Superposition of Sinusoidal Waves Let us now apply the principle of superposition to two sinusoidal waves travelling in the same direction in a linear medium. If the two waves are travelling to the right and have the same frequency, wavelength, and amplitude but differ in phase, we can express their individual wave functions as where, as usual, k = 2π/λ, ω= 2πf, and φ is the phase constant. the resultant wave function y is To simplify this expression, we use the trigonometric identity

If we let a = kx -ωt and b =kx -ωt +φ, we find that the resultant wave function y reduces to The resultant wave function y also is sinusoidal has the same frequency and wavelength as the individual waves because the sine function incorporates the same values of k and ω that appear in the original wave functions. The amplitude of the resultant wave is 2A cos(φ/2), and its phase is φ/2.

2A cos (φ/2) If the phase constant φ equals 0, then cos (φ/2) = cos 0 =1 and the amplitude of the resultant wave is 2A If the crests and troughs of the individual wave’s y 1 and y 2 occur at the same positions and combine. Because the individual waves are in phase, they are indistinguishable. they appear as a single blue curve. In general, constructive interference occurs when cos (φ/2)= ±1. when φ= 0, 2π, 4 π,... Rad

When φ is equal to π rad or to any odd multiple of π, then Cos (φ/2) = cos (π/2) = 0 the crests of one wave occur at the same positions as the troughs of the second wave the resultant wave has zero amplitude everywhere, as a consequence of destructive interference when the phase constant has an arbitrary value other than 0 or an integer multiple of π rad (Fig. 18.4c), the resultant wave has an amplitude whose value is somewhere between 0 and 2A.

Example Two travelling sinusoidal waves are described by the wave functions where x, y 1, and y 2 are in meters and t is in seconds.(a) What is the amplitude of the resultant wave? (b) What is the frequency of the resultant wave? Solution :

Plane Electromagnetic Waves conceder an electromagnetic wave that travels in the x direction (the direction of propagation). In this wave, the electric field E is in the y direction, and the magnetic field B is in the z direction, as shown in Figure

Furthermore, we assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only, and not upon the y or z coordinate imagine that the source of the electromagnetic waves is such that a wave radiated from any position in the yz plane propagates in the x direction, and all such waves are emitted in phase. If we define a ray as the line along which the wave travels, then all rays for these waves are parallel. This entire collection of waves is often called a plane wave.

A surface connecting points of equal phase on all waves, which we call a wave front is a geometric plane a point source of radiation sends waves out radially in all directions. A surface connecting points of equal phase for this situation is a sphere, so this is called a spherical wave.

For electromagnetic waves travel in empty space with the speed of light. Using these two equations and the plane-wave assumption, we obtain the following differential equations relating E and B. Where ϵ o permittivity and μ o is the conductivity of the medium Not that to evaluate both E and B, we assume that t is constant for x. Likewise, when we evaluate, x is held constant for t. Taking the derivative

Get using the same manner These two equations have the form of general wave equation Replace the speed v with c, where we find that c = x 10 8 m/s. Because this speed is precisely the same as the speed of light in empty space, we are led to believe (correctly) that light is an electromagnetic wave.

The simplest solution to both equations is a sinusoidal wave, for which the field magnitudes E and B vary with x and t according to the expressions where E max and B max are the maximum values of the fields. The angular wave number is k = 2π/λ, where λ is the wavelength. The angular frequency is ω=2πf, where f is the wave frequency. The ratio ω/k equals the speed of an electromagnetic wave, c : At every instant the ratio of the magnitude of the electric field to the magnitude of the magnetic field in an electromagnetic wave equals the speed of light.

Let us summarize the properties of electromagnetic waves as we have described them The solutions of Maxwell’s third and fourth equations are wave-like, with both E and B satisfying a wave equation. Electromagnetic waves travel through empty space at the speed of light The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of wave propagation. We can summarize the latter property by saying that electromagnetic waves are transverse waves. The magnitudes of E and B in empty space are related by the expression E/B = c. Electromagnetic waves obey the principle of superposition.

Example: Write down expressions for the electric and magnetic fields of a sinusoidal plane electromagnetic wave having a frequency of 3.00 GHz and travelling in the positive x direction. The amplitude of the electric field is 300 V/m. Solution

Example In SI units, the electric field in an electromagnetic wave is described by Find (a) the amplitude of the corresponding magnetic field oscillations, (b) the wavelength λ, and (c) the frequency f. Solution: