SINUSOIDAL WAVE

A sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. Its most basic form as a function of time (t) is:

 y(t) = A\sin(2 \pi f t + \varphi) = A\sin(\omega t + \varphi)

where:

  • A = the amplitude, the peak deviation of the function from zero.
  • f = the ordinary frequency, the number of oscillations (cycles) that occur each second of time.
  • ω = 2πf, the angular frequency, the rate of change of the function argument in units of radians per second
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
The oscillation of an undamped spring-mass system around the equilibrium is a sine wave

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(kx -\omega t + \varphi ) + D\,, if the wave is moving to the right
 y(x,t) = A\sin(kx + \omega t + \varphi ) + D\,, if the wave is moving to the left.

The wave number is related to the angular frequency by:.

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