Standing Waves With a Node on One End

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As an example of the second type, a standing wave in a transmission line is a wave in which the distribution of current, voltage, or field strength is formed by the superposition of two waves of the same frequency propagating in opposite directions. The effect is a series of nodes (zero displacement) and anti-nodes(maximum displacement) at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is reflected from the other end by an impedancemismatch, i.e., discontinuity, such as an open circuit or a short.^[1] The failure of the line to transfer power at the standing wave frequency will usually result inattenuation distortion.

Another example is standing waves in the open ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, and are the source ofmicrobaroms and microseisms.

In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is apartial standing wave, which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the standing wave ratio (SWR).^[2]

[edit]Mathematical description

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become zeroes or nodes of the wave. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the fundamental. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called harmonics or overtones. For example, a guitar player can select an overtone by putting a finger on a string to force a node at the proper position between the ends of the string, suppressing all harmonics that do not share this node.

Harmonic waves travelling in opposite directions can be represented by the equations below:

and

where:

• y₀ is the amplitude of the wave,
• ω (called angular frequency, measured in radians per second) is 2π times the frequency (in hertz),
• k (called the wave number and measured in radians per metre) is 2π divided by the wavelength λ (in metres), and
• x and t are variables for longitudinal position and time, respectively.

So the resultant wave y equation will be the sum of y₁ and y₂:

Using the trigonometric sum-to-product identity for 'sin(u) + sin(v)' to simplify:

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(kx). At locations x = 0, λ/2, λ, 3λ/2, ... called the nodes the amplitude is always zero, whereas at locations x = λ/4, 3λ/4, 5λ/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is λ/2.

Standing waves can also occur in two- or three-dimensional resonators. With standing waves on two dimensional membranes such asdrumheads, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called Chladni figures. In three-dimensional resonators, such as musical instrument sound boxes and microwave cavity resonators, there are nodal surfaces.

Eli Priyatna 17 Jun, 2011

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Source: http://basistik.blogspot.com/2011/06/standing-waves-with-node-on-one-end.html
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