Wave Interference

What is the relationship between traveling wave and standing wave solutions to the wave equation, (6.1), in an infinite medium? To help answer this question, let us form a superposition of two traveling wave solutions of equal amplitude $A$, and zero phase angle $\phi$, that have the same wavenumber $k$, but are moving in opposite directions. In other words,

$$\psi(x,t) = A\,\cos(\omega\,t-k\,x) + A\,\cos(\omega\,t+k\,x).$$ (6.19)

Because the wave equation, (6.1), is linear, the previous superposition is a valid solution provided the two component waves are also valid solutions; that is, provided $\omega=k\,v$, which we shall assume to be the case. Making use of the trigonometric identity $\cos a + \cos b\equiv 2\,\cos[(a+b)/2]\,\cos[(a-b)/2]$ (see Appendix B), the previous expression can also be written

$$\psi(x,t) = 2\,A\,\cos(k\,x)\,\cos(\omega\,t),$$ (6.20)

which is a standing wave [cf., Equation (6.2)]. Evidently, a standing wave is a linear superposition of two, otherwise identical, traveling waves that propagate in opposite directions. The two waves completely cancel one another out at the nodes, which are situated at $k\,x=(n-1/2)\,\pi$, where $n$ is an integer. This process is known as total destructive interference. On the other hand, the waves reinforce one another at the anti-nodes, which are situated at $k\,x=n\,\pi$, generating a wave whose amplitude is twice that of the component waves. This process is known as constructive interference.

As a more general example of wave interference, consider a superposition of two traveling waves of unequal amplitudes which again have the same wavenumber and zero phase angle, and are moving in opposite directions; that is,

$$\psi(x,t) = A_1\,\cos(\omega\,t-k\,x) + A_2\,\cos(\omega\,t+k\,x),$$ (6.21)

where $A_1, A_2>0$. In this case, the trigonometric identities $\cos(a-b)\equiv \cos a\,\cos b+\sin a\,\sin b$ and $\cos(a+b)\equiv \cos a\,\cos b-\sin a\,\sin b$ (see Appendix B) yield

$$\psi(x,t)= (A_1+A_2)\,\cos(k\,x)\,\cos(\omega\,t) + (A_1-A_2)\,\sin(k\,x)\,\sin(\omega\,t).$$ (6.22)

Thus, the two waves interfere destructively at $k\,x=(n-1/2)\,\pi$ [i.e., at points where $\cos(k\,x)=0$ and $\vert\sin(k\,x)\vert=1$] to produce a minimum wave amplitude $\vert A_1-A_2\vert$, and interfere constructively at $k\,x=n\,\pi$ [i.e., at points where $\vert\cos(k\,x)\vert=1$ and $\sin(k\,x)=0$] to produce a maximum wave amplitude $A_1+A_2$. It can be seen that the destructive interference is incomplete unless $A_1=A_2$. Incidentally, it is a general result that if two waves of amplitude $A_1>0$ and $A_2>0$ interfere then the maximum and minimum possible values of the resulting wave amplitude are $A_1+A_2$ and $\vert A_1-A_2\vert$, respectively.