k (float or int) – (angular) wave number

• f (float or int) – ordinary frequency

• phi (float or int) – phase

• D (float or int) – non-zero center amplitude

• degree – boolean to switch between radians and degree. If False phi is interpreted in radians and if True then phi is interpreted in degrees.

See also

function_sine_wave_degree()

sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False)

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation.

Parameters

• A (float or int) – amplitude

• k (float or int) – (angular) wave number

• f (float or int) – ordinary frequency

• phi (float or int) – phase

• D (float or int) – non-zero center amplitude

• degree – boolean to switch between radians and degree. If False phi is interpreted in radians and if True then phi is interpreted in degrees.

In general, the function is:

\[\begin{split}y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\ y(x,t) = A\sin(kx + \omega t + \varphi) + D\end{split}\]

where:

• A, amplitude, the peak deviation of the function from zero.

• 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. If ω < 0 the wave is moving to the right, if ω > 0 the wave is moving to the left.

• φ, phase, specifies (in radians) where in its cycle the oscillation is at t = 0.

• x, spatial variable that represents the position on the dimension on which the wave propagates.

• k, characteristic parameter called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν.

• D, non-zero center amplitude.

The wavenumber is related to the angular frequency by:

\[k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }\]

where λ (lambda) is the wavelength, f is the frequency, and v is the linear speed.

See also

function_cosine_wave_degree()