DHS 14 Superposition (Notes & Tutorial)

Dunman High School (Senior High Physics) 1 Topic 14 – Superposition Content • Principle of superposition • Stationary waves • Diffraction • Two-source interference • Single slit and multiple slit diffraction Learning Outcomes Candidates should be able to: (a) explain and use the principle of superposition in simple applications (b) show an understanding of the terms interference, coherence, phase difference and path difference (c) show an understanding of experiments which demonstrate stationary waves using microwaves, stretched strings and air columns (d) explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes (e) explain the meaning of the term diffraction (f) show an understanding of experiments which demonstrate diffraction including the diffraction of water waves in a ripple tank with both a wide gap and a narrow gap (g) show an understanding of experiments which demonstrate two-source interference using water waves, sound waves, light waves and microwaves (h) show an understanding of the conditions required for two-source interference fringes to be observed (i) recall and solve problems using the equation D ax=λ for double-slit interference (j) recall and use the equation b λθ =sin to locate the position of the first minima for single slit diffraction (k) recall and use the Rayleigh criterion b λθ ≈ for the resolving power of a single aperture (l) recall and use the equation d sinθ = nλ to locate the positions of the principal maxima produced by a diffraction grating (m) describe the use of a diffraction grating to determine the wavelength of light (the structure and use of a spectrometer are not required).

Dunman High School (Senior High Physics) 2 1 PRINCIPLE OF SUPERPOSITION In the previous topic, we only considered a single wave. Many interesting wave phenomena in nature are impossible to describe with a single progressive wave. In this topic, we will examine what happens when two (or more) waves pass through the same region of space at the same time. We see that after crossing each other, the waves travel as if nothing had happened and retain their original shape. Hence, pulses and waves pass through each other unaffected. When the waves meet each other, they add up vectorially to give the resultant wave. This combination of two (or more) waves to form a resultant wave follows the Principle of Superposition. Note: The two waves must be of the same type (sound, electromagnetic, etc). The Principle of Superposition states that when two or more waves of the same type meet at a point at the same time, the displacement of the resultant wave is the vector sum of the displacements of the individual waves at that point at that time. (a) explain and use the principle of superposition in simple applications. waves are some (e) The waves are receding from each other. waves waves

Dunman High School (Senior High Physics) 3 --------------------------------------------------------------------------------------------------------------------------- Solution: In this topic, we are going to examine the application of this Principle of Superposition of waves to the following: (i) formation of stationary waves (ii) diffraction of waves (iii) interference of waves 2 STATIONARY WAVES (ALSO CALLED STANDING WAVES) Vibrational energy of the wave is not transmitted from one point to another. Stationary wave has amplitudes of oscillation varying from maximum at antinodes to zero amplitude at nodes. At t = 10 ms, each wave would have moved 10 x 10-3(2.0) = 0.02 m = 2 cm At t = 15 ms (or 5 ms after 10 ms) each wave would have moved 5 x 10-3(2.0) = 0.01 m = 1 cm Example 1: Two identical waves travel at a speed of 2.0 m s-1 towards each other on a long cord. If the waves are 6.0 cm apart at t = 0 s, sketch the shape of the cord at t = 10, 15 and 20 ms. When two progressive waves of the same type of equal amplitude, equal frequency, equal speed travelling in opposite directions meet and undergo superposition with each other, a stationary wave is formed. (d) explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes. 6.0 cm 1.5 cm

Dunman High School (Senior High Physics) 4 The red curve shows a wave traveling to the left while the blue curve shows a wave with the same speed, frequency and amplitude traveling to the right. The brown curve shows the resultant wave (stationary wave) obtained by applying the principle of superposition at instants 16 T apart. There are positions on the stationary wave where the two waves meet antiphase, resulting in the am plitude always zero. These are called nodes (N). Halfway between the nodes are the antinodes (A) where the waves meet in phase, resulting in maximum amplitudes. The distance between two adjacent nodes or two adjacent antinodes is ½ λ. ½ λ ½ λ Note: Always draw stationary waves showing the two extremes of amplitude (one in solid, one in broken line.) Nodes Anti-nodes

Dunman High School (Senior High Physics) 5 2.1 Comparing Progressive and Stationary Waves Progressive Wave Stationary Wave Amplitude Same amplitude for all particles in the wave motion (if no loss of energy) The amplitude of the oscillating particles varies from zero (at node) to a maximum (at anti-node) Frequency Particles of wave vibrate in s.h.m. with frequency of progressive wave. Particles of wave vibrate in s.h.m. with frequency of stationary wave. Wavelength The wavelength of the wave is the shortest distance between any two successive points on a progressive wave which are vibrating in phase. The wavelength is twice the distance between a pair of adjacent nodes (or a pair of adjacent anti-nodes). Phase All particles within one wavelength have different phases. All particles within a loop vibrate in phase but have a phase difference of π rad with particles in the adjacent loop. Wave Profile Wave profile advances with the speed of the wave. Wave profile does not advance. Energy Energy is transported in the direction of travel of the wave. Energy is stored within the vibratory motion of the stationary wave. 2.2 Stationary Waves in Stretched Strings The set up below shows how stationary waves can be produced in strings. y x y x A string of length L is held taut by connecting a weight on one end through a pulley and the other end to a vibrator. A frequency generator is used to vary the frequency of vibration. The progressive wave produced by the vibrator moves towards the pulley with a constant velocity v. The wave is reflected at the pulley and travels backwards. The incident and reflected waves have the same speed, frequency and amplitude, moving in opposite direction and superpose, thus forming a stationary wave. (c) show an understanding of experiments which demonstrate stationary waves using microwaves, stretched strings and air columns. Fun fact: speed v of the waves on the string is given by v = µ T where T is the tension in the string and µ is the mass per unit length of the string.

Dunman High School (Senior High Physics) 6 Since the same string and same weight are used throughout the experiment, the speed of the waves along the string remains the same in all cases. A change in frequency causes the wavelength to change and different modes of vibration arise. Fundamental mode/ 1st harmonic (lowest frequency obtainable from the oscillation) 2 1λ=L or λ1 = 2L L v vf 21 1 = =λ 2nd harmonic (a harmonic is a note whose frequency is a whole number multi