Sunday, May 27, 2007
Traveling Waves
Traveling waves
Waves that remain in one place are called standing waves—e.g. vibrations on a violin string. Waves that are moving are called traveling waves, also called progressive waves, and have a disturbance that varies both with time t and distance z.
Waves that remain in one place are called standing waves—e.g. vibrations on a violin string. Waves that are moving are called traveling waves, also called progressive waves, and have a disturbance that varies both with time t and distance z.
Propagation Through Strings
Propagation through strings
The speed of a wave traveling along a string (v) is directly proportional to the square root of the tension (T) over the linear density (μ).
The speed of a wave traveling along a string (v) is directly proportional to the square root of the tension (T) over the linear density (μ).
Transmission Medium
Transmission medium
The medium that carries a wave is called a transmission medium. It can be classified into one or more of the following categories:
1. A linear medium if the amplitudes of different waves at any particular point in the medium can be added.
2. A bounded medium if it is finite in extent, otherwise an unbounded medium.
3. A uniform medium if its physical properties are unchanged at different locations in space.
4. An isotropic medium if its physical properties are the same in different directions.
The medium that carries a wave is called a transmission medium. It can be classified into one or more of the following categories:
1. A linear medium if the amplitudes of different waves at any particular point in the medium can be added.
2. A bounded medium if it is finite in extent, otherwise an unbounded medium.
3. A uniform medium if its physical properties are unchanged at different locations in space.
4. An isotropic medium if its physical properties are the same in different directions.
Mathematical Description of Sound
Mathematical description
Waves can be described mathematically using a series of parameters. The amplitude of a wave (commonly notated as A, or another letter) is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium). In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
The units of the amplitude depend on the type of wave — waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter). The amplitude may be constant (in which case the wave is a c.w. or continuous wave), or may vary with time and/or position. The form of the variation of amplitude is called the envelope of the wave.
The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum.
Waves can be represented by simple harmonic motion.
The period T is the time for one complete cycle for an oscillation of a wave. The frequency f (also frequently denoted as ν) is how many periods per unit time (for example one second) and is measured in hertz. In other words, the frequency and period of a wave are reciprocals of each other.
The angular frequency ω represents the frequency in terms of radians per second.
There are two velocities that are associated with waves. The first is the phase velocity, which gives the rate at which the wave propagates.
The second is the group velocity, which gives the velocity at which variations in the shape of the wave's amplitude propagate through space. This is the rate at which information can be transmitted by the wave. It is given by
Waves can be described mathematically using a series of parameters. The amplitude of a wave (commonly notated as A, or another letter) is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium). In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
The units of the amplitude depend on the type of wave — waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter). The amplitude may be constant (in which case the wave is a c.w. or continuous wave), or may vary with time and/or position. The form of the variation of amplitude is called the envelope of the wave.
The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum.
Waves can be represented by simple harmonic motion.
The period T is the time for one complete cycle for an oscillation of a wave. The frequency f (also frequently denoted as ν) is how many periods per unit time (for example one second) and is measured in hertz. In other words, the frequency and period of a wave are reciprocals of each other.
The angular frequency ω represents the frequency in terms of radians per second.
There are two velocities that are associated with waves. The first is the phase velocity, which gives the rate at which the wave propagates.
The second is the group velocity, which gives the velocity at which variations in the shape of the wave's amplitude propagate through space. This is the rate at which information can be transmitted by the wave. It is given by
Polarization of Sound
Polarization
A wave is polarized if it can only oscillate in one direction. The polarization of a transverse wave describes the direction of oscillation, in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization, because for these waves the direction of oscillation is along the direction of travel. A wave can be polarized by using a polarizing filter.
Examples of waves include:
Ocean surface waves, which are perturbations that propagate through water.
Radio waves, microwaves, infrared rays, visible light, ultraviolet rays, x-rays, and gamma rays make up electromagnetic radiation. In this case, propagation is possible without a medium, through vacuum. These electromagnetic waves travel at 299,792,458 m/s in a vacuum.
Sound - a mechanical wave that propagates through air, liquid or solids.
Seismic waves in earthquakes, of which there are three types, called S, P, and L.
Gravitational waves, which are fluctuations in the gravitational field predicted by general Relativity. These waves are nonlinear, and have yet to be observed empirically.Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect
A wave is polarized if it can only oscillate in one direction. The polarization of a transverse wave describes the direction of oscillation, in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization, because for these waves the direction of oscillation is along the direction of travel. A wave can be polarized by using a polarizing filter.
Examples of waves include:
Ocean surface waves, which are perturbations that propagate through water.
Radio waves, microwaves, infrared rays, visible light, ultraviolet rays, x-rays, and gamma rays make up electromagnetic radiation. In this case, propagation is possible without a medium, through vacuum. These electromagnetic waves travel at 299,792,458 m/s in a vacuum.
Sound - a mechanical wave that propagates through air, liquid or solids.
Seismic waves in earthquakes, of which there are three types, called S, P, and L.
Gravitational waves, which are fluctuations in the gravitational field predicted by general Relativity. These waves are nonlinear, and have yet to be observed empirically.Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect
Characteristic of Sound
Characteristic
Periodic waves are characterized by crests (highs) and troughs (lows), and may usually be categorized as either longitudinal or transverse. Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.
When an object bobs up and down on a ripple in a pond, it experiences an orbital trajectory because ripples are not simple transverse sinusoidal waves.
Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
All waves have common behavior under a number of standard situations.
All waves can experience the following:
Reflection - wave direction change from hitting a reflective surface
Refraction - wave direction change from entering a new medium
Diffraction - wave circular spreading from entering a hole of comparable size to their
wavelengths
Interference - superposition of two waves that come into contact with each other (collide)
Dispersion - wave splitting up by frequency
Rectilinear propagation - wave movement in straight lines
Periodic waves are characterized by crests (highs) and troughs (lows), and may usually be categorized as either longitudinal or transverse. Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.
When an object bobs up and down on a ripple in a pond, it experiences an orbital trajectory because ripples are not simple transverse sinusoidal waves.
Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
All waves have common behavior under a number of standard situations.
All waves can experience the following:
Reflection - wave direction change from hitting a reflective surface
Refraction - wave direction change from entering a new medium
Diffraction - wave circular spreading from entering a hole of comparable size to their
wavelengths
Interference - superposition of two waves that come into contact with each other (collide)
Dispersion - wave splitting up by frequency
Rectilinear propagation - wave movement in straight lines
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