Chapter 8

Mechanical waves

Waves in a string

ComplémentA video (in french) about waves in a string (with some fun science experiments)

Waves in a string

FondamentalTransverse vibrations of a string ; d'Alembert's wave equation

Consider an inextensible rope, linear density , stretched horizontally with a constant force .

At equilibrium, the cord is horizontal.

It is assumed in the following that gravity is not involved (if not, the shape of the rope is a chain).

We will study the small movements in the vicinity of this equilibrium, with the following model :

  • The element of rope located at coordinates at equilibrium is at coordinates at non equilibrium. Alternately, we can say we neglect its displacement along (Ox).

  • The angle created by the tangent to the rope at the point of abscissa at time is infinitely small ( , and , see figure).

  • If we consider a fictitious cut at the point abscissa , the action of the left side of the rope on the right is reduced to a tangential force to the rope noted .

    Similarly, the action of the right on the left side reduces to a force .

    According to the principle of reciprocal actions, .

Transverse vibrations of a string

The Newton's second law applied to a rope element located between the abscissas and gives :

In projection, and noting  :

If we limit ourselves to the order , equation provides :

Equation can be rewritten :

Yet :

Where :

We find again the d'Alembert's wave equation.

In the case of the rope, the wave is called transverse (displacement occurs along axis).

AttentionWave equation d'Alembert

We find a d'Alembert's wave equation.

The wave propagation speed is :

The tighter the rope and the lighter the rope weight are, the greater the speed is.

FondamentalThe string Melde

In the experience of Melde, the extremity abscissas of a rope is fixed ( ) and operator imposes at a harmonic movement :

The angular frequency is .

The string Melde

We are interested in forced regime (driven vibration), obtained after disappearance of the transitional regime.

It is thus sought a solution of the d'Alembert equation corresponding to a standing wave of as the same angular frequency as of that of the exciter :

The boundary conditions needed :

And :

Whence :

Is :

Therefore :

Anti-nodes and nodes of vibrations

The amplitude is maximum for :

And is, in absolute value :

This maximum amplitude becomes infinite (the rope is then in resonance) to excitatory pulses such as :

corresponding to the eigen-modes of the string.

However, the inevitable damping and stiffness of the rope make the maximum amplitude keeps a finite value.

Thus stationary wave becomes resonant (in driving regime) when the driving angular frequency coincides with a vibrational frequency (open plan) of string vibration, just like a LC series circuit where the angular frequency of vibration refers to the own frequency of free regime (free oscillation) and the angular frequency of resonance in forced regime (forced oscillation).

ComplémentA video on Melde's rope

Corde de Melde (Vidéo d'Alain Le Rille, enseignant de physique en CPGE au lycée Janson de Sailly, Paris)

ComplémentA video (in English) on mechanical waves

Un résumé sur les ondes mécaniques

ComplémentA lecture video on waves in a rope (Reference "college Physics")

Vidéo de cours sur les ondes dans une corde

SimulationJAVA animations by Jean-Jacques Rousseau (University of Le Mans)

  • Frequencies of a vibrating string : click HERE

  • Chain : click HERE

  • Elastic wave propagation : click HERE

SimulationAnimation JAVA (University of Colorado Boulder)

  • Wave on a String :

Wave on a String
Learning program
Reflection on a free mass