The balance of the forces acting in the y direction on the segment ${\displaystyle \Delta x}$ gives (see figure):

${\displaystyle F_{y}=T_{y}(x_{0})+T_{y}(x_{0}+\Delta x)=T\sin \alpha +T\sin \beta \!}$.

Remembering that we actually want to study a segment of string of infinitesimal length dx, we can approximate ${\displaystyle \sin }$ with ${\displaystyle \tan }$, and, remembering that the tangent to the curve y(x) is nothing but its first derivative with respect to x, write

${\displaystyle F_{y}\approx T\tan \alpha +T\tan \beta }$,

${\displaystyle \tan(\alpha )=-{\frac {\partial y(x_{0})}{\partial x}}}$

${\displaystyle \tan(\beta )={\frac {\partial y(x_{0}+dx)}{\partial x}}}$

and finally, replacing the infinitesimal difference of the first derivatives with the second derivative

${\displaystyle {\frac {\partial y(x_{0}+dx)}{\partial x}}-{\frac {\partial y(x_{0})}{\partial x}}={\frac {\partial ^{2}y(x_{0})}{\partial x^{2}}}dx}$

we obtain

${\displaystyle F_{y}\approx Tdx{\frac {\partial ^{2}y}{\partial x^{2}}}}$.

${\displaystyle {\frac {T}{\mu }}=c^{2}}$

has the dimensions of the square of a velocity.

Indeed, writing the wave equation for the string as

${\displaystyle {\frac {\partial ^{2}y}{\partial x^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}y}{\partial t^{2}}}}$,

we see that the numerators have the same dimensions, and the denominator on the left has the dimensions of the square of a length; therefore the denominator on the right must also have dimensions equal to the square of a length.

The constant c is important because it is precisely the speed at which waves propagate in the string. In our ideal case it is the speed at which all waves propagate in the string.

This is shown by observing that any function of the form

${\displaystyle F_{1}(x-ct)+F_{2}(x+ct)\!}$,

where ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are arbitrary functions, is a solution of our wave equation. But ${\displaystyle F_{1}(x-ct)}$ is nothing but an ${\displaystyle F_{1}}$ that moves in time at a constant speed c.

How it is solved

• The advantage of a linear equation is that a difficult solution can be written as a sum of simpler solutions (see the most important application of this principle in the page on Fourier's theorem)
• The equation alone is not enough to obtain the solution. One must in fact specify initial conditions, which fix the shape of the string at time ${\displaystyle t=0}$, and, if the string is not infinite, boundary conditions, which state what happens at the edge (see, for example, reflection in strings).
• There are two main routes to determining the solutions, which use equivalent but very different points of view, and which are examined in separate sections:
  1. The general solution is a superposition of normal modes.
  2. The general solution is a superposition of travelling waves
• Whichever point of view we choose to adopt, there are also several ways to actually solve the equation:
  1. analytical methods (the simplest cases are dealt with in natural frequencies of the string, standing waves in 1D, normal modes of a string)
  2. numerical methods: an example (albeit in two dimensions) is our app 2D Waves app.
• Finally, physics is essentially based on the method of experimental investigation, which in this case can be carried out by analysing the resonances of the system through an analysis of the normal modes, or by studying its evolution in the time domain.

"Less ideal" strings

In real strings the wave equation in general no longer has the form described above. Let us see how it changes if we drop, one at a time, the assumptions we made for the ideal case.

Dropping assumption no. 1

T is constant during the motion of the string (that is, whatever its deformation)

In real strings the tension T depends on the shape of the string itself, or at least on the maximum displacement ${\displaystyle y_{\rm {max}}}$ of the string from its equilibrium position. Assumption no. 1 is almost always valid as long as the maximum displacement of the string is not very large.

In mathematical terms, dropping this ideal assumption means making T a function of ${\displaystyle y_{\rm {max}}}$ (more generally a functional of y(x))

${\displaystyle {\frac {T(y_{\rm {max}})}{\mu }}{\frac {\partial ^{2}y}{\partial x^{2}}}={\frac {\partial ^{2}y}{\partial t^{2}}}}$.

Consequently, without ideal assumption no. 1, the speed of waves in the string is no longer a constant, but depends on the shape of the string itself, or at least on the size of its deformation.

${\displaystyle c(y_{\rm {max}})={\sqrt {\frac {T(y_{\rm {max}})}{\mu }}}}$.

This implies that for large displacements waves travel at a different speed than for small displacements.

• Important consequences follow, for example, for musical instruments: a larger displacement, in a string instrument, corresponds to a louder sound. If the tension of the string, for example, increases for large displacements, playing a string louder will also raise the frequency of the note produced. This requires intonation corrections when a string instrument is played softly or loudly. Alternatively, the slight detuning can be used as an effect.

• The tension can also depend on the frequency at which the string is set into oscillation

${\displaystyle c(\omega )={\sqrt {\frac {T(\omega )}{\mu }}}}$

In this case the phase velocity and the group velocity of the waves in the string differ. That is, high-frequency waves propagate at a different speed from low-frequency waves. (Remember that, for sound waves, higher frequencies generally correspond to higher-pitched sounds)

Dropping assumption no. 2

Part one: T has the same magnitude along the whole string...

This assumption is actually rather well justified if the string is homogeneous. The case of non-homogeneous strings is not of particular interest for musical instruments, or for oscillations, while it has important applications in signal transmission along electrical lines, and in transduction (see for example impedance, reflection in strings, refraction in one dimension (in Italian)).

Part two:...at every point, acts only in the direction tangent to the string (that is, without components perpendicular to the string).

Considering transverse components of the tension in the string means thinking of the string no longer as a one-dimensional system, but as a genuine three-dimensional system. Think, for example, not of a violin string but of a thick steel cable of a suspension bridge. Then we cannot neglect, for example, its torsional motion, which can take energy away from the transverse vibration modes, modifying them. See the section on waves in bars

Dropping assumption no. 3

all other external forces (such as gravity) are negligible compared with T.

If we include other forces in the equation, then the behaviour of the string depends on the nature of these forces.

In particular, it is straightforward to introduce friction forces that tend to damp the oscillations, and also external forces that keep the string vibrating by counteracting friction. These terms can easily be included in the wave equation while keeping it linear (e.g. we know that the friction term leads to the telegraph equation: see wave equation).

• Other possible modifications of the equation consist in making it nonlinear. To obtain a nonlinear equation it is enough, for example, not to replace the combination of sines with the corresponding first derivatives.

For waves described by a nonlinear equation, the superposition principle no longer holds: two waves that "collide" along this string can influence each other!

In this regime one observes an interesting variety of phenomena, some of which, although not specific to strings, are examined in the pages Nonlinear effects, Fermi-Pasta-Ulam chain, Intermodulation distortion and Modulational instability