User:Dave3457/Sandbox/General Purpose



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


 * $$A(x,t)=A_o \sin (k x - \omega t+\varphi)$$

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.