Physics 102 Superposition

1. Physics 102 Superposition Lecture 6 Moza M. Al-Rabban Professor of Physics mmr@qu.edu.qa

2. The Goal of this chapter is to understand and use the idea of superposition • Apply the principle of superposition. • Understand how standing waves are generated. • Calculate the allowed wavelengths and frequencies of standing waves. • Understand how waves cause constructive and destructive interference. • Calculate the beat frequency between two nearly equal frequencies.

3. Waves vs. Particles

4. Colliding Waves The Principle of Superposition When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

5. The Principle of Superposition When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

6. Stop To Think Two pulses on a string approach each other at speed of 1 m/s. What is the shape of the string at t= 6 s?

7. Standing Waves A standing wave is the superposition of two waves.

8. Standing Waves We will assume that the two waves have the same frequency, the same wavelength, and the same amplitude. They are identical waves except that one travel to the right and the other to the left. What happens as these two waves pass through each other? The blue dot shows that the wave in (b) is moving neither right nor left. This is a wave, but it is not traveling wave. It is a standing wave.

9. Nodes and Antinodes The points that never move! Spaced by /2, are called nodes. Halfway between nodes are points where the particle of medium oscillate with maximum displacement. These points of maximum amplitude are called antinodes, and they are also spaced /2 apart.

10. Two waves 1 and 2 are said to be in phase at a point where D1 is always equal to D2. This is called a point of constructive interference. The antinodes of standing wave are points of constructive interference between the two traveling waves. In contrast, two waves are said to be out of phase at points where D1 is always equal to –D2. Their superposition gives a wave with zero amplitude== no wave at all! This is a point of destructive interference. The nodes of a standing wave are points of destructive interference.

11. The points of maximum intensity occur where the standing wave oscillates with the largest amplitude (the antinodes) and that the intensity is zero at the nodes. If this is a sound wave, the loudness periodically varies from zero( no sound) to maximum and back to zero. The key idea is that the intensity is maximum at points of constructive interference and zero ( if the waves have equal amplitude) at points of destructive interference.

12. A sinusoidal wave traveling top the right along the x-axis An equivalent wave traveling to the left is The Mathematics of Standing Waves Where the amplitude function A(x) is defined as

13. Notes The amplitude reaches a maximum value Amax = 2a at points where sin kx =1. The displacement is neither a function of (x-vt) or (x+vt) , hence it is nota traveling wave. The cos t , describes a medium in which each point oscillates in simple harmonic motion with frequency f= /2. The function A(x) =2a sin kx determines the amplitude of the oscillation for a particle at position x.

14. The amplitude of oscillation, given by A(x), varies from point to point in the medium. The nodes of the standing wave are the points at which the amplitude is zero. They are located at positions x for which That is true if Thus the position xm of the mth node is Where m is an integer.

15. Example 1: Node spacing on a string • A very long string has a linear density of 5.0 g/m and is stretched with a tension of 8.0 N. 100 Hz waves with amplitudes of 2.0 mm are generated at the ends of string. • What is the node spacing along the resulting standing wave? • What is the maximum displacement of the string? MODELTwo counter-propagation waves of equal frequency create a standing wave. • SOLVE • The speed of the waves on the string is And thus the wavelength is Consequently, the spacing between adjacent nodes is/2 = 20 cm. b. The maximum displacement, at the antinodes where sin kx = 1, is

16. Reflection and Transmission When a traveling wave moving on a string reaches a point where the mass density ( and velocity) change, the wave “splits” into a reflected wave and an ongoing wave. If the mass density goes down and the velocity increases, the amplitude of the reflected wave is positive. If the mass density goes up and the velocity decreases, the amplitude of the reflected wave is negative.

17. Reflection from a Boundary When a traveling wave moving on a string reaches a fixed boundary, the wave is reflected. Because there is no transmitted wave, all the wave’s energy is reflected. Hence, the amplitude of a wave reflected from a boundary is unchanged. With respect to the incident wave, the amplitude of the reflected wave is equal in magnitude and opposite sign.

18. Standing Waves on a String If a string is plucked at the center, traveling waves move in both directions, are reflected at the boundaries, and a standing wave results.

19. But, It satisfies the boundary condition D(x = 0,t)=0 Boundary Condition • A boundary condition is a mathematical statements of any constraint that must be obeyed at the boundary or edge of a medium. Because the string is tied down at the ends, the displacement at x = 0 and x = L must be zero at all times. Thus the standing-wave boundary conditions are D(x = 0,t)=0 and D(x = L,t)=0 . Or, we requirenodes at both ends of the string.

20. The second boundary condition, at x = L, requires This condition will be met at all times if 2a sin kL = 0 ( boundary condition at x = L) This will be true if sin kL =0, i.e. D(x = L,t)=0 kL must be multiple of m, but m=0 is excluded because L can’t be zero. For a string of fixed length L, the only quantity in this equation that can be vary is . That is, the boundary condition can be satisfied only if the wavelength has one of the values A standing wave can exist on the string only if its wavelength is one of the values given by this equation.

21. The lowest allowed frequency which correspond to wavelength 1: Fundamental frequency Standing Wave Normal Modes The allowed standing wave frequencies are all integer multiples of fundamental frequency. The higher frequency standing waves are called harmonics, with m=2 wave called second harmonic, the m=3 wave called third harmonic, and so on.

22. Standing Wave Normal Modes Here are the first four possible standing waves on a string of fixed length L. The possible standing waves are called the normal modes of the string. Each mode, numbered by the integer m, has a unique wavelength and frequency. Keep in mind that These drawings simply show the envelope, or outer edge, of oscillations. The string is continuously oscillating at all positions between these edges.

23. Notes about the normal modes of a string • m is the number of antinodes on the standing wave, not the number of nodes. • The fundamental mode, with m=1, has 1 = 2L, not 1 =L. • The frequencies of the normal modes form an arithmetic series: The fundamental frequency can be found as the difference between the frequencies of any two adjacent modes.

24. Example 2: a standing wave on a string A 2.50 m long string vibrates as a 100 Hz standing wave with nodes 1.00 m and 1.50 m from one end of the string and at no points in between these two. Which harmonic is this, and what is the string’s fundamental frequency? Model: The nodes of a standing wave are spaced /2 apart. SOLVE: If there are no nodes between the two at 1.0 m and 1.5 m. then the node spacing is /2 =0.50 m. The number of 1.50 m wide segments that fit into a 2.50 m length is five, so this is m=5 mode and 100 Hz is the fifth harmonic. The harmonic frequencies are Hence, the fundamental frequency is

25. Clicker Question 2 A standing wave on a string vibrates as shown. If the tension is quadrupled while the frequency and length remain the same, which diagram represents the new vibration?

26. Standing Electromagnetic Waves Another example of a standing wave are the electromagnetic waves in a laser cavity that is bounded by two reflecting mirrors. Suppose that a laser cavity has length L=30 cm containing visible light with wavelength l=600 nm. Then the mode number m is:

27. Example:Standing Light Wave in a Laser A helium-neon laser emits red light of l = 632.9924 nm. The distance between the laser mirrors is 310.372 mm. • In what standing-wave mode does the laser operate? • What is the next longest wavelength that could make standing waves in the laser cavity?

28. Example:Cold Spots in a Microwave Oven “Cold spots”, i.e. locations where objects are not adequately heated in a microwave oven are found to be 1.25 cm apart. What is the frequency of the microwaves?

29. Standing Sound Waves and Musical Acoustics A long narrow column of air such as the air in a tube or pipe can support a longitudinal standing sound wave. Such a tube may be open or closed at the ends. The closed end of a column of air must be a node. An open end of an air column is required to be an antinode. In the example shown here, both ends are closed and the standing wave mode is m=2. There are nodes at each end and one in the center, and there are two antinodes at the quarter wave locations.

30. Example 5: Singing in the ShowerA shower stall is 2.45 m tall. For what frequencies less than 500 Hz can there be vertical standing sound waves in the shower stall? Model: The shower stall, at least to a first approximation, is a column of air 2.45 m long. It is closed at the ends by the ceiling and floor. Assume a room temperature speed of sound. The possible standing wave frequencies are integer multiples of the fundamental frequency.

31. Pipes and Modes Open-Open or Closed-Closed Open-Closed

32. Clicker Question 2 An open-open tube of air supports standing waves of frequencies of 300 Hz and 400 Hz, with no frequencies between these two. The second harmonic (m=2) of this tube has frequency: (a) 100 Hz; (b) 200 Hz; (c) 400 Hz; (d) 600 Hz; (e) 800 Hz.

33. Musical Instruments Many woodwind instruments are effectively an open-closed pipe. This means they have only odd harmonics. Their fundamental frequency will be: The vibrating string of a stringed instrument is the equivalent of a closed-closed pipe. This means it will have both odd and even harmonics. Its fundamental frequency is: Note that for wind instruments, L is the only adjustable parameter, while for stringed instruments, L, Ts and m can, in principle, be varied. However, wind instruments can be played at relatively pure harmonic frequencies, while strings cannot.

34. Example:The Notes of a Clarinet A clarinet (an open-closed instrument) is 66 cm long. The speed of sound in warm air is 350 m/s. What are the frequencies of the lowest note on a clarinet and of the next highest harmonic?

35. End of Lecture 6