User:Dave3457/Sandbox/General Purpose

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.

As a wave travels by, the value at a given point oscillates sinusoidally.

One of the simplest formalisms of a plane wave involves defining it along the direction of the x-axis.

$A(x,t)=A_{o}\sin(kx-\omega t+\varphi )$

In the above equation…

• $A(x,t)\,\!$ is the magnitude or disturbance of the wave at a given point in space and time. For example, in the case of a sound wave, $A(x,t)\,\!$ could be chosen to represent the excess air pressure.
• $A_{o}\,\!$ is the amplitude of the wave (the peak magnitude of the oscillation).
• $k\,\!$ is the wave’s wave number or more specifically the angular wave number and equals $2\pi /\lambda \,\!$, where $\lambda \,\!$ is the wavelength of the wave. $k\,\!$ has the units of radians per unit distance.
• $x\,\!$ is a given point along the x-axis. $y\,\!$ and $z\,\!$ are not part of the equation because the wave's magnitude is the same at every point on any given y-z plane. This equation defines what that magnitude is.
• $\omega \,\!$ is the wave’s angular frequency which equals $2\pi /T\,\!$, where $T\,\!$ is the period of the wave. $\omega \,\!$ has the units of radians per unit time.
• $t\,\!$ is a given point in time
• $\varphi \,\!$ is the phase shift of the wave and has the units of radians. Note that a positive phase shift, at a given moment of time, shifts the wave in the negative x-axis direction. A phase shift of $2\pi \,\!$ radians shifts it exactly one wavelength.

Other formalisms which directly use the wave’s wavelength $\lambda \,\!$, period $T\,\!$, frequency $f\,\!$ and velocity $c\,\!$ are below.

$A=A_{o}sin[2\pi (x/\lambda -t/T)+\varphi ]\,\!$

$A=A_{o}sin[2\pi (x/\lambda -ft)+\varphi ]\,\!$

$A=A_{o}sin[(2\pi /\lambda )(x-ct)+\varphi ]\,\!$

With regards to the above set of equations it is noteworthy that $f=1/T\,\!$ and $c=\lambda /T\,\!$

For a plane wave the velocity $c\,\!$ is equal to both its phase velocity and its group velocity.