Superposition Principle Chapter 18 Superposition and Standing Waves If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves Waves that obey the superposition principle are linear waves For mechanical waves, linear waves have amplitudes much smaller than their wavelengths Superposition and Interference Two traveling waves can pass through each other without being destroyed or altered A consequence of the superposition principle The combination of separate waves in the same region of space to produce a resultant wave is called interference Superposition Example Two pulses are traveling in opposite directions The wave function of the pulse moving to the right is y 1 and for the one moving to the left is y 2 The pulses have the same speed but different shapes The displacement of the elements is positive for both 1
Superposition Example, cont When the waves start to overlap (b), the resultant wave function is y 1 + y 2 When crest meets crest (c ) the resultant wave has a larger amplitude than either of the original waves Superposition Example, final The two pulses separate They continue moving in their original directions The shapes of the pulses remain unchanged Constructive Interference, Summary Use the active figure to vary the amplitude and orientation of each pulse Observe the interference between them Types of Interference Constructive interference occurs when the displacements caused by the two pulses are in the same direction The amplitude of the resultant pulse is greater than either individual pulse Destructive interference occurs when the displacements caused by the two pulses are in opposite directions The amplitude of the resultant pulse is less than either individual pulse 2
Destructive Interference Example Two pulses traveling in opposite directions Their displacements are inverted with respect to each other When they overlap, their displacements partially cancel each other Use the active figure to vary the pulses and observe the interference patterns Superposition of Sinusoidal Waves Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude The waves differ only in phase y 1 = A sin (kx - ωt) y 2 = A sin (kx - ωt + φ) y = y 1 +y 2 = 2A cos (φ /2) sin (kx - ωt + φ /2) Superposition of Sinusoidal Waves, cont 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 (φ / 2) The phase of the resultant wave is φ / 2 Sinusoidal Waves with Constructive Interference When φ = 0, then cos (φ/2) = 1 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 3
Sinusoidal Waves with Destructive Interference When φ = π, then cos (φ/2) = 0 Also any odd multiple of π The amplitude of the resultant wave is 0 Crests of one wave coincide with troughs of the other wave The waves interfere destructively Sinusoidal Waves, General Interference When φ is other than 0 or an even multiple of π, the amplitude of the resultant is between 0 and 2A The wave functions still add Use the active figure to vary the phase relationship and observe resultant wave Sinusoidal Waves, Summary of Interference Constructive interference occurs when φ = nπ where n is an even integer (including 0) Amplitude of the resultant is 2A Destructive interference occurs when φ = nπ where n is an odd integer Amplitude is 0 General interference occurs when 0 < φ < nπ Amplitude is 0 < A resultant < 2A Interference in Sound Waves Sound from S can reach R by two different paths The upper path can be varied Whenever Δr = r 2 r 1 = nλ (n = 0, 1, ), constructive interference occurs 4
Interference in Sound Waves, 2 Whenever Δr = r 2 r 1 = (nλ)/2 (n is odd), destructive interference occurs A phase difference may arise between two waves generated by the same source when they travel along paths of unequal lengths Standing Waves Assume two waves with the same amplitude, frequency and wavelength, traveling in opposite directions in a medium y 1 = A sin (kx ωt) and y 2 = A sin (kx + ωt) They interfere according to the superposition principle Standing Waves, cont The resultant wave will be y = (2A sin kx) cos ωt This is the wave function of a standing wave There is no kx ωt term, and therefore it is not a traveling wave In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves Standing Wave Example Note the stationary outline that results from the superposition of two identical waves traveling in opposite directions The envelop has the function 2A sin kx Each individual element vibrates at ω 5
Note on Amplitudes There are three types of amplitudes used in describing waves The amplitude of the individual waves, A The amplitude of the simple harmonic motion of the elements in the medium, 2A sin kx The amplitude of the standing wave, 2A A given element in a standing wave vibrates within the constraints of the envelope function 2A sin kx, where x is the position of the element in the medium Standing Waves, Particle Motion Every element in the medium oscillates in simple harmonic motion with the same frequency, ω However, the amplitude of the simple harmonic motion depends on the location of the element within the medium Standing Waves, Definitions A node occurs at a point of zero amplitude These correspond to positions of x where An antinode occurs at a point of maximum displacement, 2A These correspond to positions of x where Features of Nodes and Antinodes The distance between adjacent antinodes is λ/2 The distance between adjacent nodes is λ/2 The distance between a node and an adjacent antinode is λ/4 6
Standing Waves in a String Consider a string fixed at both ends The string has length L Standing waves are set up by a continuous superposition of waves incident on and reflected from the ends There is a boundary condition on the waves Standing Waves in a String, 2 The ends of the strings must necessarily be nodes They are fixed and therefore must have zero displacement The boundary condition results in the string having a set of normal modes of vibration Each mode has a characteristic frequency The normal modes of oscillation for the string can be described by imposing the requirements that the ends be nodes and that the nodes and antinodes are separated by λ/4 We identify an analysis model called waves under boundary conditions model Standing Waves in a String, 3 This is the first normal mode that is consistent with the boundary conditions There are nodes at both ends There is one antinode in the middle This is the longest wavelength mode ½λ = L so λ = 2L Standing Waves in a String, 4 Consecutive normal modes add an antinode at each step The section of the standing wave from one node to the next is called a loop The second mode (c) corresponds to to λ = L The third mode (d) corresponds to λ = 2L/3 7
Standing Waves on a String, Summary The wavelengths of the normal modes for a string of length L fixed at both ends are λ n = 2L / n n = 1, 2, 3, n is the nth normal mode of oscillation These are the possible modes for the string The natural frequencies are Also called quantized frequencies Notes on Quantization The situation where only certain frequencies of oscillations are allowed is called quantization It is a common occurrence when waves are subject to boundary conditions It is a central feature of quantum physics With no boundary conditions, there will be no quantization 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 Musical Note of a String The musical note is defined by its fundamental frequency The frequency of the string can be changed by changing either its length or its tension 8
Harmonics, Example Standing Wave on a String, Example Set-Up A middle C on a piano has a fundamental frequency of 262 Hz. What are the next two harmonics of this string? ƒ 1 = 262 Hz ƒ 2 = 2ƒ 1 = 524 Hz ƒ 3 = 3ƒ 1 = 786 Hz One end of the string is attached to a vibrating blade The other end passes over a pulley with a hanging mass attached to the end This produces the tension in the string The string is vibrating in its second harmonic 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 Resonance, cont This phenomena is called resonance 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 If the system is driven at a frequency that is not one of the natural frequencies, the oscillations are of low amplitude and exhibit no stable pattern 9
Resonance Example Standing waves are set up in a string when one end is connected to a vibrating blade When the blade vibrates at one of the natural frequencies of the string, largeamplitude standing waves are produced 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 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 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 10
Standing Waves in an Open Tube Standing Waves in a Tube Closed at One End 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, The closed end is a displacement node The open end is a displacement antinode The fundamental corresponds to ¼λ The frequencies are ƒ n = nƒ = n (v/4l) where n = 1, 3, 5, Example16.1 A piano string of density 0.0050 kg/m is under a tension of 1350 N. Find the velocity with which a wave travels on the string. a. 260 m/s b. 520 m/s c. 1040 m/s d. 2080 m/s e. 4160 m/s Example16.2 The lowest A on a piano has a frequency of 27.5 Hz. If the tension in the 2.00 m string is 308 N, and one-half wavelength occupies the string, what is the mass of the wire? a. 0.025 kg b. 0.049 kg c. 0.051 kg d. 0.081 kg e. 0.037 kg 11
Example16.3 If y = 0.02 sin (30x 400t) (SI units), the frequency of the wave is a. 30 Hz b. 15/π Hz c. 200/π Hz d. 400 Hz e. 800p Hz Example16.4 If y = 0.02 sin (30x 400t) (SI units), the wavelength of the wave is a. π/15 m b. 15/π m c. 60π m d. 4.2 m e. 30 m Example16.5 If y = 0.02 sin (30x 400t) (SI units), the velocity of the wave is a. 3/40 m/s b. 40/3 m/s c. 60π/400 m/s d. 400/60π m/s e. 400 m/s Example16.6 If y = 0.02 sin (30x - 400t) (SI units) and if the mass density of the string on which the wave propagates is.005 kg/m, then the transmitted power is a. 1.03 W b. 2.13 W c. 4.84 W d. 5.54 W e. 106 W 12
Example16.7 Write the equation of a wave, traveling along the +x axis with an amplitude of 0.02 m, a frequency of 440 Hz, and a speed of 330 m/sec. a. y = 0.02 sin [880π (x/330 t)] b. y = 0.02 cos [880π x/330 440t] c. y = 0.02 sin [880π(x/330 + t)] d. y = 0.02 sin [2π(x/330 + 440t)] e. y = 0.02 cos [2π(x/330 + 440t)] Example16.8 For the transverse wave described by y = 0.15 sin[π/16(2x-64t)] (SI units), determine the maximum transverse speed of the particles of the medium. a. 0.192 m/s b. 0.6π m/s c. 9.6 m/s d. 4 m/s e. 2 m/s Example18.1 Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s. What is the wavelength of the interfering waves? a. 3.14 m b. 1.00 m c. 6.28 m d. 12.0 m e. 2.00 m Example18.2 Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 4 sin (5x) cos (6t) where x is in m and t is in s. What is the approximate frequency of the interfering waves? a. 3 Hz b. 1 Hz c. 6 Hz d. 12 Hz e. 5 Hz 13
Example18.3 A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the wavelength (in cm) of the first harmonic? a. 600 b. 400 c. 800 d. 1000 e. 200 Example18.4 A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the fundamental frequency? a. 316 Hz b. 632 Hz c. 158 Hz d. 215 Hz e. 79 Hz Example18.5 A stretched string is observed to vibrate in three equal segments when driven by a 480 Hz oscillator. What is the fundamental frequency of vibration for this string? a. 480 Hz b. 320 Hz c. 160 Hz d. 640 Hz e. 240 Hz 14