PHY385 Module 8: Superposition Principle and Energy Density
Table of contents
Activity 8.1 – Superposition principle
8.1.1. You will watch the superposition of two disturbances with animations created by Prof. Daniel Russell from Pennsylvania State University. The two pulses are travelling

in the opposite direction;

in the same direction
What is the amplitude of the resulting disturbance in any location of space in both cases (1) and (2)?
Give your estimate of a speed of propagation of the disturbance in the cases (1) and (2)?
The two waves are traveling
3. in the opposite direction;
4. in the same direction with an observer at rest in the reference frame of one of the wavefronts; and
5. in the same direction with slightly different frequency
Treating the amplitude as the maximum possible magnitude of the disturbance (displacement), what is
 the amplitude of the resulting wave in the cases (3), (4) and (5)?
 the wavelength of the resultant in the cases (3)  (5)?
 frequency of the resulting disturbance in the cases (3)  (5)?
8.1.2. The superposition principle states that the superposition of the waves is also a wave.

Determine which of the following describe traveling waves:
\(\Psi(y.t) = e^{(a^2y^2+b^2t^22abty)}\)
\(\Psi(z,t)=A\sin(Az^2bt^2)\)
\(\Psi(x,t)=A\cos^2[2\pi(tx)]\)
2. For the traveling wave(s) from the above list, write the direction of propagation and the speed of the wave.
3. Choose any one of the traveling waves determined above. Prove that the superposition of the two of such waves is also a wave.
Activity 8.2 – Electromagnetic Wave
A 550nm harmonic EM wave whose electric field is in the zdirection is traveling in the ydirection in vacuum.
 What is the frequency of the wave?
 Determine ω and k of the wave.
 If the electric field amplitude is 600 V/m, what is the amplitude of the magnetic field?
 Write an expression for E(t) and B(t) given that each is zero at x = 0 and t = 0. Put in the appropriate units
Activity 8.3 – Complex form of the wave function
An electromagnetic wave is specified (in SI units) by the following equation:
\(E = (6\hat{i}+3\sqrt{5}\hat{j})\bigg(10^4\frac{V}{m}\bigg)e^{i\bigg[\frac{1}{3}(\sqrt{5}x+2y)10^7\pi  9.42\times10^{15}t\bigg]}\)
Find:
 the direction along which the electric field oscillates;
 the scalar value of amplitude of the electric field;
 the direction of propagation of the wave
 the propagation number;
 the frequency and the angular frequency; and
 the speed of the wave.
Activity 8.4 – Energy Density & Poynting Vector
Consider a linearly polarized plane electromagnetic wave traveling in the + x direction in free space having as its plane of vibration the xy plane. Given that its frequency is10 MHz and its amplitude is E_{0} = 0.08 V/m,
 Find the period and wavelength of the wave.
 Write an expression for E(t) and B(t).
 Find the expression for the Poynting vector S.
 The flux density of the wave is the time average of the magnitude of the Poynting vector S. Find the flux density <S> of the wave.
 Find the energy density of the wave.
 Prove that the energy densities of the electric and magnetic fields are equal ( u_{E} =u_{B} ) for electromagnetic field.