# Waves

A sine wave is a function of the time \(t\):

\[y(t)=A\sin(\omega t - \phi)\]

The **phase** of a sine wave, \(\phi\), is the displacement of the wave when \(t=0\).

The **angular frequency** of a sine wave, \(\omega\), has the unit rad/s.

The **frequency** of a sine wave, \(f\), denotes how many revolutions there are per unit time. The unit for
frequency is Hz=1/s (Hertz). The correspondence between angular frequency and frequency is:

\[\omega= 2\pi f\]

A one dimensional sine wave that propagates in space, is a function of two variables \(x\) and \(t\), where \(x\) is the position and \(t\) the time.

\[y(x,t)=A\sin(kx+\omega t- \phi)\]

\(k\) denotes the **speed of propagation**, i.e. how fast the wave is propagating.

The difference between changing the speed of propagation and changing the frequency, when a wave is played as music, is demonstrated in the recording below.

The wave shown in the recording is not a simple sine wave but a superposition of waves.

## Superposition of Waves

The superposition principle of waves is: the resulting wave formed by several waves overlapping in space and time, can be found by adding the waves.

When \(k_1=2\) and \(k_2=1\), the blue wave propagates twice as fast as the red wave.

If the waves move with the same speed of propagation but in opposite directions (\(k_1=-k_2\)),
and if they have the same angular frequency (\(\omega_1=\omega_2\)), then the resulting wave
is a **standing wave**.

If the waves moves with the same speed of propagation and in the same direction (\(k_1=k_2\)),
and if they have the same angular frequency, then they cause **interference**.
If the waves have the same phase, they cause **constructive interference**, i.e. the amplitude
of the resulting wave is the sum of the amplitudes. If one of the waves is shifted by \(\phi=\pi\), then they
cause **destructive interference**.

# references:

Audacity (from the recording) can be downloaded from: http://audacity.sourceforge.net/

Empty by Tryad (from the recording) can be downloaded from: tryad::long live free music

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License