H. SAIBI November 25, 2015

Outline Generalities Superposition of waves Superposition of the wave equation Interference of harmonic waves

A light wave travels through three transparent materials of equal thickness. Rank is order, from the largest to smallest, the indices of refraction n 1, n 2, and n 3. A. n 2 > n 1 > n 3 B. n 3 > n 1 > n 2 C. n 1 > n 2 > n 3 D. n 3 > n 2 > n 1 E. n 1 = n 2 = n 3

Reflection of Transverse Wave Pulse A pulse traveling to the right on a heavy string attached to a lighter string Speed suddenly increases

Reflection of Transverse Wave Pulse A pulse traveling to the right on a light string attached to a heavier string Speed suddenly decreases

Superposition of waves ©2008 by W.H. Freeman and Company After two pulses traveling in opposite direction “collide”, they each continue moving with the same speed, size, and shape that they had before the “collision”. When two or more waves overlap, the resultant wave is the algebraic sum of the individual waves: principle of superposition. Figure 1. Wave pulses moving in opposite directions on a string. The shape of the string when the pulses overlap is found by adding the displacements due to each separate pulse. (a) Superposition of two pulses having displacements in the same direction (upward). The figure shows the shape of the string at equal time intervals of duration  t. Each pulse travels the length of pulse 2 during time  t. (b) Superposition of two pulses having equal displacements in opposite directions. Here the algebraic addition of the displacement amounts to the subtraction of the magnitudes.

Superposition and the wave equation The principle of superposition follows from the fact that the wave equation is linear for small transverse displacements. That is, the function y(x,t) and its derivatives occur only to the first power. The defining property of a linear equation is that if y 1 and y 2 are two solutions of the equation, then the linear combination: Where C 1 and C 2 are any constants, is also a solution. The linearity of the wave equation can be shown by the direct substitution of y 3 into the wave equation. The result is the mathematical statement of the principle of superposition. If any two waves satisfy a wave equation, then their algebraic sum also satisfies the same wave equation. (1)

Interference of harmonic waves The result of the superposition of two harmonic waves of the same frequency depends on the phase difference  between the waves. Let y 1 (x,t) be the wave function for a harmonic wave traveling to the right with amplitude A, angular frequency , and wave number k: For this wave function, we have chosen the phase constant to be zero. If we have another harmonic wave also traveling to the right with the same amplitude, frequency, and wave number, then the general equation for its wave function can be written: Where  is the phase constant. The two waves described by Eq.2-3 differ in phase by . Figure 2 shows a plot of the two wave functions versus position at time t=0. The resultant wave is the sum We can simplify Eq.4 by using the trigonometric identity (2) (3) (4) (5)

©2008 by W.H. Freeman and Company Figure 2 Displacement versus position at a given instant for 2 harmonic waves having the same amplitude, frequency, and wavelength, but differing in phase by 

Interference of harmonic waves For this case,  1 =kx-  t, and  2 =kx-  t+ , so that And Thus, Eq.4 becomes Superposition of two wave of the same amplitude and frequency (6) Fig. 3. Constructive interference. If 2 harmonic waves of the same frequency and in phase, the amplitude of the resultant wave is the sum of the amplitudes of the individual waves. Waves 1 and 2 are identical, so they appear as a single harmonic wave. Fig.4. Destructive interference. If 2 harmonic waves of the same frequency differ in phase by 180 o, the amplitude of the resultant wave is the difference between the amplitudes of the individual waves. If the original wave have equal amplitudes, they cancel completely.

Beats The interference of 2 sound waves with slightly different frequencies produces the interesting phenomenon known as beats. Consider 2 sound waves that have angular frequencies of  1 and  2 and the same pressure amplitude p0. What do we hear? At a fixed point, the spatial dependence of the wave merely contributes a phase constant, so we can neglect it. The pressure at the ear due to either wave acting alone will be a simple harmonic function of the type: Where we have chosen sin functions, rather than cosine functions for convenience, and have assumed that the waves are in phase at time t=0. Using the trigonometry identity For the sum of 2 functions, we obtain for the resultant wave If we write  av=(  1+  2)/2 for the average angular frequency and  =  1-  2 for the difference in angular frequencies, the resultant wave function is Where:and (7)

Beats Figure 5 shows a plot of pressure variations as a function of time. The waves are initially in phase. Thus, they add constructively at time t=0. Because their frequencies differ, the waves gradually become out of phase and interfere destructively. An equal time interval later (time t2 in this figure), the two waves are again in phase and interfere constructively. The greater the difference in the frequencies of the two waves, the more rapidly they oscillate in and out of phase. ©2008 by W.H. Freeman and Company Fig. 5. Beats. a) two harmonic waves of different but nearly equal frequencies that are in phase at t=0 are 180o out of phase at some later time t1. At a still later time t2, they are back in phase. b) the resultant of the two waves shown in (a). The frequency of the resultant wave is about the same as the frequencies of the original waves, but the amplitude is modulated as indicated. The intensity is maximum at times 0 and t2, and zero at times t1 and t3. Beat frequency (8)

Principle of Superposition If two or more waves combine at a given point, the resulting disturbance is the sum of the disturbances of the individual waves. Two traveling waves can pass through each other without being destroyed or even altered!

Some Results of Superposition: Two waves, same wavelength and frequency, opposite direction: Standing Wave Two waves, same wavelength and frequency, similar direction, different phase: Interference Two waves, same direction, slightly different frequency and wavelength: Beats!

Standing Wave: The superposition of two 1-D sinusoidal waves traveling in opposite directions.

Standing Waves Are a form of “resonance” There are multiple resonant frequencies called harmonics The boundary conditions and speed of waves determine which frequencies are allowed. The ends of the resonant cavity have forced nodes or antinodes With a wave on a string, it is possible to force an intermediate node

Harmonic frequencies Transverse standing wave on a string clamped at both ends: there are nodes in displacement at both ends. Standing sound wave in a tube open at both ends: there are nodes in pressure both ends.

Harmonic frequencies Standing sound wave in a tube closed at one end: there is a node in pressure at the open end, and an anti-node at the closed end.

Waves Constructive and Destructive Interference Interference Patterns Beats

Wave Interference Two waves moving in the same direction with the same amplitude and same frequency form a new wave with amplitude: where a is the amplitude of either of the individual waves, and is their phase difference.

Beat frequency Beats are loud sounds separated by soft sounds The beat frequency is the difference of the frequencies of the two waves that are being added: The frequency of the actual sound is the average of the frequencies of the two waves that are being added:

let’s look at the superposition of some simple combinations of two waves.

The first addition of waves that will be described involves two waves that are in phase. This is referred to as constructive interference. A crest of one wave is positioned with the crest of the other wave. The same can be said for troughs.

This represents the displacement by the black wave alone. This represents the displacement by the blue wave alone. Since they are both displacements on the same side of the baseline, they add together. Just repeat this step for several points along the waves.

The next addition of waves that will be described involves two waves that are out of phase. This is referred to as destructive interference. A crest of one wave is positioned with a trough of the other wave.

This represents the displacement by the black wave alone. This represents the displacement by the blue wave alone. Since the two displacements are on opposite sides of the baseline, the top one should be considered positive and the bottom one negative. Just add the positive and negatives together like this. Repeat this step for several points along the waves.

Finally we observe two waves that are partially in phase. A different method of adding the waves will be demonstrated.

By overlaying the constructive interference curve from a previous slide you can tell that the curve of this slide is not fully constructive interference.