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2. Faculty of Computer and Information Fayoum University 20-4-20162FCI

3. Standing Waves The student will be able to: Define the standing wave. Describe the formation of standing waves. Describe the characteristics of standing waves. 20-4-20163FCI

4. Objectives: the student will be able to: - Define the resonance phenomena. - Define the standing wave in air columns. - Demonstrate the beats 20-4-20164FCI

5. 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 20-4-20165FCI

6. 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 The maximum amplitude is limited by friction in the system Resonance, 20-4-20166FCI

7. Example : 20-4-20167 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

8. 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. 20-4-20168FCI

9. 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. 20-4-20169FCI

10. 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. 20-4-201610FCI

11. 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. 20-4-201611FCI

12. 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. 20-4-201612FCI

13. 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. 20-4-201613FCI

14. 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. 20-4-201614FCI

15. 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. 20-4-201615FCI

16. 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 20-4-201616FCI

17. 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 |. 20-4-201617 Note that a maximum in the amplitude of the resultant sound wave is detected when, FCI

18. 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 20-4-201618 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

19. 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. 20-4-201619FCI

20. 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. 20-4-201620FCI

21. 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 20-4-201621FCI

22. 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 20-4-201622FCI

23. 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. 20-4-201623FCI