Editing Standing wave (section)

=== Standing wave in a pipe ===
{{See also|Acoustic resonance#Resonance of a tube of air}}

Consider a standing wave in a pipe of length ''L''. The air inside the pipe serves as the medium for [[longitudinal wave|longitudinal]] [[sound wave]]s traveling to the right or left through the pipe. While the transverse waves on the string from the previous examples vary in their displacement perpendicular to the direction of wave motion, the waves traveling through the air in the pipe vary in terms of their pressure and longitudinal displacement along the direction of wave motion. The wave propagates by alternately compressing and expanding air in segments of the pipe, which displaces the air slightly from its rest position and transfers energy to neighboring segments through the forces exerted by the alternating high and low air pressures.{{sfn|Halliday|Resnick|Walker|2005|p=450}} Equations resembling those for the wave on a string can be written for the change in pressure Δ''p'' due to a right- or left-traveling wave in the pipe.
:<math> \Delta p_\text{R}(x,t) =  p_\text{max}\sin \left({2\pi x \over \lambda} - \omega t \right), </math>
:<math> \Delta p_\text{L}(x,t) = p_\text{max}\sin \left({2\pi x \over \lambda} + \omega t \right), </math>

where
*''p''<sub>max</sub> is the pressure amplitude or the maximum increase or decrease in air pressure due to each wave,
*''ω'' is the [[angular frequency]] or equivalently ''2π'' times the [[frequency]] ''f'',
*''λ'' is the [[wavelength]] of the wave.

If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum
:<math> \Delta p(x,t) = \Delta p_\text{R}(x,t) + \Delta p_\text{L}(x,t) = 2p_\text{max}\sin \left({2\pi x \over \lambda} \right) \cos(\omega t).</math>

This formula for the pressure is of the same form as Equation ({{EquationNote|1}}), so a stationary pressure wave forms that is fixed in space and oscillates in time.

If the end of a pipe is closed, the pressure is maximal since the closed end of the pipe exerts a force that restricts the movement of air. This corresponds to a pressure anti-node (which is a node for molecular motions, because the molecules near the closed end cannot move). If the end of the pipe is open, the pressure variations are very small, corresponding to a pressure node (which is an anti-node for molecular motions, because the molecules near the open end can move freely).<ref name="HyperPhyiscs Standing Waves">{{cite web| last=Nave| first= C. R.| title = Standing Waves| series = HyperPhysics| publisher = Georgia State University| year = 2016| url = http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/standw.html| access-date = August 23, 2020}}</ref>{{sfn|Streets|2010|p=6}} 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.{{sfn|Halliday|Resnick|Walker|2005|p=457}} This difference in length is ignored in this example. In terms of reflections, open ends partially reflect waves back into the pipe, allowing some energy to be released into the outside air. Ideally, closed ends reflect the entire wave back in the other direction.{{sfn|Halliday|Resnick|Walker|2005|p=457}}{{sfn|Streets|2010|p=15}}

First consider a pipe that is open at both ends, for example an open [[organ pipe]] or a [[recorder (musical instrument)|recorder]]. Given that the pressure must be zero at both open ends, the boundary conditions are analogous to the string with two fixed ends,
:<math> \Delta p(0,t) = 0,</math>
:<math> \Delta p(L,t) = 2p_\text{max}\sin \left({2\pi L \over \lambda} \right) \cos(\omega t) = 0,</math>

which only occurs when the wavelength of standing waves is{{sfn|Halliday|Resnick|Walker|2005|p=457}}
:<math> \lambda = \frac{2L}{n}, </math>
:<math> n = 1, 2, 3, \ldots, </math>

or equivalently when the frequency is{{sfn|Halliday|Resnick|Walker|2005|p=457}}{{sfn|Serway|Faughn|1992|p=478}}
:<math> f = \frac{nv}{2L},</math>

where ''v'' is the [[speed of sound]].

Next, consider a pipe that is open at {{nowrap|''x'' {{=}} 0}} (and therefore has a pressure node) and closed at {{nowrap|''x'' {{=}} ''L''}} (and therefore has a pressure anti-node). The closed "free end" boundary condition for the pressure at {{nowrap|''x'' {{=}} ''L''}} can be stated as {{nowrap|''∂(Δp)/∂x'' {{=}} 0}}, which is in the form of [[Wave equation#The Sturm–Liouville formulation|the Sturm–Liouville formulation]]. The intuition for this boundary condition {{nowrap|''∂(Δp)/∂x'' {{=}} 0}} at {{nowrap|''x'' {{=}} ''L''}} is that the pressure of the closed end will follow that of the point to its left. Examples of this setup include a bottle and a [[clarinet]]. This pipe has boundary conditions analogous to the string with only one fixed end. Its standing waves have wavelengths restricted to{{sfn|Halliday|Resnick|Walker|2005|p=457}}
:<math> \lambda = \frac{4L}{n}, </math>
:<math> n = 1, 3, 5, \ldots, </math>

or equivalently the frequency of standing waves is restricted to{{sfn|Halliday|Resnick|Walker|2005|p=458}}{{sfn|Serway|Faughn|1992|p=478}}
:<math> f = \frac{nv}{4L}.</math>

For the case where one end is closed, ''n'' only takes odd values just like in the case of the string fixed at only one end.

[[File:Molecule2.gif|thumb|200px|upright|Molecular representation of a standing wave with {{nowrap|''n'' {{=}} 2}} for a pipe that is closed at both ends. Considering longitudinal displacement, the molecules at the ends and the molecules in the middle are not displaced by the wave, representing nodes of longitudinal displacement. Halfway between the nodes there are longitudinal displacement anti-nodes where molecules are maximally displaced. Considering pressure, the molecules are maximally compressed and expanded at the ends and in the middle, representing pressure anti-nodes. Halfway between the anti-nodes are pressure nodes where the molecules are neither compressed nor expanded as they move.]]

So far, the wave has been written in terms of its pressure as a function of position ''x'' and time. Alternatively, the wave can be written in terms of its longitudinal displacement of air, where air in a segment of the pipe moves back and forth slightly in the ''x''-direction as the pressure varies and waves travel in either or both directions. The change in pressure Δ''p'' and longitudinal displacement ''s'' are related as{{sfn|Halliday|Resnick|Walker|2005|p=451}}
:<math> \Delta p = -\rho v^2 \frac{\partial s}{\partial x}, </math>

where ''ρ'' is the [[density]] of the air. In terms of longitudinal displacement, closed ends of pipes correspond to nodes since air movement is restricted and open ends correspond to anti-nodes since the air is free to move.{{sfn|Halliday|Resnick|Walker|2005|p=457}}{{sfn|Serway|Faughn|1992|p=477}} A similar, easier to visualize phenomenon occurs in longitudinal waves propagating along a spring.<ref>{{cite AV media| last = Thomas-Palmer| first = Jonathan| date = October 16, 2019| title = Longitudinal Standing Waves Demonstration| url = https://www.flippingphysics.com/standing-wave-longitudinal.html | access-date = August 23, 2020 | publisher = Flipping Physics| time = 4:11| id = YouTube video ID: 3QbmvunlQR0}}</ref>

We can also consider a pipe that is closed at both ends. In this case, both ends will be pressure anti-nodes or equivalently both ends will be displacement nodes. This example is analogous to the case where both ends are open, except the standing wave pattern has a {{frac|π|2}} phase shift along the ''x''-direction to shift the location of the nodes and anti-nodes. For example, the longest wavelength that resonates–the fundamental mode–is again twice the length of the pipe, except that the ends of the pipe have pressure anti-nodes instead of pressure nodes. Between the ends there is one pressure node. In the case of two closed ends, the wavelength is again restricted to
:<math> \lambda = \frac{2L}{n}, </math>
:<math> n = 1, 2, 3, \ldots, </math>

and the frequency is again restricted to
:<math> f = \frac{nv}{2L}.</math>

A [[Rubens tube]] provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends.<ref>{{cite AV media| last = Mould| first = Steve| date = April 13, 2017| title = A better description of resonance| url = https://www.youtube.com/watch?v=dihQuwrf9yQ | access-date = August 23, 2020 | publisher = YouTube| time = 6:04| id = YouTube video ID: dihQuwrf9yQ}}</ref>