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### Transcript of University Physics: Waves and Electricity Ch16. Transverse Waves Lecture 3 Dr.-Ing. Erwin Sitompul ...

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University Physics: Waves and ElectricityCh16. Transverse WavesLecture 3Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com20133/#Erwin SitompulUniversity Physics: Wave and Electricity1A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. (a)How far apart are two points that differ in phase by /3 rad?(b)What is the phase difference between two displacements at a certain point at times 1 ms apart?Homework 2: Phase Differences3/#Erwin SitompulUniversity Physics: Wave and Electricity

(a)

(b)

Solution of Homework 2: Phase Differences3/#Erwin SitompulUniversity Physics: Wave and Electricity(a)What is u, the transverse velocity of the element of the string, at x = 22.5 cm and t = 18.9 s? A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x2.72t),in which the numerical constants are in SI units.

Example 13/#Erwin SitompulUniversity Physics: Wave and Electricity(b)What is the transverse acceleration ay of the same element of the spring at that time? A wave traveling along a string is described byy(x,t) = 0.00327sin(72.1x2.72t),in which the numerical constants are in SI units.

Example 13/#Erwin SitompulUniversity Physics: Wave and Electricity

It often happens that two or more waves pass simultaneously through the same region (sound waves in a concert, electromagnetic waves received by the antennas).Suppose that two waves travel simultaneously along the same stretched string, the displacement of the string when the waves overlap is then the algebraic sum.The Principle of Superposition for Waves3/#Erwin SitompulUniversity Physics: Wave and Electricity

Let y1(x,t) and y2(x,t) be two waves travel simultaneously along the same stretched string, then the displacement of the string is given by:

Overlapping waves algebraically add to produce a resultant wave (or net wave).Overlapping waves do not in any way alter the travel of each other.The Principle of Superposition for Waves3/#Erwin SitompulUniversity Physics: Wave and Electricity

Suppose there are two sinusoidal waves of the same wavelength and the same amplitude, and they are moving in the same direction, along a stretched string.The resultant wave depends on the extent to which one wave is shifted from the other.We call this phenomenon of combining waves as interference.

Interference of Waves3/#Erwin SitompulUniversity Physics: Wave and ElectricityInterference of WavesThe resultant wave as the superposition of y1(x,t) and y2(x,t) of the two interfering waves is:

The resultant sinusoidal wave which is the result of an interference travels in the same direction as the two original waves.3/#Erwin SitompulUniversity Physics: Wave and Electricity Interference of Waves

Fully constructiveinterferenceFully destructiveinterferenceIntermediateinterference

3/#Erwin SitompulUniversity Physics: Wave and ElectricityPhase Difference and Resulting Interference Types

3/#Erwin SitompulUniversity Physics: Wave and ElectricityCheckpointHere are four possible phase differences between two identical waves, expressed in wavelengths: 0.2, 0.45, 0.6, and 0.8.Rank them according to the amplitude of the resultant wave, greatest first.Rank: 0.2 and 0.8 tie, 0.6, 0.45

3/#Erwin SitompulUniversity Physics: Wave and ElectricityExample 2Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude ym of each wave is 9.8 mm, and the phase difference between them is 100. (a)What is the amplitude ym of the resultant wave due to the interference, and what is the type of this interference?

The interference is intermediate, which can be deducted in two ways:1. The phase difference is between 0 and radians.2. The amplitude ym is between 0 and 2ym.3/#Erwin SitompulUniversity Physics: Wave and ElectricityExample 2(b)What phase difference, in radians and wavelengths, will give the resultant wave an amplitude of 4.9 mm?Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude ym of each wave is 9.8 mm, and the phase difference between them is 100.

3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding WavesThe following figures shows the superposition of two waves of the same wavelength and amplitude, traveling in opposite direction.There are places along the string, called nodes, where the string never moves. Halfway between adjacent nodes, we can see the antinodes, where the amplitude of the resultant wave is a maximum.The resultant wave is called standing waves because the wave pattern do not move left or right.

Where?Where?3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves

3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding WavesTo analyse a standing wave, we represent the two combining waves with the equations:

The principle of superposition gives:

3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves

For a standing wave, the amplitude 2ymsinkx varies with position.For a traveling wave, the amplitude ym is the same for all position.NNNNANANANx03/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding WavesIn the standing wave, the amplitude is zero for values of kx that give sinkx = 0.

NodesIn the standing wave, the amplitude is zero for values of kx that give sinkx = 1

Antinodes

3/#Erwin SitompulUniversity Physics: Wave and ElectricityCheckpointTwo waves with the same amplitude and wavelength interfere in three different situations to produce resultant waves with the following equations:(a) y(x,t) = 4sin(5x4t)(b) y(x,t) = 4sin(5x)cos(4t) (c) y(x,t) = 4sin(5x+4t)

In which situation are the two combining waves traveling (i) toward positive x, (ii) toward negative x, and (ii) in opposite directions?Toward positive x: (a), the sign before t is negativeToward negative x: (c), the sign before t is positiveIn opposite directions: (b), resulting standing wave3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves and ResonanceConsider a string, such as a guitar string, that is stretched between two clamps.If we send a continuous sinusoidal wave of a certain frequency along the string, the reflection and interference will produce a standing wave pattern with nodes and antinodes like those in the figure.Such a standing wave is said to be produced at resonance. The string is said to resonate at a certain resonant frequencies.

3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves and Resonance

For a string stretched between two clamps, we note that a node must exist at each of its end, because each end is fixed and cannot oscillate.The simplest patterns that meets this requirement is a single-loop standing wave, with two nodes and one antinode.A second simple pattern is the two loop pattern. This pattern has three nodes and two antinodes.A third pattern has four nodes, three antinodes, and three loops3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves and ResonanceThus, a standing wave can be set up on a string of length L by a wave with a wavelength equal to one of the values:

The resonant frequencies that correspond to these wavelengths are:

The last equation tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, f = v/2L, for n = 1.The oscillation mode with the lowest frequency is called the fundamental mode or the first harmonic.3/#Erwin SitompulUniversity Physics: Wave and ElectricityStanding Waves and Resonance

The second harmonic is the oscillation mode with n = 2, the third harmonic is that with n = 3, and so on.The collection of all possible oscillation modes is called the harmonic series.n is called the harmonic number.3/#Erwin SitompulUniversity Physics: Wave and ElectricityCheckpointIn the following series of resonant frequencies, one frequency (lower than 400 Hz) is missing: 150, 225, 300, 375 Hz. (a) What is the missing frequency? (b) What is the frequency of the seventh harmonic?

The most possible value for v/2L from the above series is 75 Hz?The missing frequency below 400 Hz is thus 75 Hz.The seventh harmonic has the frequency of f5 + 2 v/2L = 375 + 275 Hz = 520 Hz.

3/#Erwin SitompulUniversity Physics: Wave and ElectricityHomework 3: Standing WavesTwo identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation(a)What are the amplitude and speed of the two waves?(b)What is the distance between nodes?(c)What is the transverse speed of a particle of the string at the position x = 1.5 cm when t = 9/8 s?

3/#Erwin SitompulUniversity Physics: Wave and ElectricityHomework 3A: Standing WavesTwo identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation(a)What are the amplitude and speed of the two waves?(b)What is the distance between nodes?(c)What is the transverse speed of a particle of the string at the position x = 2.70 m when t = 0.25 min?

Two waves propagate in one direction on a stretched rope. The frequency of the waves is 120 Hz. Both have the same amplitude of 4 cm and wavelength of 0.04 m. (a) Determine the amplitude of the resultant wave if the two original waves differ in phase by /3? (b) What is the phase difference between the two waves if the amplitude of the resultant wave is 0.05 cm?3/#Erwin SitompulUniversity Physics: Wave and Electricity