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Standing Waves

We normally think of waves as moving. Ripples in a pond, tsunamis washing over cities, or waves crashing onto a beach. This isn't always the case though. Standing waves are waves that have a specific pattern and are formed under certain conditions. The conditions and formation process will be explained in the article. Harmonics and fundamental frequencies will also be…

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Standing Waves

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Jetzt kostenlos anmeldenWe normally think of waves as moving. Ripples in a pond, tsunamis washing over cities, or waves crashing onto a beach. This isn't always the case though. Standing waves are waves that have a specific pattern and are formed under certain conditions. The conditions and formation process will be explained in the article. Harmonics and fundamental frequencies will also be explored.

A **standing** or **stationary** wave is a wave whose peak amplitude does not move along through a medium as a traveling wave.

A standing wave consists of **nod****es** and **antinodes **as seen in the figure below where$n$are nodes and$A$are antinodes.

nodes are fixed points on a wave that do not vibrate. Antinodes are maximum amplitude points where the wave is oscillating vertically. Antinodes form the maxima and minima of the wave. Vaia Originals.

Because standing waves do not travel, there is no transfer of energy. The energy of the traveling waves that formed the standing wave is **stored **within the standing wave.

The period of the standing wave is the time needed for an antinode to complete one full cycle of vibration. This means that the antinode oscillated from the maximum amplitude above the center line to the maximum value below the center line and back. The frequency of a wave is the number of full cycles per second.

The frequency velocity and wavelength of a wave can be found using the wave equation below, where$f$is frequency in $\mathrm{Hz}$,$v$is the velocity in$\mathrm{m}/\mathrm{s}$and$\lambda $is the wavelength in$\mathrm{m}$.

$v=f\lambda \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}velocity=frequency\xb7wavelength\phantom{\rule{0ex}{0ex}}$

The phase difference of different points on a stationary wave depends on the number of nodes between those two points.

- If the number of nodes between two points is odd, then the points are out of phase.
- If the number of nodes between the points is even then the points are in phase.

A standing wave is a wave pattern formed by the** superposition** of two or more traveling waves moving in opposite directions along the same line. **The waves must have the same frequency or wavelength, and amplitude. **Usually, a standing wave is formed by a traveling wave that reflects off a boundary and begins moving in the opposite direction. The original wave and the reflected wave interfere and create a standing wave.

**Superposition **occurs when two or more waves with the same frequency interfere at a certain point in space, the resultant displacement is the sum of the displacements of each wave.

Two waves can superpose in two ways, constructively and destructively.

- Destructive interference is the interference of two waves that are in anti-phase. This means a peak and a trough between the two waves line up which cancel each other out producing a resulting wave with a smaller amplitude.
- Constructive interference occurs between two waves in phase. This means that two peaks or two troughs line up forming a resulting wave with a larger amplitude.

Consider a wave traveling down a string with both ends of the string fixed. As it hits the end it will reflect and begin traveling in the opposite direction. If we keep sending waves down the string, the reflected waves will begin constructively and destructively interfering with the waves we are sending. If we send our waves at specific frequencies, the constructive and destructive interference will form a standing wave, where the nodes and antinodes are formed at equal spacing over the length of the string. Nodes on a standing wave are formed by destructive interference between the traveling waves and Antinodes are formed by constructive interference.

Now that we understand how standing waves are formed, we need to know how to work with them. To do this we need to figure out how to mathematically describe them. Consider two traveling waves that are moving in opposite directions but are otherwise equivalent:

${y}_{1}(x,t)=A\mathrm{sin}(kx-\omega t)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{y}_{2}(x,t)=A\mathrm{sin}(kx+\omega t)\begin{array}{}\end{array}$

Recall that$k=\frac{2\pi}{\lambda}$is the wave number, and$\omega =2\pi f$is the angular frequency. To get a standing wave we can simply take the superposition of these two waves.

$y(x,t)={y}_{1}(x,t)+{y}_{2}(x,t)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}y(x,t)=A\mathrm{sin}(kx-\omega t)+A\mathrm{sin}(kx+\omega t)$

Now recall that the sum of angles trigonometric identity is given as

$\mathrm{sin}(\alpha \pm \beta )=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)\pm \mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$

Next, we apply this identity to our equation where$\alpha =kx,$and$\beta =\omega t.$Thus

$y(x,t)=A\left[\mathrm{sin}\right(kx\left)\mathrm{cos}\right(\omega t)-\mathrm{cos}(kx\left)\mathrm{sin}\right(\omega t\left)\right]+A\left[\mathrm{sin}\right(kx\left)\mathrm{cos}\right(\omega t)+\mathrm{cos}(kx\left)\mathrm{sin}\right(\omega t\left)\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)y(x,t)=2A\mathrm{sin}\left(kx\right)\mathrm{cos}\left(\omega t\right)}$

This is our formula for a standing wave as a function of position and time.

The amplitude of a standing wave at a given time is dependent on the position you are at on the standing wave. This ranges from the maximum amplitude being a superposition of the two amplitudes of the traveling waves that make up our standing wave, or 2A, then we pass through 0 and eventually get to a minimum of -2A. You can probably see where this is going. Our Standing wave amplitude contains the sin term from our formula for a standing wave. Thus we have

${A}_{standing}=2A\mathrm{sin}\left(kx\right)$

Below we will look at two examples: sound waves and vibrating strings.

Sound waves can be produced as a result of the formation of standing waves inside an air column

Sound waves can produce standing waves in air columns. This can be visualized by placing a powder inside the air column and a loudspeaker on one end which is open. The loudspeaker will produce sound waves which will be reflected once they reach the boundary. With a traveling and a reflected wave, we get standing waves at certain frequencies. The powder inside the air column will be spaced evenly indicating visually the position of nodes.

This is how musical instruments, such as clarinets work.

Standing waves can be formed in stretched strings fixed at both ends which are subjected to tension. Consider a uniform string of length L. The fixed ends cannot move, so our standing wave must have nodes at the two ends. Thus our amplitude must be 0 on the boundaries. Recall our formula for a standing wave:

$y(x,t)=2A\mathrm{sin}\left(kx\right)\mathrm{cos}\left(\omega t\right)$

As before, our standing wave amplitude is given as

$2A\mathrm{sin}\left(kx\right)$

Thus for our amplitude to be 0 at the boundaries, we must have that

$2A\mathrm{sin}(k\xb70)=0$

and

$2A\mathrm{sin}\left(kL\right)=0$

The first equation always holds, so this does not tell us much, but the second equation implies that

$kL=n\pi \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{2\pi}{\lambda}L=n\pi \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\lambda =\frac{2L}{n}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

where $n=1,2,3,...$

Thus we have restrictions on our wavelength where standing waves will form. Due to the relationship between the wavelength and the frequency, we also have restrictions on the frequencies.

$v=f\lambda \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}f=\frac{v}{\lambda}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)f=\frac{nv}{2L}}$

Thus standing waves will only form at specific frequencies.

Vibrating strings produce sound, which is how some musical instruments work such as violins, pianos, guitars, etc.

Harmonics are different wave patterns of standing waves formed on strings with two fixed ends. The pattern of the standing wave depends on its frequency. The higher the frequency of the wave, the more harmonics appear on the wave. The simplest form of harmonic shown in the figure below is formed by the lowest frequency which is known as the** first harmonic or fundamental harmonic. **This consists of one loop formed by two nodes at its ends and a single antinode as shown below. The frequency of the first harmonic is dependent on the length of the string L, and the speed of the wave.

Similarly, **a second harmonic or a first overtone** is formed by a higher frequency and consists of three nodes and two antinodes. Finally, a third harmonic or a **second overtone** is formed by an even higher frequency and consists of four nodes and 3 antinodes.

By using our restriction on the wavelength and frequency, we can calculate the harmonics.

Harmonic | Wavelength, $\lambda \left(\mathrm{m}\right)$ | Frequency, $f\left(\mathrm{Hz}\right)$ |

1st | $2L$ | $\frac{v}{2L}$ |

2nd | $L$ | $\frac{v}{L}$ |

3rd | $\frac{2L}{3}$ | $\frac{3v}{2L}$ |

Find the fourth harmonic frequency.

Solution:

This is a straightforward application of our formula.

$f=\frac{vn}{2L}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}f=\frac{4v}{2L}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)f=2\frac{v}{L}}$

While standing and traveling waves have some similar properties, they also have several differences. The table below summarizes the differences between a standing and traveling wave.

Standing wave | Traveling wave |

There is no energy transfer as energy is stored within nodes. | Energy is propagated through a medium by oscillating particles from the equilibrium position. |

Has nodes and antinodes. | Does not have nodes or antinodes. |

The wave does not move. | The wave is propagating in a medium. |

Only antinodes vibrate, nodes are fixed points. | All particles of the wave vibrate. |

Points are either in phase or anti-phase. | Points can have a phase difference between 0 and 360 degrees. |

- Standing waves are formed by the superposition of two traveling waves that are moving in opposite directions.
- Standing waves have stationary points known as nodes and vibrate only at specific points known as antinodes.
- Standing waves can form harmonics, which is a specific wave pattern that is formed by two fixed ends.

Standing waves are caused by superposition of two traveling waves.

- Standing waves do not transfer energy .
- They have nodes and antinodes.
- They points that form a standing wave are either in phase or anti phase.

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