mathematical description of a wave

Before we start describing a wave mathematically, it is useful to recall that if \[f(x)\] is some function, then \[f(x-d)\] is the same function translated in the positive \[x\]-direction by a distance \[d\] while \[f(x+d)\] translates it in the negative \[x\]-direction by a distance \[d\].

To construct the model of the wave using a periodic function, we first start with the relationship \[\frac{\Delta \theta}{2\pi}=\frac{\Delta x}{\lambda}\]. This equation expresses the fact that when the spatial coordinate \[x\] increases by one wavelength \[\lambda\], the phase \[\theta\] advances by a full cycle (i.e. \[2\pi\] radians). Rearranging this equation we get \[\Delta \theta=\frac{2\pi}{\lambda}\Delta x\].

To model the \[y\]-position of the wave as a function of the \[x\]-position, let \[A\] be the amplitude, we know that in a sine wave, \[y=A\sin\theta\]. Substituting \[\theta\], we get \[y(x)=A\sin \left( \frac{2\pi}{\lambda}x \right)\]. However, this equation just describes the wave at time \[t=0\], i.e. a stationary wave. For it to be time-dependent and "travel" in the positive direction, we know that after some time \[t\], the \[x\] position will have shifted to the right by \[\lambda\cdot \frac{t}{T}\], assuming it takes time \[T\] to complete one oscillation. Notice that the definition of wave speed is \[v=\frac{\lambda}{T}\], which allows us to replace \[\lambda\cdot \frac{t}{T}=vt\]. Then as mentioned above, to shift a function to the right, we do \[f(x-d)\], similar here, \[y(x-vt)=y(x,t)=A\sin \left( \frac{2\pi}{\lambda}(x-vt) \right)=A\sin \left( \frac{2\pi}{\lambda}x-\frac{2\pi}{\lambda}vt \right)\].

Replace \[k=\frac{2\pi}{\lambda}\] where \[k\] is the wavenumber and \[v=\frac{\lambda}{T}\], we get \[A\sin \left( kx-\frac{2\pi}{\lambda}\left( \frac{\lambda}{T} \right)t \right)=A\sin(kx-\omega t)\] where \[\omega\] is the angular frequency. To generalize this formula, \[y(x,t)=A\sin(kx\mp \omega t)\], where the minus sign is for waves moving in the positive x-direction, and the plus sign is for waves moving in the negative x-direction.

To allow for a "head-start" for waves, so that they don't necessarily have to start at the zero angle, we add a constant \[\phi_{0}\], known as the initial phase or phase constant, which is an angle that tells us at what angle this wave is at when \[x=0\] and \[t=0\]. Combine this, we get our final equation \[y(x,t)=A\sin(kx\mp\omega t+\phi_{0})\].

The phase of a wave is a measure of where the wave is in its cycle at a given moment (or more simply its angle), and is the \[kx\mp \omega t+\phi_{0}\] portion of our wave equation.

The phase function is mathematically defined as \[\phi(x,t)=\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t+\phi_{0}\]. The result we obtain from this formula tells us the angle (in radians) of the wave at position \[x\] and time \[t\] on a sine graph.

The phase difference or known as phase shift is the difference in the phase angles of two periodic signals. In mathematical terms, if you have two waves described by \[y_{1}(x,t)=A_{1}\sin(k_{1}x-\omega_{1}t+\phi_{01})\] and \[y_{2}(x,t)=A_{2}\sin(k_{2}x-\omega_{2}t+\phi_{02})\]. Then the phase difference \[\Delta\phi\] is given by \[\Delta\phi=\phi_{2}-\phi_{1}\], evaluated at the same \[x\] and \[t\].

If these two waves have the same frequency \[\omega\] and wavenumber \[k\], it simplifies to \[\Delta\phi=\phi_{02}-\phi_{01}\].