7.2 Superposition of Waves

Since the wave equation is linear, the solutions follow the principle of linear superposition. This means that when multiple waves come together, the amplitude at every point in space and time is determined by the sum of all the waves at that point.

Example 7.2.

Consider two sinusoidal waves with the travelling with the same frequency and direction. Then the superposition is given by

	$\displaystyle y(x,t)$	$\displaystyle=A\cos(kx-\omega t)+A\cos(kx-\omega t)$		(7.15)
		$\displaystyle=2A\cos(kx-\omega t).$		(7.16)

So, the resultant wave has the same frequency and direction but double the amplitude.

Now consider what happens if one of the waves has a phase shift of $\pi$ radians. The resultant wave is

$\displaystyle y(x,t)$	$\displaystyle=A\cos(kx-\omega t)+A\cos(kx-\omega t+\pi)$	(7.17)
	$\displaystyle=A\cos(kx-\omega t)-A\cos(kx-\omega t)$	(7.18)
	$\displaystyle=0.$	(7.19)

The two waves cancel each other out completely.

In the general case with a phase shift $\Omega$ , we get

$\displaystyle y(x,t)$	$\displaystyle=A\cos(kx-\omega t)+A\cos(kx-\omega t+\Omega)$	(7.20)
	$\displaystyle=2A\cos\left(kx-\omega t+\frac{\Omega}{2}\right)\cos\left(-\frac{% \Omega}{2}\right)$	(7.21)
	$\displaystyle=\underbrace{2A\cos\left(\frac{\Omega}{2}\right)}_{\text{% Amplitude }\leq 2A}\underbrace{\cos\left(kx-\omega t+\frac{\Omega}{2}\right)}_% {\text{Time-dependent part}}.$	(7.22)

Note that we have used the identity $\cos(\alpha)+\cos(\beta)=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(% \frac{\alpha-\beta}{2}\right)$ .

Most superpositions of two sinusoidal functions do not have a nice simplification which is easily interpreted like this, but from these basic examples we can build an intuition for what happens when two waves meet.

7.2.1 Standing Waves

Now consider what happens if the waves still have the same frequency but are moving in opposite directions. In this case the superposition is

$\displaystyle y(x,t)$	$\displaystyle=A\cos(kx-\omega t)+A\cos(kx+\omega t)$	(7.23)
	$\displaystyle=2A\cos\left(\frac{2kx}{2}\right)\cos\left(-\frac{2\omega t}{2}\right)$	(7.24)
	$\displaystyle=\underbrace{2A\cos(kx)}_{A(x)}\cos(\omega t).$	(7.25)

So we have a spatially varying amplitude $A(x)$ multiplied by a time-dependent variation. This is known as a standing wave.

Example 7.3.

In the case where one of the waves has a phase shift $\Omega$ . The relation above becomes

$\displaystyle y(x,t)$	$\displaystyle=A\cos(kx-\omega t)+A\cos(kx+\omega t+\Omega)$	(7.26)
	$\displaystyle=2A\cos\left(\frac{2kx+\Omega}{2}\right)\cos\left(-\frac{w\omega t% +\Omega}{2}\right)$	(7.27)
	$\displaystyle=2A\cos\left(kx+\frac{\Omega}{2}\right)\cos\left(\omega t+\frac{% \Omega}{2}\right).$	(7.28)

The points on the standing wave which don’t move at all are known as nodes, and points which move up the twice the ampltide of the waves are called antinodes. Nodes are separated by half a wavelength. In a standing wave, the whole string oscillates in simple harmonic motion.

The standing waves allowed in a one-dimensional region of length $L$ are given by

\lambda=\frac{2L}{p},\quad f_{p}=p\frac{v}{2L}=pf_{1},

(7.29)

where $p\in\mathbb{Z}$ .

7.2.2 Describing Waves with Complex Exponentials

Note that sometimes it is more convenient to express waves in terms of complex exponential functions according to Euler’s formula:

e^{i\theta}=\cos\theta+i\sin\theta,

(7.30)

so a general sinusoidal wave like above would be written as

y(x,t)=Ae^{i(kx-\omega t+\phi)}.

(7.31)

We recover the trigonometric form (cosine in this case) by taking the real part of this function. For the standing wave example above, we have

$\displaystyle y(x,t)$	$\displaystyle=Ae^{i(kx-\omega t)}+Ae^{i(kx+\omega t)}$	(7.32)
	$\displaystyle=Ae^{ikx}\left[e^{-i\omega t}+e^{i\omega t}\right]$	(7.33)
	$\displaystyle=2Ae^{ikx}\cos(\omega t),$	(7.34)

where in the last line we have used Euler’s identity to get $e^{-i\theta}+e^{i\theta}=2\cos\theta$ . Taking the real part of this, we recover $y(x,t)=2A\cos(kx)\cos(\omega t)$ which we found above.

Example 7.4.

Given a periodic wave, what is the phase difference between two points on the wave separated by a distance $\Delta x$ ?

\Delta\phi=2\pi\frac{\Delta x}{\lambda}=k\Delta k.

(7.35)

What is the phase difference between a single point over a interval of time $\Delta t$ ?

\Delta\phi=2\pi\frac{\Delta t}{T}=\omega\Delta t.

(7.36)