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Standing waves in one dimension
Keywords: standing waves, boundary conditions, resonant modes, waves on a string experiment, nonharmonic resonances, wire hoop resonances, animated waves, Flash animation
Contents of this posting 

Standing waves in one dimension 
1. Mathematical expressions for standing waves in one dimension
In a one dimensional system, standing waves can be expressed in real form as:
or in complex form as:
where A, A_{1}, B, C, D, E, F, φ, and θ are real constants determining the amplitude and phase of the oscillating factors. φ is a real constant setting the phase of the oscillations in time in the complex form while θ sets the phase of the position sinusoid. E_{1} is a complex constant setting the amplitude of the standing wave and also the phase of the time oscillating factor.
We show the times symbol × to emphasize that for the standing waves, the time and space sinusoids multiply each other in contrast to the traveling wave case where both time t and position x are found together in the argument of a single sinusoidal.
2. Boundary conditions
For standing waves in enclosed spaces there are two boundary conditions that are commonly used to approximate the reflection properties occurring in the real world. These are:
 Requiring that the wave amplitude be zero at the boundary (Dirichlet), and
 Requiring that the derivative of the wave be zero at the boundary (Neumann).
Both types of boundaries require that there are standing waves as opposed to traveling waves, because with a traveling wave every point experiences the whole wave as the wave sweeps past. Thus traveling waves will not have special end points at which the wave is always zero or its derivative is always zero. Figs. 5 and 6 show typical standing waves that meet the two types of boundary conditions at their ends. Fig. 7 shows a typical standing wave for a Dirichlet boundary at one end and a Neumann boundary at the other end.
In the animations below we show a time changing graph of the wave as a function of position. This can alternately be thought of as a wave on a string. Also we include a false color plot of the wave which can be thought of as the acoustical pressure (the time varying part of the pressure) versus time inside an acoustically excited pipe or the height of water waves in a channel.
Table 2. Animated linear wave resonators with various boundary conditions. 

Fig. 5. Typical wave resonance fulfilling the Dirichlet type boundary condition at the two ends. This boundary condition requires that the waves have zero amplitude at the end points (the boundaries). These waves are seen in the waves on a string experiment done in many college physics courses, as well as in stringed musical instruments. This form of resonance is also present in pipes which are open at both ends. This boundary condition requires that an integer number of half wavelengths equal the length ℓ of the resonator, i.e. nλ/2 = ℓ where n is a positive integer greater than zero. We can translate this into wavenumber as κ = nπ/ℓ from which we can calculate the frequency using ω = κc (Equation (3) of the previous posting) to get f_{n} = n (c/2ℓ) = n f_{1}. This means that for media of constant phase velocity c (nondispersive media) the resonant frequencies for this type of resonator occur at all the integer multiples of the lowest frequency, i.e. at f_{1}, 2f_{1}, 3f_{1}, etc. The lowest resonant frequency is given by: f_{1} = c/2ℓ. If we assume the left end of the resonator is at x = 0, then the equation of the wave will be: Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it. 
Fig. 6. Typical wave fulfilling the Neumann type boundary condition at the two ends. This boundary condition requires zero slope (horizontal) amplitude at the two ends. This is the wave type found in water waves slushing back and forth along a narrow channel with closed ends or acoustical resonance inside a length of pipe with the two ends closed off.
Like the setup in Fig. 5, this boundary condition requires that an integer number of half wavelengths equal the length ℓ of the resonator, i.e. nλ/2 = ℓ , where the waves are shifted over from those shown in Fig. 5. We can translate this into wavenumber as κ = nπ/ℓ from which we can calculate the frequency using ω = κc to get f_{n} = n (c/2ℓ) = n f_{1}. This means that the resonant frequencies for this type of resonator with nondispersive media occur at all the integer multiples of the lowest frequency, i.e. at f_{1}, 2f_{1}, 3f_{1}, etc. The lowest resonant frequency is f_{1} = c/2ℓ. If we assume the left end of the resonator is at x = 0, then the equation of the wave will be: Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it. 
Fig. 7. Typical wave having a Dirichlet boundary at one end and
a Neumann boundary at the other end. This would be the setup for most organ pipes which have the upper end open and the lower end closed.
This system requires that exactly n − ½ half wave lengths fit into the length of the resonator. That is to say (n − ½)λ/2 = ℓ . We can translate this into wavenumber as κ = (π/ℓ) (n−½) from which we can calculate the frequency using ω = κc to get f_{n} = (2n − 1) (c/4ℓ) = (2n − 1) f_{1}. This means that the resonant frequencies for this type of resonator with nondispersive media occur at odd integer multiples of the lowest frequency, i.e. at f_{1}, 3f_{1}, 5f_{1}, etc. The lowest resonant frequency f_{1} = c/4ℓ is half that of the two resonators above given the same length and wave speed. If we assume the left end of the resonator is the closed end and is at x = 0, then the equation of the wave will be the same as (8a) but with a different value of κ: Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it. 
3. Resonant Modes
Below we show animations of the first seven resonant modes of a linear wave resonator for the three different boundary conditions introduced above. At the bottom of each animation we see a spectrum of the resonant modes: a linear graph of the frequencies of the modes for a fixed density of string with a particular tension.
Table 3. Resonant modes of linear wave resonators.  

⇐ Fig. 8. Animation of linear resonator with Dirichlet boundary conditions at both ends forcing the wave to be zero at the ends. We see the resonant modes are integer multiples of the lowest frequency (the fundamental  shown at the top) as discussed in Fig. 5 above. Resonators such as this one which have the resonator frequencies at integer multiples of the fundamental are referred to as harmonic. When a Dirichlet resonator is excited in the lowest frequency mode, i.e. its fundamental, the resonator is referred to as a half wave resonator, since at this frequency one half wave precisely fits into its length. A nonsinusoidal exciting force, such as a hammer striking a string inside a piano, a plucking of a guitar string, or a bowing of a violin string, will excite a number of these modes simultaneously. Fourier decomposition can be applied to the exciting force as discussed in an earlier blog to resolve how much of each resonator mode is excited. Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it.  
⇐ Fig. 9. Animation of linear resonator with Neumann boundary conditions at both ends forcing the derivative of the wave to be zero at the ends. Just like in the previous case, we see the resonant modes are integer multiples of the lowest (the fundamental) frequency as discussed in Fig. 6 above. When excited in the lowest mode, this resonator can be referred to as a half wave resonator as in the resonator of Fig. 8.
Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it.  
⇐ Fig. 10. Animation of linear resonator with Dirichlet boundary condition at the left end and Neumann boundary condition at the right end. We see that the lowest frequency mode is half the lowest frequency modes in Fig. 8 and 9 above, and that the higher frequency modes in Fig. 10 are odd integer multiples of the lowest frequency in Fig. 10 as discussed in Fig. 7 above. When excited in the lowest mode, this resonator is often referred to as a quarter wave resonator.
Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it. 
4. Relationship between traveling and standing waves
Traveling waves and standing waves are mathematically interrelated, that is one can be expressed in terms of the other. A standing wave can be thought of as a sum of two oppositely propagating traveling waves in a resonant or trappedwave situation. As illustrated in Figs. 5 through 10 the traveling wave is reflected back on itself multiple times. Animations demonstrating this idea are found in the first animation of an earlier blog (click the first animation on the page to "continuous" and "components") and in the first two animations of a second earlier blog (click "total reflection+" or "total reflection−" and also "show components"). A third blog showing traveling and standing waves is here (in the second animation and associated text).
The math demonstration of this is:
which is equivalent to (7). In deriving (9) we have used standard trig equations:
5. Experimenting with standing wave resonators
In many ways wave resonators behave pretty much as a collection of simple resonators, one for each mode. Like simple resonators they can be either shock excited and the decay observed, or continuously excited with the standard resonance curve seen for each mode. Continuous sinusoidal excitation allows the excitation of specific modes, one at a time, while shock excitation usually causes the excitation of many modes simultaneously. Playing musical instruments often involves simultaneous excitation of a chain of harmonic modes of frequencies which are integer multiples of the lowest mode excited.
In contrast with the idealized spectrum showing the frequencies of the various modes at the bottom of Figs. 8, 9 and 10, the animation in Fig. 11 shows a typical experimental resonance curve for a wave resonator. Even without internal loss the coupling strength will often vary from mode to mode giving the different peaks various heights and widths.
Fig. 11. Animation showing a frequency scan with a wavesonastring apparatus. Mouse over the image to see the animation, mouse off to suspend it and click on it to restart it. The purple line on the graph shows the up and down motion at the site of the "sensor" while the red line is the envelope of these oscillations, more like what we expect in a spectrum. Spectrums are often shown in terms of the energy of the resonance instead of the amplitude, in which case we would have the square of the envelope shown above which would make the peaks stand out more.
Comparing the results of this scan with the spectrum in Fig. 8 above, we see that the real resonant peaks are not nearly as sharp as the idealized lines of Fig. 8 and that there is a "floor" to the resonances. This floor is due to the sum of slight responses of the many resonant modes to an offresonance exciting force. 
Also shown in Fig. 11 is the physical setup that might be used to observe the resonance curve shown. A video of another such setup is shown here. These are basically the same setups as shown in the animation of an earlier blog with an exchange of a wave resonator for the elementary resonator of the earlier animation. A variable frequency generator and "driver" is used to excite the wave resonator at some specific place on the resonator. A sensor element is used to sense the degree of excitation at another point. Depending on the resonator type, another setup might combine the exciting and sensing elements.
The locations of the exciting and sensing points will affect the heights and widths of the observed resonance curves. For example if either exciter or sensor is located at a node (a zero) of a particular oscillation mode, that mode will not be present in the observed frequency sweep. In general, the coupling strength for the exciter will be proportional to the strength of the mode at the excitation point. Stronger coupling does not necessarily mean greater excitation. The maximum excitation is achieved when there is a "match" between coupling strength and the losses (internal and coupling) in the resonant mode in question (see Fig. 31 of this blog).
6. Example of a nonharmonic standing wave resonator  oscillations of a slender beam
To give an example of a wave oscillator that is nonharmonic (the oscillation frequency of its modes are not integer multiples of the lowest resonant frequency) we briefly discuss the oscillations of a slender "beam". A suitable "beam" would be a stiff piece of wire or thin flat ruler with basically the same equipment as shown in Fig. 11 with the string replaced by such a slender beam.
The equations used are taken from Elmore and Heald, Physics of waves, Section 4.6 and are considerably more complex than those for waves on a string. The displacement function shown in Fig. 12 is a mix of regular sinusoids and hyperbolic functions and is valid for a setup in which the ends of the beam are clamped, i.e. forced to stay horizontal.
Table 4. Relative frequencies of the first five modes of a slender beam clamped at the ends  

Mode number:  1  2  3  4  5 
Relative frequency  1.000  2.757  5.404  8.934  13.346 
The normalized frequencies of these first five modes are shown in Table 4, normalized by the frequency of the first mode. They are clearly not integer multiples of the lowest frequency mode. Somewhat surprisingly, these frequencies are the same as for the situation in which the ends of the beam are free (the case covered in detail in Elmore and Heald, referenced above).
Math for clamped modes of slender beam 

We follow the nomenclature of Elmore and Heald, Section 4.6. We start with the mathematical equations (4.6.30) of Elmore and Heald: and where η(x,t) = X(x)⋅T(t) . Requiring both η and η' to equal 0 at x = 0 and x = ℓ (the right end of the beam), gives us four equations. We have five unknowns A, B, C, D and k; however we are free to choose an arbitrary overall amplitude of the resonance. So we choose A = 1, thereby getting rid of one of the unknowns. Substituting x = 0 into (12) and (13) and setting both equations equal to zero yields C = − A = − 1 and D = − B . We then substitute these and x = ℓ into (12) and (13) and set both of these equations equal to zero. We get two equations for B: Setting these equal to each other and using trig identities, we get: which is the same equation Elmore and Heald came up with for the case of freefree ends. So the wavenumber κ allowed for resonance will be the same in the two cases. Because the other equations are different, the two setups with different end conditions will have different values for B, C and D and thus different appearances of the modes. Elmore and Heald list the values of κℓ as being: (3/2)π(1.0037), (5/2)π(0.9999), (7/2)π(1.0000) and (n + ½)π for higher order modes. The resonant frequencies are proportional to the square of κ and are reported in Table 4 above. 
Fig. 13. The first mode of oscillation of a stiff wire hoop. 
7. Oscillations of a wire hoop
Similar oscillations can be set up if the "beam" is bent and joined into a circle, such as a stiff wire hoop. Such a setup is shown in Fig. 13. The math is covered in Elmore and Heald, referenced above. In this case, since there are no ends for end conditions, we can just use the continuous traveling wave case, more accurately the rotating wave case. In this case the displacements are simple sinusoids, as in the waves on a string. The rotating wave modes will be given by:
ψ(φ,t) = A cos(mφ − ωt) , (11)
where A is the amplitude of the deviations from a perfect circle, m is the mode number (an integer, positive or negative), φ is the angular coordinate measuring position around the circle, and ω is the angular frequency of the mode. Since the restoring force of this wave media is due to bending, and mode indices of m = −1, 0, and +1 do not involve bending, these modes will not be present. The standing wave modes, such as the m = 2 mode shown in Fig. 13, are just sums of pairs of rotating wave modes as discussed in the next posting which is on ring resonators. Exciting the mode with a single driver will most likely only excite standing wave modes unless there is an asymmetry in the ring (having a segment that is different from the rest) in which case a rotating wave can be excited by a single driver.
Finally as shown in Elmore and Heald for traveling waves on a slender beam, we expect the frequency to be proportional to the wavenumber squared (see Eqn 4.6.10 in Elmore and Heald). Requiring an integer number of wavelengths to exactly fit into the circumference of the hoop fixes the wavenumber κ = 2π/λ = 2πm/2πr = m/r where λ is the wavelength and r is the radius of the hoop. This will mean that the resonance frequencies will be proportional to the mode index squared. It is not clear if this resonator would be classified as harmonic or inharmonic, since the various modes are integer multiples of a nonexistent m = 1 mode and not all possible integers have modes associated with them.
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