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A wave is defined as a disturbance in some physical system that is periodic in both space and time. In one dimension, a wave is generally represented in terms of a wavefunction: for instance, $\label{ew} \psi(x,t) = A\,\cos(k\,x-\omega\,t+\varphi),$ where $$x$$ is a position coordinate, $$t$$ represents time, and $$A$$, $$k$$, $$\omega >0$$. For example, if we are considering a sound wave then $$\psi(x,t)$$ might correspond to the pressure perturbation associated with the wave at position $$x$$ and time $$t$$. On the other hand, if we are considering a light-wave then $$\psi(x,t)$$ might represent the wave’s transverse electric field. As is well known, the cosine function, $$\cos \theta$$, is periodic in its argument, $$\theta$$, with period $$2\pi$$: in other words, $$\cos(\theta+2\pi)=\cos\theta$$ for all $$\theta$$. The function also oscillates between the minimum and maximum values $$-1$$ and $$+1$$, respectively, as $$\theta$$ varies. It follows that the wavefunction (2.1.1) is periodic in $$x$$ with period $$\lambda=2\pi/k$$. In other words, $$\psi(x+\lambda,t)=\psi(x,t)$$ for all $$x$$ and $$t$$. Moreover, the wavefunction is periodic in $$t$$ with period $$T=2\pi/\omega$$. In other words, $$\psi(x,t+T)=\psi(x,t)$$ for all $$x$$ and $$t$$. Finally, the wavefunction oscillates between the minimum and maximum values $$-A$$ and $$+A$$, respectively, as $$x$$ and $$t$$ vary. The spatial period of the wave, $$\lambda$$, is known as its wavelength, and the temporal period, $$T$$, is called its period. Furthermore, the quantity $$A$$ is termed the wave amplitude, the quantity $$k$$ the wavenumber, and the quantity $$\omega$$ the wave angular frequency. Note that the units of $$\omega$$ are radians per second. The conventional wave frequency, in cycles per second (otherwise known as hertz), is $$\nu=1/T=\omega/2\pi$$. Finally, the quantity $$\varphi$$, appearing in expression (2.1.1), is termed the phase angle, and determines the exact positions of the wave maxima and minima at a given time. In fact, the maxima are located at $$k\,x-\omega\,t+\varphi = j\,2\pi$$, where $$j$$ is an integer. This follows because the maxima of $$\cos\theta$$ occur at $$\theta=j\,2\pi$$. Note that a given maximum satisfies $$x=(j-\varphi/2\pi)\,\lambda+ v\,t$$, where $$v=\omega/k$$. It follows that the maximum, and, by implication, the whole wave, propagates in the positive $$x$$-direction at the velocity $$\omega/k$$. Analogous reasoning reveals that $\psi(x,t) = A\,\cos(-k\,x-\omega\,t+\varphi)=A\,\cos(k\,x+\omega\,t-\varphi),$ is the wavefunction of a wave of amplitude $$A$$, wavenumber $$k$$, angular frequency $$\omega$$, and phase angle $$\varphi$$, that propagates in the negative $$x$$-direction at the velocity $$\omega/k$$.