VISUAL PHYSICS ONLINE

WAVES

RESONANCE

Forced Vibrations of a String

 

 

STANDING WAVES

 

When a taut string fixed at both ends is disturbed, travelling waves propagate until they reach the ends where they are reflected and travel back along the string. Each such reflection gives rise to waves travelling in opposite directions. The waves along the string interfere with each other according to the principle of superposition.

 

Standing waves can be setup along the string where all points execute simple harmonic motion with the same phase and frequency.

 

A standing wave is described by

(1)

For a string of length , to satisfy the boundary conditions at and , the propagation must have a set of discrete values

(2)

Hence, the allowed wavelengths of the string are

(3) condition for standing waves

For a standing wave, multiples of must fit into the length . Fixed points that have the largest displacement along the string are called antinodes and points which have the zero amplitude are called nodes. The antinodes are spaced at intervals apart and so the nodes are also spaced at intervals.

The speed of propagation of the wave along the string depends upon the string tension and linear density

(4)

 

Using equations (1), (2) and (3), it is clear that the frequencies for the vibration the standing waves in the string form a harmonic series where is called the fundamental or the 1st harmonic

(5)

 

and is the nth harmonic. The harmonics are referred to as the natural frequencies of vibration of the system.

 

 

RESONANCE

 

In general, whenever a system capable of oscillating is acted upon by a series of impulses (driving force) having frequency equal to nearly equal to one of the natural frequencies of oscillation of the system, then, the system can oscillate with relatively large amplitude. This phenomenon is called resonance and the system is said to resonate.

 

 

SIMULATIONS

 

Image a taut string being clamped at one end (node) but shaken up and down at a constant frequency with a very small amplitude at the other end so that we can approximate this end as a node as well.

In the modelling of the vibrating string driven by an external force acting on it, the driven end of the string behaves approximately as a node and all dissipative forces are ignored. When the driving force matches any of the natural frequencies of vibration of the string, energy is continually being transferred from the driver to the string, as a consequence the amplitude of the oscillation grows and grows with time.

 

 

Exercise

 

String parameters:

length L = 0.800 m

string tension FT = 400 N

Linear density m = 4.00x10-4 kg.m-3

 

        Show that the fundamental frequency is f1 = 625 Hz and the speed of the transverse waves along the string is v = 1000 m.s-1.

 

        Calculate the natural frequencies of vibration for the first 6 harmonics.

 

        Sketch the shape of the taut string for these 6 normal modes.

 

        If a string is driven at a frequency , what is the frequency of vibration of the string ?

 

PREDICT OBSERVE EXPLAIN (P O E)

Predict the responses of the string driven at a frequency of 500 Hz, 625, Hz, 1000 Hz, 1250 Hz, 1875 Hz, 2000 Hz, 2500 Hz, 3125 Hz, and 3750 Hz.

 

Observe the animations of the driven string.

 

Compare your predictions with your observations and Explain any discrepancies.

 

 

 

Make your predictions before viewing the animation below

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For a true standing wave, the nodes occur at fixed locations along the string and are points where the string is permanently at rest. Hence, energy is not transported along the string to the right or left. The energy remains standing in the string as energy is continually being transferred between elastic potential energy and vibrational kinetic energy in the region between the nodes. Also, between nodes, all point vibrate in-phase.

 

In our simulation of the strings vibrations the end at x = 0 is not a true node since this end of the string does not remain permanently at rest. If you carefully observe the animated motion of the string you will also see that there are no true nodes. There cant be true nodes as energy must flow along the string from the stimulus driving the oscillations at the end of the string. Energy is continually added to the string by a vibrator, building up the amplitude of the oscillation only at or near one of the natural frequencies of vibration. When the string is excited at a frequency well away from a natural frequency, the reflected waves reaching the end of the string at x = 0 are out of phase with the external vibrator, hence energy can be transferred from the string back in to the vibrator. The wave pattern is not fixed but wiggles about. On average, the amplitude of the oscillation is small and not much different from that of the vibrator, and the string vibrates at the driving frequency. Hence, the string absorbs maximum energy from the vibrator at resonance. Tuning a radio is an analogous process: the circuit resonates with the transmitted signal and absorbs peak energy from the radio signal.

 

 

 

 

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If you have any feedback, comments, suggestions or corrections please email:

Ian Cooper School of Physics University of Sydney

ian.cooper@sydney.edu.au