• An extremely important case of wave reflection is the reflection of a pulse travelling along a string when it reaches the end of the string. Indeed, this is the prototype of wave reflection in one dimension. For many kinds of waves one can reason in ways similar to those found in the following sections.
• Note that there is obviously reflection when the end is fixed, but also (perhaps less obviously) when it is completely free to move. However, the ways in which reflection occurs differ:

1. from the constrained end the wave is reflected with phase inversion, that is, upside down
2. from the free end the wave is reflected without phase inversion, that is, unchanged

• In the following sections we examine these types of reflection, and for each we propose three ways of explaining the phenomenon, all equivalent to the analysis of the boundary conditions:
  1. the balance of forces at the boundary
  2. the point of view of D'Alembert, or of travelling waves
  3. the point of view of impedance

Reflection from a constrained end

As can be seen in the animation, the wave is not only reflected but also undergoes a change of "phase" (mathematicians say: "the phase has changed by 180°"). Here are the explanations of the phenomenon according to the three approaches:

1. as the pulse approaches the fixed end, the tension that the string exerts on the end point (and hence on the wall holding it still) is a force directed upwards. The wall, by the principle of action and reaction, reacts by exerting on the string a force directed downwards, which causes the phase inversion.
2. Alternatively, we can use "d'Alembert's solution": the solution of the wave equation can always be written as the sum of a wave propagating forwards and one propagating backwards. Calling c the propagation speed along the string, ${\displaystyle \psi }$ the shape of the string in time, and F and G two arbitrary functions, we have (see D'Alembert's solution and wave equation in a string)

   ${\displaystyle F(x-ct)+G(x+ct)=\psi (x,t)}$.

   We place the origin of the coordinates ${\displaystyle x=0}$ at the fixed end. It must stay fixed at every instant, so we have, for every t,

   ${\displaystyle F(-ct)+G(ct)=0}$,

   from which

   ${\displaystyle \psi (x,t)=F(x-ct)-F(-x-ct)}$.

   So the wave can be thought of, at every point and at every instant, as made up of two superposed waves: ${\displaystyle F(x-ct)}$ represents a wave travelling forwards (or progressive wave), while ${\displaystyle -F(-x-ct)}$ represents a wave travelling backwards (or regressive wave), with opposite phase (the minus sign in front) and with its shape reversed from right to left. On the page interference between reflected waves you can see an animation of how this superposition takes place in time.
3. we can consider the string with a fixed end as a limiting case of two strings 1 and 2 joined at a point, where string 2 has infinite density. But the propagation speed of waves in a medium of infinite density is zero (see speed of mechanical waves). We can then treat the problem as a limiting case of refraction, in which medium 2 has infinite impedance. For the details of the calculation see rifrazione in una dimensione (in Italian).

Reflection from a free end

To simulate the free end, imagine that the string ends with a ring of negligible mass that can slide without friction along the vertical direction. Here are the explanations of the phenomenon according to the three approaches:

1. the end of the string is completely free to move in the vertical direction; in this direction it is not subject to any force from the wall. Since that force arises as a reaction to the tension of the string exerted on the free end, we conclude that this tension must always be horizontal. Since this tension is directed along the tangent to the string, we conclude that at every instant the free end must have a horizontal tangent.
2. Alternatively, we can again use "d'Alembert's solution": the solution of the wave equation can always be written as the sum of a wave propagating forwards and one propagating backwards. Calling c the propagation speed along the string, ${\displaystyle \psi }$ the shape of the string in time, and F and G two arbitrary functions, we have (see D'Alembert's solution and wave equation in a string)

   ${\displaystyle F(x-ct)+G(x+ct)=\psi (x,t)}$.

   We place the origin of the coordinates ${\displaystyle x=0}$ at the free end. This time the condition to be imposed to always have a horizontal tangent is

   ${\displaystyle F^{\prime }(-ct)+G^{\prime }(ct)=0}$, that is, ${\displaystyle G(u)=F(-u)+\text{<mrow data-mjx-texclass="ORD">const</mrow>}}$,

   from which, apart from the constant, which only indicates the vertical position of the whole string,

   ${\displaystyle \psi (x,t)=F(x-ct)+F(-x-ct)}$.

   So the wave can be thought of, at every point and at every instant, as made up of two superposed waves: ${\displaystyle F(x-ct)}$ represents a wave travelling forwards (or progressive wave), while ${\displaystyle F(-x-ct)}$ represents the same wave, reversed from right to left, travelling backwards (or regressive wave). On the page interference between reflected waves you can see an animation of how this superposition takes place in time.
3. we can consider the string with a free end as a limiting case of two strings 1 and 2 joined at a point, where string 2 has zero density. But the propagation speed of waves in a medium of zero density is infinite (see speed of mechanical waves). We can then treat the problem as a limiting case of refraction, in which medium 2 has zero impedance. For the details of the calculation see rifrazione in una dimensione (in Italian).

How do standing waves form in a string?

Frequency of vibration of the standing wave

The standing wave oscillates with a well-defined frequency. What does this frequency depend on? The animation leaves no doubt:

• on the distance between two successive maxima of each of the two travelling waves (that is, on the wavelength ${\displaystyle \lambda }$; in the standing wave, two consecutive antinodes are instead ${\displaystyle \lambda /2}$ apart)
• on the speed c with which the waves "come towards each other".

The oscillation has a period equal to the time needed for each of the two waves to advance by one wavelength, that is

${\displaystyle T=\frac{\lambda }{c}}$

and therefore, once again, a frequency equal to

${\displaystyle f=\frac{c}{\lambda }}$

Check your understanding

Further reading and links

• The result just obtained is fundamental because it shows that the oscillation frequency of a string depends only on the wavelength and on the propagation speed of the wave. In musical instruments, the oscillation frequency of the string determines the pitch of the note produced.
• To see how the size of the string selects the wavelengths, visit the sections interference between reflected waves and Normal modes of a string.
• To see how the physical properties of the string (tension, thickness) determine the speed of the wave, visit the section speed of mechanical waves or, for further details, the section wave equation in a string
• For the frequencies selected by a string, visit the section natural frequencies of the string
• If you are interested in seeing the incredible ways in which a string can vibrate, you cannot miss the section Wave synthesis
• Reflection of sound waves also occurs in tubes, which are the basic building block of many wind instruments. The reflection of the pressure wave at an open end of a tube is equivalent to reflection from a fixed end of the string, while reflection at a closed end of a tube is equivalent to reflection from a free end of a string. If, instead of the pressure, we consider the displacement of the air, an open tube corresponds to a free string, and a closed tube corresponds to a fixed string. All this is explained in detail on the page reflection of sound. You can visualize and interact in real time with these configurations using the 2D Waves app, or by following one of our guided paths.