Presentation on theme: "ISAT 241 ANALYTICAL METHODS III Fall 2004 D. J. Lawrence"— Presentation transcript:
1 ISAT 241 ANALYTICAL METHODS III Fall 2004 D. J. Lawrence Wave MotionISAT 241ANALYTICAL METHODS IIIFall 2004D. J. Lawrence
2 Wave Motion -- Examples Water wavesEarthquake wavesMechanical waves in large structures(e.g., bridges and skyscrapers)Waves on stretched stringsSound wavesElectromagnetic waves(e.g., radio, TV, and light)
3 Wave Motion Wave -- the motion or “propagation” of a disturbance. Mechanical waves require a body or “medium” that can be disturbed.Electromagnetic waves do not require a medium. They can travel through vacuum.All waves carry energy.
4 Wave MotionWaves travel or “propagate” with a specific speed that depends on the properties of the “medium” through which they are traveling, for example:velocitySound in air at 20°C m/s = 767 mi/hSound in air at 0°C m/sSound in water at 25°C m/sSound in aluminum m/sLight in vacuum 3´108 m/s = 186,000 mi/sLight in diamond ´108 m/s
5 Wave MotionTransverse Wave -- a wave in which the particles of the “disturbed medium” move perpendicular to the wave velocity, e.g.,mechanical wave on a rope or a stringelectromagnetic waves (e.g., radio, TV, and light; although these waves do not require a medium, the associated electric and magnetic fields vary in a direction perpendicular to the wave velocity)
8 Wave MotionLongitudinal Wave -- a wave in which the particles of the “disturbed medium” move in a direction parallel to the wave velocity, e.g.,sound waves in airSome waves are partly transverse and partly longitudinal, e.g.,water waves
10 Traveling WavesConsider a wave pulse on a string, moving from left to right (along x-direction) with speed = vxyvvtAPulse at t = Pulse at time tThis is a transverse wave <=> the displacement of the string (the medium) is in the y-direction.
11 Traveling WavesxyvvtAPulse at t = 0 Pulse at time tA is called the amplitude of the wave = maximum displacement.At t = 0, peak of pulse is at x = 0. At a later time, t, peak of pulse is at x = vt.
12 Traveling WavesA mathematical function that describes a wave is called a wave function.We can describe the wave pulse that we have been considering by a function of the formdisplacement(along y-axis)position of pulse(along x-axis)y is a function of the quantity (x - vt).y is a “function of two variables”.
13 Traveling WavesA wave traveling to the left with speed v can be described by a wave function of the form:A wave traveling to the right with speed v can be described by a wave function of the form:
14 Traveling Waves -- Example Consider the following wave function that describes a pulse traveling along the x-axis:where x and y are measured in centimeters and t in seconds.Is this pulse moving to the left or to the right?What is the speed of the pulse?Is the wave transverse or longitudinal?What direction does a “particle” of the rope move and what is its speed?
16 Superposition and Interference of Waves The Superposition Principle:If two or more traveling waves are moving through some medium, the resultant wave function at any point is the algebraic sum of the wave functions of the individual waves.The combination of separate waves in the same region of space to produce a resultant wave is called interference.
19 Reflection and Transmission of Waves Whenever a traveling wave reaches a boundary (e.g., the end of the string, or a location where the medium changes in some way), part or all of the wave is reflected. Any part of the wave that is not reflected is said to be transmitted through the boundary.See the Figures in your text.
34 Interference of Sinusoidal Waves The superposition principle can be applied to two or more sinusoidal waves traveling simultaneously through the same medium.The term interference is also used to describe the result of combining two or more waves.
36 Standing Waves / Modes of Vibration At certain frequencies, a stationary pattern of vibration or oscillation is produced when two or more sinusoidal waves interfere (combine) in the medium. This stationary pattern is called a standing wave.Such a pattern is also called a “mode of vibration” or a “normal mode”.These modes of vibration can be found in musical instruments, bridges, buildings, molecules ...
38 Standing Waves in a String If a stretched string is clamped at both ends, waves traveling in both directions can be reflected from the ends.The incident and reflected waves combine according to the superposition principle.Consider these two waves:y1 = A sin (kx - wt) y2 = A sin (kx + wt)Adding these two functions givesy = y1 + y2 = (2A sin kx) cos (wt)This is the wave function of a standing wave.
41 Standing Waves in a String y = y1 + y2 = (2A sin kx) cos (wt)Standing wave has angular frequency = wEvery particle of string vibrates (in SHM) with the same frequency, f = w/2p.Amplitude = 2A sin kxAmplitude of motion of a particle of string depends on x.
42 Standing Waves in a String Consider a string of length L that is clamped at both ends.The ends of the string cannot move.Points along the string that do not move are called nodes.If the string is displaced at its midpoint and released, a vibration is produced in which the center of the string undergoes the greatest movement.The center of the string is called an antinode.
43 Standing Waves in a String For this pattern of vibration, called a normal mode, the length of the string equals l/2 , i.e.,L = l/ >>>>> l =2LWe can produce vibrations (normal modes) in which there is more than one antinode.In general, the wavelengths of the various normal modes can be written
44 Standing Waves in a String The frequencies of the normal modes can be writtenIf T is the tension in the string and m is its mass per unit length, then the speed of the wave is given bySo we can write
45 Standing Waves in a String We get the lowest frequency when n = 1. This is called the fundamental frequency (or the first harmonic) and it isThe remaining modes are integral multiples of the fundamental frequency and are called higher harmonics, e.g., 2nd harmonic, 3rd harmonic, etc.
47 Modes of Vibration / Natural Frequencies / Resonance These normal modes of vibration occur in stringed musical instruments.There are natural frequencies, normal modes, or resonant frequencies associated with many objects and phenomena, in addition to strings.Natural frequencies can be found in air columns (as in organ pipes and wind instruments), tuning forks, bridges, buildings, automobile suspensions, molecules, antennas, radio and TV tuners, playground swings ...
48 Modes of Vibration / Natural Frequencies / Resonance If a periodic force is applied to such a system, the amplitude of the resulting motion will be large if the frequency of the applied force is equal or nearly equal to one of the natural frequencies of the system.The natural frequencies of oscillation of a system are often referred to as resonant frequencies.