Wave on a vibrating cord

The vibrating cord is the physical model making it possible to represent the movements of oscillation of a tended wire. One will suppose here that it is held by its two ends, which is not always the case (in the Pendule S or wire with lead, for example, the end of bottom is free).

Being held by its two ends, the vibrations are thought of each end, there is thus a phenomenon of Standing wave .

This model makes it possible to include/understand the its S emitted by the string instruments, but also the movements which can agitate the mechanical structures like the Câble S, Caténaire S and sling S.

This simple model is also a good introduction to similar but more complex phenomena, like the sound pipes, the phenomena of vibration of the plates…

Phenomenologic approach

Let us consider a cord maintained by its two ends. In the mode of the vibration simplest, known as “fundamental”, it forms an arc at every moment, and the arrow of this arc varies in a periodic way (the curve increases, then decreases, then is reversed, then increases in the other direction…). One can thus define a frequency of vibration, and one notices that this frequency depends:

• of the linear mass of the cord (noted µ and expressed in Kilogram by Meter, kg/m);
• of the force with which one tightens this cord (tension noted T and expressed in Newton, NR);
• length of the cord (noted L and expressed in Meter, m).

If the influence of each parameter is sought:

• more the cord is light (µ is weak), plus the frequency is high (this is why the acute cords are finer);
• more the cord is tended, plus the frequency of vibration is high (from an acoustic point of view, the note rises when one tightens the cord);
• more the cord is long, plus the sound is serious.

On an instrument, each cord has a different linear mass, and one adjusts the tension for to grant. To play, one exploits the choice of the cord, and when the instrument has a handle, over the length of the cord by gripping the cord against the handle with the finger.

With regard to the length: the frequency varies like the reverse length. Thus, if one divides the length by two, one multiplies the frequency by two i.e. one assembles of a octave. One notices thus that the twelfth Frette of a guitar is in the middle of the cord (since an octave makes twelve Demi-ton S in the moderate Gamme).

But a cord can vibrate in other manners: if the ends remain fixed, the form which it takes can have two, three,… N arcs head-digs. One speaks about “mode of vibration”. If one is with the mode N , one thus has N arcs, and each arc has as a length L / N . It thus vibrates with a frequency N time higher than the fundamental one. Thus a cord can emit sounds several different heights.

In fact, the real vibration is a combination of the various modes; one speaks about “harmonics”. The amplitude of the various harmonics and a characteristic of the instrument, and determines its sonority.

Let us note that it is not only the vibration of the cord which imports, but that of all the instrument, in particular of the Caisse of resonance.

Equation of wave for a tended cord

All that follows supposes that the sound cord is without stiffness and of null diameter, which is never rigorously checked.

The cord initially at rest occupies a segment along the x axis. It is tended with a tension T (Force) applied at its 2 ends. One deforms the cord in the direction there and one releases it. Let us call there (X, T) the displacement of the cord to X-coordinate X and.

Let us write the equation of Newton (Lois of Newton) for a portion of cord plumb with the segment + \ delta x. At the ends there are the forces $\backslash \; overrightarrow\; \{F\_1\}$ and $\backslash \; overrightarrow\; \{F\_2\}$ of module T being tangentially exerted.

$\backslash \; overrightarrow\; \{F\_1\}\; =\; -\; T\; (\backslash \; cos\; \backslash \; alpha\; \backslash \; overrightarrow\; \{e\_x\}\; +\; \backslash \; sin\; \backslash \; alpha\; \backslash \; overrightarrow\; \{e\_y\})$

The small deformation is supposed, so that the angles are small:

$\backslash \; cos\; (\backslash \; alpha)\; \backslash \; simeq\; 1$

$\backslash \; sin\; (\backslash \; alpha)\; \backslash \; simeq\; \backslash \; tan\; (\backslash \; alpha)$

$\backslash \; simeq\; \{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; X\}\; (X,\; T)$ (tan = sin/cos)

$\backslash \; overrightarrow\; \{F\_1\}\; =\; -\; T\; (\backslash \; overrightarrow\; \{e\_x\}\; +\; \{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; X\}\; (X,\; T)\; \backslash \; overrightarrow\; \{e\_y\})$

Same manner, at the other end located in $x\; +\; \backslash \; delta\; x$

$\backslash \; overrightarrow\; \{F\_2\}\; =\; +\; T\; (\backslash \; overrightarrow\; \{e\_x\}\; +\; \{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; X\}\; (X\; +\; \backslash \; delta\; X,\; T)\; \backslash \; overrightarrow\; \{e\_y\})$

From where: $\backslash \; overrightarrow\; \{F\_2\}\; +\; \backslash \; overrightarrow\; \{F\_1\}\; \backslash \; simeq\; T\; \{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; X\}\; (X\; +\; \backslash \; delta\; X,\; T)\; -\; \{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; X\}\; (X,\; T))\; \backslash overrightarrow\{e\_y\}$

By the Théorème of Taylor limited to the 1st order ($\backslash \; delta\; x$ is supposed very small) one obtains:

$\backslash \; simeq\; T\; \{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; x^2\}\; (X,\; T)\; \backslash \; delta\; X\; \backslash \; overrightarrow\; \{e\_y\}$

By the equation of Newton (F=ma second law of Newton)), one will have by neglecting the force of the Pesanteur:

$\backslash \; delta\; m\; \{\backslash \; delta\; \backslash \; overrightarrow\; \{v\}\; \backslash \; over\; \backslash \; delta\; T\}\; =\; T\; \{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; x^2\}\; (X,\; T)\; \backslash \; delta\; X\; \backslash \; overrightarrow\; \{e\_y\}$

Where:

$\backslash \; delta\; m\; =\; \backslash \; driven\; \backslash \; delta\; X$

is the mass of the element of cord. At first approximation all the elements of the portion of cord have same speed

$\{\backslash \; delta\; there\; \backslash \; over\; \backslash \; delta\; T\}\; (X,\; T)$ in the direction there

one from of deduced:

$\backslash \; driven\; \{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; t^2\}\; =\; T\; \{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; x^2\}$ or:

; Equation of wave of Alembert to a dimension

$\{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; x^2\}\; =\; \{1\; \backslash \; over\; v^2\}\; \{\backslash \; delta\; ^2\; there\; \backslash \; over\; \backslash \; delta\; t^2\}$

where $v\; =\; \backslash \; sqrt\; \{T\; \backslash \; over\; \backslash \; driven\}$ µ: mass per unit of length.

It sagit of an partial derivative equation for the function with two variable there (X, T) called equation of wave.

Significance of v like propagation velocity of a deformation.

Let us suppose that the cord is infinite. In this case, a possible solution of the equation of wave is: $there\; (X,\; T)\; =\; F\; (x-vt)$ where F is an arbitrary function of a variable which is x-vt.

Note: $there\; (X,\; T)\; =\; F\; (x+vt)$ is also an acceptable solution, but corresponds to a wave being propagated in the direction of X negative.

The equation is indeed satisfied for any F. In particular, if t=0 is made, one has

$F\; (X)\; =\; there\; (X,\; 0)$: initial deformation of the cord in t=0

At the moment $t\_1$, one finds the same form but moved in $vt\_1$.

The deformation was propagated of $vt\_1$ during time $t\_1$, with a speed v, without undergoing deformation.

If, at the moment t=0, one deals with deformation of the cosine type: $F\; (X)\; =\; has\; \backslash \; cos\; (2\; \backslash \; pi\; \{X\; \backslash \; over\; \backslash \; lambda\})$ or $\backslash \; lambda$ is the wavelength, one finds at every moment:

$there\; (X,\; T)\; =\; has\; \backslash \; cos\; (2\; \backslash \; pi\; \{X\; -\; vt\; \backslash \; over\; \backslash \; lambda\})$

that one can rewrite in the form:

$there\; (X,\; T)\; =\; has\; \backslash \; cos\; (kx\; -\; \backslash \; Omega\; T)\; =\; Re\; e^\; \{(I\; K\; X\; -\; I\; \backslash \; Omega\; T)\}$

where $K\; =\; \{2\; \backslash \; pi\; \backslash \; over\; \backslash \; lambda\}$ is the number of wave and $\backslash \; Omega\; =\; \{2\; \backslash \; pi\; v\; \backslash \; over\; \backslash \; lambda\}$ is the pulsation.

The frequency is given: $\backslash \; naked\; =\; \{v\; \backslash \; over\; \backslash \; lambda\}$ and $\{\backslash \; Omega\; =\; K\; v\}$

Clean modes of vibration of a cord

Let us seek a solution of the equation of wave which is Harmonique in time, while posing

$there\; (X,\; T)\; =\; U\; (X)\; \backslash \; cos\; \backslash \; Omega\; T\; =\; Re\; (U\; (X)\; e^\; \{-\; I\; \backslash \; Omega\; T\})$

One finds thus like equation:

$v^2\; \{d^2\; U\; \backslash \; over\; dx^2\}\; =\; -\; \backslash \; Omega\; ^2\; U\; (X)$

from where $\{d^2u\; \backslash \; over\; dx^2\}\; +\; k^2\; U\; (X)\; =\; 0$

with $K\; =\; \{\backslash \; Omega\; \backslash \; over\; v\}$. The general solution of the equation above is:

$u\; (X)\; =A\; \backslash \; cos\; kx\; +\; B\; \backslash \; sin\; kx$

where has and B are two constants of integration. If the cord is length L and fixed at its 2 ends (x=0 and x=L), one must impose commes boundary conditions that U (0) = U (L) = 0. The first condition imposes which = 0 have and the second gives B sin kL = 0.

With share the commonplace solution B=0 (=> u=0, which does not have any interest), this condition is also satisfied if $kL\; =\; \backslash \; pi,\; 2\; \backslash \; pi,\dots$. A family of solutions thus is found: $u\_n\; (X)\; =\; B\_n\; \backslash \; sin\; (N\; \backslash \; pi\; \{X\; \backslash \; over\; L\})$.

For lequelles the pulsation are $\backslash \; Omega\; =\; N\; \backslash \; pi\; \{v\; \backslash \; over\; L\}$.

The corresponding frequencies are $\backslash \; naked\; =\; N\; \{v\; \backslash \; over\; 2L\}$, i.e. multiple of a fundamental frequency $\{v\; \backslash \; over\; 2L\}$ (opposite of the time of an outward journey and return along the cord).

There thus exists an infinity of clean modes of vibration, described by:

$y\; (X,\; T)\; =\; B\_n\; \backslash \; sin\; (N\; \backslash \; pi\; \{X\; \backslash \; over\; L\})\; \backslash \; cos\; (N\; \backslash \; pi\; \{vt\; \backslash \; over\; L\})$

The Amplitudes $B\_n$ are arbitrary.

The general solution of the equation of wave can be written in the form of a superposition of all the clean modes:

$there\; (X,\; T)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \{\backslash \; infty\}\; B\_n\; \backslash \; sin\; (N\; \backslash \; pi\; \{X\; \backslash \; over\; L\})\; \backslash \; cos\; (N\; \backslash \; pi\; \{vt\; \backslash \; over\; L\})$

At the moment t=0, in particular,

$there\; (X,\; 0)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \{\backslash \; infty\}\; B\_n\; \backslash \; sin\; (N\; \backslash \; pi\; \{X\; \backslash \; over\; L\})$

If one gives oneself the initial shape of the cord, i.e. if one supposes like the function $F\; (X)\; =\; there\; (X,\; 0)$, the $B\_n$ represent the coefficients of a Fourier series in sine of F (X):

$B\_n\; =\; \{2\; \backslash \; over\; L\}\; \backslash \; int\_\; \{0\}\; ^\; \{L\}\; F\; (X)\; \backslash \; sin\; (N\; \backslash \; pi\; \{X\; \backslash \; over\; L\})\; dx$.

See too

Related articles

• progressive mechanical Wave

• Standing wave
• vibrating Cord
• Experiment of Melde

External bonds

• Applet Java simulating the vibration of a cord (in English)

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