1. 22-4-20151FCI

2. Prof. Nabila M. Hassan Faculty of Computers and Information Fayoum University 2014/2015 22-4-20152FCI

3. Chapter 4 - Superposition and Standing Waves: Superposition and Interference: Interference of Sound Waves: Standing Waves: Standing Waves in String Fixed at Both Ends: Resonance: standing Waves in Air Columns: Beats: 22-4-20153FCI

4. Superposition of Sinusoidal Waves Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude The waves differ in phase  Where : sin a +sin b = 2 cos [(a-b)/2] sin [(a+b)/2] y 1 = A sin (kx – ωt) & y 2 = A sin (kx – ωt +  )  y = y 1 +y 2 = 2A cos (  /2) sin (kx – ωt +  /2) The resultant wave function, y, is also sinusoidal The resultant wave has the same frequency and wavelength as the original waves The amplitude of the resultant wave is 2A cos (  /2) The phase of the resultant wave is  /2 22-4-20154FCI

5. Sinusoidal Waves with Constructive Interference If  = 0, 2 , 4 ,… (even multiple of  ), then: cos(  /2) = ±1  y = ± 2A(1) sin(kx – ωt + 0/2)  y = ± 2A sin(kx – ωt ) The amplitude of the resultant wave is ± 2A The crests of one wave coincide with the crests of the other wave The waves are everywhere in phase The waves interfere constructively 22-4-20155FCI

6. Sinusoidal Waves with Destructive Interference: If  = , 3 , 5 ,… (odd multiple of  ), then: cos(  /2 ) = 0  y = 2A(0)sin(kx – ωt +  /2)  The amplitude of the resultant wave is 0 Crests of one wave coincide with troughs of the other wave The waves interfere destructively 22-4-20156FCI

7. Sinusoidal Waves, General Interference When  is other than 0 or an even multiple of , the amplitude of the resultant is between 0 and 2A. The wave functions still add The interference is neither constructive nor destructive. 22-4-20157FCI

8. Summary of Interference Constructive interference occurs when  = 0 Amplitude of the resultant is 2A Destructive interference occurs when  = n  where n is an odd integer Amplitude is 0 General interference occurs when 0 <  < n  Amplitude is 0 < A resultant < 2A 22-4-20158FCI

9. Interference in Sound Waves Sound from S can reach R by two different paths The upper path(r 2) can be varied (r 1 is fixed) Whenever Δr = |r 2 – r 1 | = n (n = 0, 1, …) constructive interference occurs Whenever  Δr = |r 2 – r 1 | = (n + ½ ) (n = 0, 1, …) destructive interference occurs 22-4-20159FCI

10. Standing Waves The student will be able to: Define the standing wave. Describe the formation of standing waves. Describe the characteristics of standing waves. 22-4-201510FCI

11. Standing Waves on a String, Summary The wavelengths of the normal modes for a string of length L fixed at both ends are n = 2L / n n = 1, 2, 3, … n is the n th normal mode of oscillation These are the possible modes for the string: The natural frequencies are Also called quantized frequencies 22-4-201511FCI

12. Waves on a String, Harmonic Series The fundamental frequency corresponds to n = 1. It is the lowest frequency, ƒ 1 The frequencies of the remaining natural modes are integer multiples of the fundamental frequency. ƒ n = nƒ 1 Frequencies of normal modes that exhibit this relationship form a harmonic series. The normal modes are called harmonics. 22-4-201512FCI

13. Objectives: the student will be able to: - Define the resonance phenomena. - Define the standing wave in air columns. - Demonstrate the beats 22-4-201513FCI

14. 4 - Resonance A system is capable of oscillating in one or more normal modes If a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system 22-4-201514FCI

15. Because an oscillating system exhibits a large amplitude when driven at any of its natural frequencies, these frequencies are referred to as resonance frequencies!!! The resonance frequency is symbolized by ƒ o Resonance, 22-4-201515FCI

16. Example : 22-4-201516 An example of resonance. If pendulum A is set into oscillation, only pendulum C, whose length matches that of A, eventually oscillates with large amplitude, or resonates. The arrows indicate motion in a plane perpendicular to the page FCI

17. Resonance A system is capable of oscillating in one or more normal modes. Assume we drive a string with a vibrating blade. If a periodic force is applied to such a system, the amplitude of the resulting motion of the string is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system. This phenomena is called resonance. 22-4-201517FCI

18. Standing Waves in Air Columns Standing waves can be set up in air columns as the result of interference between longitudinal sound waves traveling in opposite directions. The phase relationship between the incident and reflected waves depends upon whether the end of the pipe is opened or closed. Waves under boundary conditions model can be applied. 22-4-201518FCI

19. Standing Waves in Air Columns, Closed End A closed end of a pipe is a displacement node in the standing wave. The rigid barrier at this end will not allow longitudinal motion in the air. The closed end corresponds with a pressure antinode. It is a point of maximum pressure variations. The pressure wave is 90 o out of phase with the displacement wave. 22-4-201519FCI

20. 1-Standing Waves in a Tube Closed at One End The closed end is a displacement node. The open end is a displacement antinode. The fundamental corresponds to ¼ The frequencies are ƒ n = nƒ = n (v/4L) where n = 1, 3, 5, … In a pipe closed at one end, the natural frequencies of oscillation form a harmonic series that includes only odd integral multiples of the fundamental frequency. 22-4-201520FCI

21. Standing Waves in Air Columns, Open End The open end of a pipe is a displacement antinode in the standing wave. As the compression region of the wave exits the open end of the pipe, the constraint of the pipe is removed and the compressed air is free to expand into the atmosphere. The open end corresponds with a pressure node. It is a point of no pressure variation. 22-4-201521FCI

22. 2- Standing Waves in an Open Tube Both ends are displacement antinodes. The fundamental frequency is v/2L. This corresponds to the first diagram. The higher harmonics are ƒ n = nƒ 1 = n (v/2L) where n = 1, 2, 3, … In a pipe open at both ends, the natural frequencies of oscillation form a harmonic series that includes all integral multiples of the fundamental frequency. 22-4-201522FCI

23. Notes About Musical Instruments As the temperature rises: Sounds produced by air columns become sharp Higher frequency Higher speed due to the higher temperature Sounds produced by strings become flat Lower frequency The strings expand due to the higher temperature. As the strings expand, their tension decreases. 22-4-201523FCI

24. Quiz : A pipe open at both ends resonates at a fundamental Frequency f open. When one end is covered and the pipe is again made to resonate, the fundamental frequency is f closed. Which of the following expressions describes how these two resonant frequencies compare? (a) f closed = f open (b) f closed = 1/2 f open (c) f closed = 2 f open (d) f closed = 3/2 f open 22-4-201524FCI

25. (b). With both ends open, the pipe has a fundamental frequency given by Equation : f open = v/2L. With one end closed, the pipe has a fundamental frequency given by: 22-4-2015FCI25

26. Resonance in Air Columns, Example A tuning fork is placed near the top of the tube. When L corresponds to a resonance frequency of the pipe, the sound is louder. The water acts as a closed end of a tube. The wavelengths can be calculated from the lengths where resonance occurs. 22-4-201526FCI

27. Beats and Beat Frequency Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies. The number of amplitude maxima one hears per second is the beat frequency. It equals the difference between the frequencies of the two sources. The human ear can detect a beat frequency up to about 20 beats/sec. 22-4-201527FCI

28. Consider two sound waves of equal amplitude traveling through a medium with slightly different frequencies f 1 and f 2. The wave functions for these two waves at a point that we choose as x = 0 Using the superposition principle, we find that the resultant wave function at this point is 22-4-201528FCI

29. Beats, Equations The amplitude of the resultant wave varies in time according to Therefore, the intensity also varies in time. The beat frequency is ƒ beat = |ƒ 1 – ƒ 2 |. 22-4-201529 Note that a maximum in the amplitude of the resultant sound wave is detected when, FCI

30. This means there are two maxima in each period of the resultant wave. Because the amplitude varies with frequency as ( f1 - f2)/2, the number of beats per second, or the beat frequency f beat, is twice this value. That is, the beats frequency 22-4-201530 For example, if one tuning fork vibrates at 438 Hz and a second one vibrates at 442 Hz, the resultant sound wave of the combination has a frequency of 440 Hz (the musical note A) and a beat frequency of 4 Hz. A listener would hear a 440-Hz sound wave go through an intensity maximum four times every second. FCI

31. 1.The sound waves that humans cannot hear are those with frequencies a.from 20 to 20,000 Hz. b.below 20 Hz. c.above 20,000 Hz. d.both B and C 2.Sound travels in air by a series of a.compressions. b.rarefactions. c.both compressions and rarefactions. d.pitches. Assessment Questions 22-4-201531FCI

32. 3.Sound travels faster in a.a vacuum compared to liquids. b.gases compared to liquids. c.gases compared to solids. d.solids compared to gases. 4.The speed of sound varies with a.amplitude. b.frequency. c.temperature. d.pitch. Assessment Questions 22-4-201532FCI

33. 5- When an object is set into vibration by a wave having a frequency that matches the natural frequency of the object, what occurs is a.forced vibration. b.resonance. c.refraction. d.amplitude reduction. 6- The phenomenon of beats is the result of sound a.destruction. b.interference. c.resonance. d.amplification. Assessment Questions 22-4-201533FCI

34. Non-sinusoidal Wave Patterns The wave patterns produced by a musical instrument are the result of the superposition of various harmonics. The human perceptive response to a sound that allows one to place the sound on a scale of high to low is the pitch of the sound. Pitch vs. frequency Frequency is the physical measurement of the number of oscillations per second. Pitch is a psychological reaction to the sound. Frequency is the stimulus and pitch is the response. The human perceptive response associated with the various mixtures of harmonics is the quality or timbre of the sound. 22-4-201534FCI

35. Quality of Sound – Tuning Fork A tuning fork produces only the fundamental frequency. 22-4-201535FCI

36. Quality of Sound – Flute The same note played on a flute sounds differently. The second harmonic is very strong. The fourth harmonic is close in strength to the first. 22-4-201536FCI

37. Analyzing Non-sinusoidal Wave Patterns If the wave pattern is periodic, it can be represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series. Any periodic function can be represented as a series of sine and cosine terms. This is based on a mathematical technique called Fourier’s theorem. A Fourier series is the corresponding sum of terms that represents the periodic wave pattern. If we have a function y that is periodic in time, Fourier’s theorem says the function can be written as: ƒ 1 = 1/T and ƒ n = nƒ 1 A n and B n are amplitudes of the waves. 22-4-201537FCI

38. Fourier Synthesis of a Square Wave In Fourier synthesis, various harmonics are added together to form a resultant wave pattern. Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f. In (a) waves of frequency f and 3f are added. In (b) the harmonic of frequency 5f is added. In (c) the wave approaches closer to the square wave when odd frequencies up to 9f are added. 22-4-201538FCI

39. Summary: 1- When two traveling waves having equal amplitudes and frequencies superimpose, the resultant waves has an amplitude that depends on the phase angle φ between the resultant wave has an amplitude that depends on the two waves are in phase, two waves. Constructive interference occurs when the two waves are in phase, corresponding to rad, Destructive interference occurs when the two waves are 180 o out of phase, corresponding to rad. 2- Standing waves are formed from the superposition of two sinusoidal waves having the same frequency, amplitude, and wavelength but traveling in opposite directions. the resultant standing wave is described by 22-4-201539FCI

40. 3- The natural frequencies of vibration of a string of length L and fixed at both ends are quantized and are given by Where T is the tension in the string and µ is its linear mass density.The natural frequencies of vibration f 1, f 2,f 3,…… form a harmonic series. 4- Standing waves can be produces in a column of air inside a pipe. If the pipe is open at both ends, all harmonics are present and the natural frequencies of oscillation are 22-4-201540FCI

41. If the pipe is open at one end and closed at the other, only the odd harmonics are present, and the natural frequencies of oscillation are 5- An oscillating system is in resonance with some driving force whenever the frequency of the driving force matches one of the natural frequencies of the system. When the system is resonating, it responds by oscillating with a relatively large amplitude. 6- The phenomenon of beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies. 22-4-201541FCI