Wave Motion and String Waves Concept Page - 13

Result

Forced vibrations vs resonant vibrations

Forced vibrations	Resonant vibrations
Vibrations under external periodic force.	Vibrations under external periodic force and natural frequency of vibrations.
Amplitude is usually small.	Amplitude is very large.
Vibrations are not in phase with the driving force.	Vibrations are in phase with the external periodic force.
Vibrations last for a small time.	Vibrations last very long.

Diagram

Amplitude v/s frequency curve of a system

Definition

Describe the disadvantages of resonance

The disadvantages of resonance are as follows:

• A negative effect of resonance could be the effect of waves hitting a rock face. The vibration of kinetic energy from the wave resonates through the rock face causing cracks and eventually great slabs of the cliff fall into the sea.
• In a car crash a passenger may be injured because there chest is thrown hard against the seat belt. the vibration can burst blood vessels. Bruising of the skin or internal organs such as concussion of the brain can be caused because of the vibration of a movement. 
• If you are shot with a bullet the resonance of the bullet hitting the body can cause liquefaction of the internal organs, this can also occur if you are near a loud explosion. The vibration of the explosion will apparently burst blood vessels and liquefy some of the organs.
• When bridge builders get it wrong and the wind causes it to resonate at its own frequency causing it to tear its self apart like the Tacoma Bridge in 1940.
• Shattering glass when a high pitched sound is played.

Definition

Characteristics of a standing wave

• In standing waves, there are certain points called nodes where the particles are permanently at rest and certain other points called antinodes where the particles vibrate with maximum amplitude. The nodes and antinodes are formed alternately.

• The amplitude of vibration increases gradually from zero to maximum from a node to an antinode.

• Particles in same phase reach same transverse displacement simultaneously

• Velocity and acceleration of all the particles separated by a distance of wavelength, are the same at a given instant.

• There is no net transport of energy in the medium.

• Compressions and rarefactions do not travel forward as in progressive waves. They appear and disappear alternately, at the same place.

• During each vibration, all the particles pass simultaneously through their mean positions twice, with maximum velocity which is different for different particles.

Formula

State the parameters in the equation of a standing wave

Equation of standing wave is:
x=Asinkx coswt$x=A\sin kx\cos wt$
Where
A$A$ : amplitude
k$k$ : wave number
w$w$ : frequency

Definition

Progressive Waves

The disturbance produced in the medium travels onward, it being handed over from one particle to the next. Each particle executes the same type of vibration as the preceding one, though not at the same time, the amplitude is same but phase changes continuously.

Example

Properties of Waves

The particles of the medium vibrate simple harmonically along the direction of propagation of the wave.

1. All the particles have the same amplitude, frequency and time period.

There is a gradual phase difference between the successive particles.

All the particles vibrating in phase will be at a distance equal to nA.. Here n = 1,2, 3,

etc. It means the minimum distance between two particles vibrating in phase is equal to the wavelength-

The velocity of the particles is maximum at their mean position and it is zero at their  Extreme positions.

Example

Solve problems on frequencies of stringed musical instruments

Example: A string is stretched between fixed points separated by 75.0 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz. There are no other resonant frequencies between these two. Then, what is the lowest resonant frequency for this string ?

Solution:
f1=n2lTμ−−√$f_1={\displaystyle \frac{n}{2l}}\sqrt{{\displaystyle \frac{T}{\mu }}}$
f1−f2=12lTμ−−√$f_1-f_2={\displaystyle \frac{1}{2l}}\sqrt{{\displaystyle \frac{T}{\mu }}}$
105=12lTμ−−√$105={\displaystyle \frac{1}{2l}}\sqrt{{\displaystyle \frac{T}{\mu }}}$
lowest frequency is f=12lTμ−−√=105 Hz$f={\displaystyle \frac{1}{2l}}\sqrt{{\displaystyle \frac{T}{\mu }}}=105H_z$

Diagram

Experimental setup of the Sonometer

A Sonometer is a device for demonstrating the relationship between the frequency of the sound produced by a plucked string, and the tension, length and mass per unit length of the string.

Definition

Use the Laws of Transverse Vibrations of a String in the setup of a sonometer

Velocity of a transverse wave travelling in stretched string is given by: v=Tμ−−√$v=\sqrt{{\displaystyle \frac{T}{\mu }}}$, where T$T$ is the tension in the string and μ$\mu$ is the mass per unit length.
Now, the fundamental frequency of the stretched string is given by: f=12lTμ−−√

BookMarks

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