Standing wave
[1][2] Franz Melde coined the term "standing wave" (German: stehende Welle or Stehwelle) around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings.
For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy.
As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges.
Standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom.
A requirement for this in river currents is a flowing water with shallow depth in which the inertia of the water overcomes its gravity due to the supercritical flow speed (Froude number: 1.7 – 4.5, surpassing 4.5 results in direct standing wave[7]) and is therefore neither significantly slowed down by the obstacle nor pushed to the side.
[8] The failure of the line to transfer power at the standing wave frequency will usually result in attenuation distortion.
In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved.
This section includes a two-dimensional standing wave example with a rectangular boundary to illustrate how to extend the concept to higher dimensions.
To begin, consider a string of infinite length along the x-axis that is free to be stretched transversely in the y direction.
Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping in the string equals the power supplied by the driving force so the waves have constant amplitude.
However, at x = L where the string can move freely there should be an anti-node with maximal amplitude of y. Equivalently, this boundary condition of the "free end" can be stated as ∂y/∂x = 0 at x = L, which is in the form of the Sturm–Liouville formulation.
Reviewing Equation (1), for x = L the largest amplitude of y occurs when ∂y/∂x = 0, or This leads to a different set of wavelengths than in the two-fixed-ends example.
where If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum This formula for the pressure is of the same form as Equation (1), so a stationary pressure wave forms that is fixed in space and oscillates in time.
This corresponds to a pressure anti-node (which is a node for molecular motions, because the molecules near the closed end cannot move).
[16][17] The exact location of the pressure node at an open end is actually slightly beyond the open end of the pipe, so the effective length of the pipe for the purpose of determining resonant frequencies is slightly longer than its physical length.
The closed "free end" boundary condition for the pressure at x = L can be stated as ∂(Δp)/∂x = 0, which is in the form of the Sturm–Liouville formulation.
The intuition for this boundary condition ∂(Δp)/∂x = 0 at x = L is that the pressure of the closed end will follow that of the point to its left.
[18][23] A similar, easier to visualize phenomenon occurs in longitudinal waves propagating along a spring.
Likewise, the y term equals a constant with respect to y that we can define as and the dispersion relation for this wave is therefore Solving the differential equation for the y term, Multiplying these functions together and applying the inverse Fourier transform, z(x,y,t) is a superposition of modes where each mode is the product of sinusoidal functions for x, y, and t, The constants that determine the exact sinusoidal functions depend on the boundary conditions and initial conditions.
For the x dependence, z(x,y,t) must vary in a way that it can be zero at both x = 0 and x = Lx for all values of y and t. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is with kx restricted to Likewise, the y dependence of z(x,y,t) must be zero at both y = 0 and y = Ly, which is satisfied by Restricting the wave numbers to these values also restricts the frequencies that resonate to If the initial conditions for z(x,y,0) and its time derivative ż(x,y,0) are chosen so the t-dependence is a cosine function, then standing waves for this system take the form So, standing waves inside this fixed rectangular boundary oscillate in time at certain resonant frequencies parameterized by the integers n and m. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both the x- and y-directions such that they satisfy the boundary conditions.
Even though the SWR is now finite, it may still be the case that no energy reaches the destination because the travelling component is purely supplying the losses.
One easy example to understand standing waves is two people shaking either end of a jump rope.
This effect is most noticeable in musical instruments where, at various multiples of a vibrating string or air column's natural frequency, a standing wave is created, allowing harmonics to be identified.
[32] The wavelength of light is very short (in the range of nanometers, 10−9 m) so the standing waves are microscopic in size.
[33] Because of the short wavelength of X-rays (less than 1 nanometer), this phenomenon can be exploited for measuring atomic-scale events at material surfaces.
This shift can be analyzed to pinpoint the location of a particular atomic species relative to the underlying crystal structure or mirror surface.
The XSW method has been used to clarify the atomic-scale details of dopants in semiconductors,[34] atomic and molecular adsorption on surfaces,[35] and chemical transformations involved in catalysis.
If they shake in sync, the rope will form a regular pattern with nodes and antinodes and appear to be stationary, hence the name standing wave.
Similarly a cantilever beam can have a standing wave imposed on it by applying a base excitation.
In this case the free end moves the greatest distance laterally compared to any location along the beam.
