Waves in a string

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, .

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).

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

The angular frequency is .

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 :

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).