Standing waves
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A standing wave is a vibration of a medium where some points remain fixed while others vibrate at maximum amplitude, with points of zero displacement called nodes. The formula for a standing wave on a string relates the length, number of antinodes, and wavelength. For a 1.5m string vibrating at its first harmonic with 33 cycles in 10 seconds, the frequency is 3.3Hz, period is 0.3s, wavelength is 3m, and speed is 9.9m/s.
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Standing waves
A standing wave is a vibration where some points in a medium remain fixed while other points vibrate at maximum amplitude, with points of zero displacement called nodes. The formula for a standing wave on a string relates the length of the string, number of antinodes, and wavelength. For a 2 meter string vibrating at its first harmonic with 55 cycles in 10 seconds, the frequency is 5.5 Hz, period is 0.18 seconds, wavelength is 4 meters, and speed is 22 meters per second.
Standing Waves
Standing waves are caused by two waves interfering with each other, usually a wave and its reflection. A standing wave on a string is formed when the string length is a multiple of half wavelengths. The fundamental frequency is the lowest frequency at which a standing wave can form, which is equal to the wave speed divided by twice the string length. Harmonics are integer multiples of the fundamental frequency. Nodes are points that always have zero amplitude, while antinodes are points that have maximum amplitude.
Waves
Progressive waves transfer energy through a medium by vibrating particles in the medium. There are two main types of waves: longitudinal waves where particles vibrate parallel to the wave direction, and transverse waves where particles vibrate perpendicular to the wave direction. The speed of a wave can be calculated using the formula that the speed equals the wavelength multiplied by the frequency.
Standing waves
1. A standing wave is formed by two waves of equal amplitude, wavelength, and frequency travelling in opposite directions in the same medium.
2. Nodes occur at positions where the amplitude is zero, while antinodes occur at positions of maximum amplitude. The distance between nodes is half the wavelength, and between a node and adjacent antinode is a quarter wavelength.
3. For a string fixed at both ends, standing waves can form with wavelengths of 2L/m, where L is the string length and m is a positive integer. The lowest frequency is called the fundamental frequency. Higher integer multiples of this frequency are the harmonics.
Stationary waves
Stationary waves are produced by the superposition of two progressive waves of equal amplitude and frequency travelling in opposite directions. They have nodes where there is no displacement and antinodes where the displacement is at a maximum. In a stationary wave, the waveform does not move through the medium and energy is not carried away from the source. All particles between two nodes vibrate in phase.
Stationary Waves
Stationary waves are produced by the superposition of two progressive waves of equal amplitude and frequency traveling in opposite directions. They have nodes where there is no displacement and antinodes where displacement is at maximum. A stationary wave's waveform does not move through the medium. Particles within a stationary wave vibrate in phase but with varying amplitudes, reaching maximum at antinodes and resting at nodes. The fundamental frequency and overtones of vibrating strings and air columns can be determined from measurements of stationary waves produced in them. The timbre of a sound depends on its harmonic content and amplitudes.
Waves_and_superposition.pdf
This document discusses waves and wave properties. It defines waves as progressive periodic oscillations that transfer energy from one place to another without transferring matter. Waves can be longitudinal or transverse. Key wave properties discussed include amplitude, wavelength, frequency, period, and speed. The relationship between these properties is defined by the wave equation. Longitudinal and transverse waves are illustrated and the differences between them are explained. Sound waves are described as longitudinal waves involving compression and rarefaction. Interference and superposition of waves are also discussed.
Standing Waves
Standing waves occur when two waves of equal amplitude, frequency, and wavelength travel in opposite directions and combine. When this happens, the result is a wave that oscillates in simple harmonic motion where the string moves only up and down. Nodes occur where the amplitude is zero and anti-nodes where it is maximum. Standing waves can form on a string with fixed ends, with wavelengths that are integer divisions of the string length and resonant frequencies dependent on tension and mass density. This principle applies to stringed musical instruments where the string forms standing waves at harmonic frequencies.
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Standing waves
A standing wave is a vibration where some points in a medium remain fixed while other points vibrate at maximum amplitude, with points of zero displacement called nodes. The formula for a standing wave on a string relates the length of the string, number of antinodes, and wavelength. For a 2 meter string vibrating at its first harmonic with 55 cycles in 10 seconds, the frequency is 5.5 Hz, period is 0.18 seconds, wavelength is 4 meters, and speed is 22 meters per second.
Standing Waves
Standing waves are caused by two waves interfering with each other, usually a wave and its reflection. A standing wave on a string is formed when the string length is a multiple of half wavelengths. The fundamental frequency is the lowest frequency at which a standing wave can form, which is equal to the wave speed divided by twice the string length. Harmonics are integer multiples of the fundamental frequency. Nodes are points that always have zero amplitude, while antinodes are points that have maximum amplitude.
Waves
Progressive waves transfer energy through a medium by vibrating particles in the medium. There are two main types of waves: longitudinal waves where particles vibrate parallel to the wave direction, and transverse waves where particles vibrate perpendicular to the wave direction. The speed of a wave can be calculated using the formula that the speed equals the wavelength multiplied by the frequency.
Standing waves
1. A standing wave is formed by two waves of equal amplitude, wavelength, and frequency travelling in opposite directions in the same medium.
2. Nodes occur at positions where the amplitude is zero, while antinodes occur at positions of maximum amplitude. The distance between nodes is half the wavelength, and between a node and adjacent antinode is a quarter wavelength.
3. For a string fixed at both ends, standing waves can form with wavelengths of 2L/m, where L is the string length and m is a positive integer. The lowest frequency is called the fundamental frequency. Higher integer multiples of this frequency are the harmonics.
Stationary waves
Stationary waves are produced by the superposition of two progressive waves of equal amplitude and frequency travelling in opposite directions. They have nodes where there is no displacement and antinodes where the displacement is at a maximum. In a stationary wave, the waveform does not move through the medium and energy is not carried away from the source. All particles between two nodes vibrate in phase.
Stationary Waves
Stationary waves are produced by the superposition of two progressive waves of equal amplitude and frequency traveling in opposite directions. They have nodes where there is no displacement and antinodes where displacement is at maximum. A stationary wave's waveform does not move through the medium. Particles within a stationary wave vibrate in phase but with varying amplitudes, reaching maximum at antinodes and resting at nodes. The fundamental frequency and overtones of vibrating strings and air columns can be determined from measurements of stationary waves produced in them. The timbre of a sound depends on its harmonic content and amplitudes.
Waves_and_superposition.pdf
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Standing Waves
Standing waves occur when two waves of equal amplitude, frequency, and wavelength travel in opposite directions and combine. When this happens, the result is a wave that oscillates in simple harmonic motion where the string moves only up and down. Nodes occur where the amplitude is zero and anti-nodes where it is maximum. Standing waves can form on a string with fixed ends, with wavelengths that are integer divisions of the string length and resonant frequencies dependent on tension and mass density. This principle applies to stringed musical instruments where the string forms standing waves at harmonic frequencies.
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This document discusses standing waves, which occur when two waves of equal amplitude, wavelength, and frequency travel in opposite directions in a medium and interfere. At points called nodes, the amplitude is minimum, while at points called antinodes the amplitude is maximum. A standing wave can be described mathematically as the sum of two traveling waves. The amplitude of a standing wave oscillates periodically with distance from nodes of zero amplitude to antinodes of maximum amplitude. The distance between consecutive nodes is half the wavelength.
Standing waves
Standing waves are harmonic waves with equal amplitude, wavelength, and frequency that are moving in opposite directions. They have a phase constant of 0. Standing waves have nodes where the amplitude is 0 and antinodes where the amplitude is at a maximum of 2A. The distance between nodes is half the wavelength, as is the distance between antinodes. An example problem calculates the length of a water pipe with a standing wave pattern by determining the wavelength from the given frequency and speed, and using the node and antinode spacing to relate the length to the wavelength.
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1) Standing waves are formed by the superposition of two waves traveling in opposite directions. They have points called nodes where the disturbance is always zero and amplitudes vary along the waveform.
2) Standing waves on a string or open pipe have harmonic frequencies where the wavelength is related to the length of the string or pipe by integer multiples. Closed pipes have a different harmonic series.
3) The fundamental frequency and harmonics determine the quality of the sound from different instruments. Resonance occurs at the natural harmonic frequencies of the system.
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1) A string fixed at both ends can only oscillate in standing wave patterns with wavelengths given by λ=2L/m, where m is a positive integer.
2) The lowest frequency, called the fundamental frequency, corresponds to the longest wavelength of λ=2L.
3) Higher frequencies called harmonics have wavelengths given by λ=2L/m, where m is greater than 1.
Harmonic Waves
A harmonic wave is a wave that undergoes simple harmonic motion as it travels. It can be described by the equation D(x,t)=Asin(kx-wt+φ), where D is displacement, A is amplitude, k is wave number, w is angular frequency, x is position, t is time, and φ is phase. The document provides examples of how to use this equation to calculate properties like wavelength, frequency, and acceleration of a harmonic wave traveling through different media. It also includes sample problems solving for these wave properties given values for variables like velocity, frequency, amplitude, and position over time.
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- 1. STANDING WAVES Steven Dang
- 2. WHAT IS A STANDING WAVE? A vibration of a medium in which some points in the medium remain fixed while other points between them vibrate at a maximum amplitude. Points of zero displacement are called nodes.
- 3. STANDING WAVE CALCULATIONS The formula for standing wave on a string is: 𝐿 = 𝑛 2 Λ Where L – length of the string Where n - # of antinodes (must be an integer) Where Λ - wavelength
- 4. CHECK YOUR UNDERSTANDING A string is 1.5 meters long and is vibrating as the first harmonic. The string vibrates with 33 complete vibrational cycles in 10 seconds. Determine frequency, period, wavelength and speed for this wave. 1.5m
- 5. ANSWER Frequency – Frequency is how often a point vibrates in a unit of time and this is given to us as 33 vibrations in 10 seconds. 𝑓 = 33 10 = 3.3𝐻𝑧 Period – The formula for period is 𝑇 = 1 𝑓 and since we solved for f, we can find period. 𝑇 = 1 3.3𝐻𝑧 = 0.30𝑠 Wavelength – Using the formula for a standing wave on a string 𝐿 = 𝑛 2 Λ we can determine what it’s wavelength is. We know that this string is vibrating in the first harmonic so n = 1 and we are given the length of the string. 1.5 = 1 2 Λ 3.0𝑚 = Λ Speed – Speed can be calculated using the formula 𝑣 = 𝑓Λ so we simply plug in the values we solved a moment ago. 𝑣 = 3.3𝐻𝑧 ∗ 3.0𝑚 = 9.9𝑚/𝑠
- 6. Thank You!