P210 10d (1)
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The document discusses the speed of waves on a string. It provides an equation that shows the speed depends on the tension force (F) and the mass per unit length (μ) of the string. To slow a pulse moving along a string, students could loosen the tension or use a heavier string, as both would decrease the speed according to the equation.
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Harmonic Waves
The speed of a wave on a string depends on the tension force (F) and the mass per unit length (μ) of the string. Tighter string tension and lighter strings result in faster wave speeds, while looser tension and heavier strings produce slower waves. To slow a wave pulse on a string for a photo, students should loosen the string tension or use a heavier string.
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This document discusses stationary waves and their properties. It describes how stationary waves are produced by the superposition of two progressive waves of equal amplitude traveling in opposite directions. Stationary waves have nodes where there is no displacement and antinodes where displacement is at a maximum. The document provides examples of investigating stationary waves using sound waves and microwaves. It also discusses the factors that determine the fundamental frequency of a vibrating string and shows illustrations of standing wave patterns in a string.
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Harmonic Waves
The speed of a wave on a string depends on the tension force (F) and the mass per unit length (μ) of the string. Tighter string tension and lighter strings result in faster wave speeds, while looser tension and heavier strings produce slower waves. To slow a wave pulse on a string for a photo, students should loosen the string tension or use a heavier string.
Wave Speed on a String
This is a Power Point presentation about wave speed on a string, where I demonstrate how to solve practice problems and explain how the equations are used.
STATIONARY WAVES
Stationary waves are produced when two waves of the same frequency moving in opposite directions superimpose. When the waves are 180 degrees out of phase, they cancel each other out, and when they are in phase, they add together. A stationary wave appears stationary and consists of nodes where there is no displacement and antinodes where displacement is at a maximum. In contrast, a progressive wave transfers energy as it moves in one direction with uniform pressure and density changes.
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1) Standing waves are produced by the superposition of two waves of equal amplitude and wavelength travelling in opposite directions. The displacements of the two waves add vectorially, resulting in areas of constructive and destructive interference.
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This document discusses standing waves on violin strings. It explains that standing waves are stationary waves formed by the superposition of two harmonic waves moving in opposite directions. Standing waves can be generated when a string is plucked with both ends fixed. The string will form nodes at points of zero amplitude and antinodes at points of maximum amplitude. For a string with two fixed ends, only certain integral multiples of half wavelengths are allowed as normal modes of vibration. The fundamental frequency of the string depends on the string length, tension, and linear mass density.
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The speed of transverse waves on a string depends only on the ratio of tension to linear mass density. Doubling the tension and halving the linear mass density results in the same ratio, so the wave speed is unchanged. The speed in the second string is 2v.
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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.
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- Period, frequency, wavelength, speed, and phase are defined for waves.
- The speed of a transverse wave depends on the tension and linear mass density of the string.
- Standing waves occur at resonant frequencies when the string length is an integer multiple of half wavelengths.
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- Amplitude is the maximum displacement from equilibrium.
- Wavelength is the distance between two consecutive crests or troughs.
- Frequency is the number of waves passing a point per unit time. Period is the inverse of frequency.
- Wave speed can be calculated as wavelength multiplied by frequency.
When waves encounter boundaries, they can be reflected, transmitted, or some combination of the two depending on whether the boundary is fixed, free, or a change in medium.
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Waves are disturbances that transfer energy from one place to another without transferring matter. There are two main types of waves: mechanical waves, which need a medium to travel and can be transverse or longitudinal; and electromagnetic waves, which can travel through a vacuum. Key parts of a wave include the crest, trough, amplitude, and wavelength. The wavelength is the distance between crests or troughs, while the amplitude is half the distance between crest and trough. Frequency refers to how often a vibration occurs.
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This document discusses standing waves on violin strings. It explains that standing waves are stationary waves formed by the superposition of two harmonic waves moving in opposite directions. Standing waves can be generated when a string is plucked with both ends fixed. The string will form nodes at points of zero amplitude and antinodes at points of maximum amplitude. For a string with two fixed ends, only certain integral multiples of half wavelengths are allowed as normal modes of vibration. The fundamental frequency of the string depends on the string length, tension, and linear mass density.
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The document discusses the wave equation. It begins by defining the 1D wave equation and extending it to 2D. It then derives the wave equation for an ideal string by considering tension, mass per unit length, and applying Newton's second law. Taking derivatives, the document arrives at the wave equation and shows its solution describes waves moving left and right. It concludes that the most general solution to the wave equation is the sum of left- and right-moving waves, making it linear.
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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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The document discusses standing waves, which are created when two waves of the same amplitude and frequency traveling in opposite directions interfere. A series of diagrams show how the overlapping crests and troughs of the two waves create a standing wave pattern with fixed nodes and antinodes. Key properties of a standing wave are that the wave pattern remains fixed in space, while the displacement changes over time, and it has maximum amplitude at antinodes and zero amplitude at nodes. Standing waves differ from normal traveling waves in their amplitude, wavelength, phase, and energy distribution patterns.
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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.
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The speed of transverse waves on a string depends only on the ratio of tension to linear mass density. Doubling the tension and halving the linear mass density results in the same ratio, so the wave speed is unchanged. The speed in the second string is 2v.
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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.
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This document is a unit on waves and optics from a physics course. It covers several topics:
- Demonstrating wave motion using a rope with one end fixed and the other end moved up and down, forming crests and troughs.
- Standing waves that form on a string fixed at both ends from the interference of progressive waves moving back and forth.
- Transverse and longitudinal waves, with sound waves being longitudinal and requiring compression.
- The particle velocity, wave or phase velocity, and group velocity in wave motion. Only the group velocity transmits energy in a wave group.
- Deriving the wave equation for transverse waves on a string and considering forces from the tension.
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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 longitudinal disturbance along a stretched string or bar can propagate as a wave.
2. The wave equation derived to describe this propagation is analogous to the wave equation for transverse waves on a string.
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1) Stress is defined as force per unit area and can be tensile, compressive, or shear. Strain is the change in dimension due to stress and can be tensile, compressive, or shear.
2) Hooke's law states that within the elastic limit, stress is proportional to strain, with the constant of proportionality being Young's modulus.
3) The document provides examples of stress and strain terminology like stressor, stress capacity, stress-load, homogeneous and inhomogeneous strain, and discusses stress-strain diagrams.
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P210 10d (1)
- 2. 2 The speed of a wave on a string • When a portion of a string under tension is displaced, a wave will be launched along the string. • It is probably not surprising that the velocity will depend on the tension force F and on the weight of the string (actually the mass/unit length µ). Think of a guitar whose strings are all at the same tension. The thicker strings will produce a lower tone due to the slower speed of the wave (we will discuss standing waves shortly). • Here is the expression for the propagation speed:
- 3. 3 • Example ： Two children are playing with a string that is fixed at one end. By moving their hand up and down they can generate a wave pulse. They want to take a picture of it, but it's too fast. How can they make the pulse move slower? • (choose ALL that apply) • They could move their hands up and down a larger distance as they generate the pulse. • They could loosen the string tension. • They could tighten the string tension. • They could move their hand up and down slower. • They could use a heavier string. • They could use a lighter string.
- 4. 4 According to the formula depends on F(Ts) andAccording to the formula depends on F(Ts) and µ.µ. Μορεοϖερ, µ=Μ/ΛΜορεοϖερ, µ=Μ/Λ 1.Ιν ορδερ το σλοω τηε σπεεδ οφ τηε πυλσε, τηε στρινγ τενσιον1.Ιν ορδερ το σλοω τηε σπεεδ οφ τηε πυλσε, τηε στρινγ τενσιον Τσ σηουλδ δεχρεασεσ, σο ς δεχρεασεσ.Τσ σηουλδ δεχρεασεσ, σο ς δεχρεασεσ. 2 Τηε σπεεδ οφ τηε πυλσε αλσο δεπενδσ ον µ, ωηεν µ ισ2 Τηε σπεεδ οφ τηε πυλσε αλσο δεπενδσ ον µ, ωηεν µ ισ ραισεδ, συχη ασ ωε υσε α ηεαϖιερ στρινγ, τηε σπεεδ οφ τηεραισεδ, συχη ασ ωε υσε α ηεαϖιερ στρινγ, τηε σπεεδ οφ τηε πυλσε ωουλδ βε λοωερ.πυλσε ωουλδ βε λοωερ. σο τηε χορρεχτ ανσωερ σηουλδ βε τηε σεχονδ ονε ανδ τηεσο τηε χορρεχτ ανσωερ σηουλδ βε τηε σεχονδ ονε ανδ τηε φιφτη ονε.φιφτη ονε.