Vibration of Strings and Air Columns

The particles of a medium undergo multiple simultaneous displacements when multiple waves arrive at the exact location in the medium simultaneously.

A new wave motion is created due to the vector sum of all the displacements (since displacement is a vector). A wave superposition happens when two or more waves are combined to create a new wave. When two waves are superimposed on top of one another, a particle’s final position is determined by the vector sum of the individual displacement each wave imparts. This is the principle of superposition. 

We will discuss stationary waves and other vital concepts in the following segment.

What are Stationary waves?

• Static waves are produced by superimposing two waves travelling in opposite directions with the same frequency and amplitude.

• There are nodes and antinodes in the human body. The displacement and strain of medium particles are equal in some places. It is referred to as a “node”.

Particles of a medium in other places exhibit maximal displacement and zero strain. 

Antinodes are the technical term for these areas. They occupy a node-intermediate position.

The stationary wave’s half-wavelength is the distance between any two adjacent nodes or antinodes. One-fourth of the stationary wave’s wavelength is the distance between a node and the nearest antinode. 

As a result, it is denoted as:

NN = AA = λ/2, NA = λ/4.

String stretching produces transverse stationary waves:

• Assume that a string is fixed at both ends A and B. Pluck it at the centre at C. A commotion is sparked.

• Left untied, the string falls, and the disturbances travel to the endpoints A and B. Reflections from the fixed ends A and B go back through the string.

Nodes are generated at the fixed ends A and B, where displacement is zero and strain is at its highest. You get an anode when you pull the string down in the middle because the displacement and strain are at their highest points.

A string vibrates in two segments when held at one point and plucked at another—the halfway point of C and B. In the same way, it can vibrate in three distinct sections. 

Overtones, harmonics, and the fundamental:

• As the body vibrates, it emits multiple frequencies. Each one of the known frequencies is a multiple of some fundamental frequency.

• The first harmonic of fundamental refers to the frequency with the lowest magnitude. The second harmonic is twice the fundamental frequency; the third harmonic is three times the fundamental frequency, and so on; when the fundamental frequency is divided by n, the nth harmonic result.

• Overtones refer to all frequencies other than the fundamental. The first, second, and third overtones are the first, second, and third overtones. As a result, the second harmonic produces the first overtone, while the second overtone is produced by the third. Without a second harmonic, the third harmonic acts as the primary overtone.

The length of time and the frequency of vibrations of a stretched string:

Tension T can be applied to the string at its two ends to make it longer. You can pluck it in the middle at C and leave it free. A string of transverse stationary waves travels over their entire length. The formula, which gives the transverse wave velocity, is mentioned below:

This frequency is known as the second harmonic. It becomes the first overtone. When the string vibrates in three segments, then the length of each segment becomes l/3.

and  

V2 = 12l3(T/m) = 32l (T/m) = 3 v

This frequency is called the third harmonic. It becomes the second overtone. In general, when using the string it vibrates in p segments, then.

V= P2l (T/m)

Which becomes harmonic (p).

Therefore, both even and odd harmonics are produced in a stretched string.

A stretched string’s transverse vibrations:

We know that the fundamental vibrational frequency of a string that has a mass per unit length of length m and is stretched at a constant T is v.

The formula is given by:

v = 12l (T/m)

Vibrations that are unforced, forceful or resonant:

1. Free vibrations:  Free vibrations are the self-repeating vibrations of a vibrating body. When this occurs, the body’s natural frequency describes its vibrating frequency.

2. Forced vibrations: Forced vibrations result from a periodic force acting on a vibrating body. When a periodic force is applied, the body vibrates at the same frequency as the force and only as long as the force is applied. When the periodic force is removed, the body either ceases to vibrate or vibrates at a frequency consistent with its own.

3. Vibrations that resonate: Resonant vibrations result from a vibrating body’s vibrations being influenced by another vibrating body of the same frequency that is kept close by. Resonance is the name given to this phenomenon. Even if the second body is removed, the primary vibrating body will continue to vibrate. Example: When a paper rider falls on a sonometer’s wire, the vibrations are resonant (the wire and tuning fork are in perfect harmony). The frequency of both is equal.

Conclusion

To summarise, everything has a natural frequency or a group of natural frequencies it vibrates at when struck, plucked, strummed or otherwise disturbed. The material determines an object’s constructed frequency and length, which impacts its travel speed (this affects the wave’s wavelength).