Fundamental Modes of Vibration

Introduction

Standing Waves (Waves that are still in motion) 

When a wave (sound, heat, light, etc.) is restricted to a finite region of space (string, pipe, hollow, etc. ), it produces a spectrum of vibrating patterns known as “standing waves”. A wave’s frequency is quantised when it is constrained.

The existence of stationary states (energy levels) in atoms and molecules is explained by standing waves, which explain how musical instruments produce sound. When a guitar string is plucked, a violin string is bowed, and a piano string is hammered, standing waves are generated.

The most minor component of a complicated vibration, or the lowest frequency at which an oscillation occurs. Overtones or harmonics are whole-number multiples of the fundamental frequency seen in various pulsing variables. The simple periodic expansion and contraction of the star’s outer layers, for example, is the primary mode of radial pulsation in stars like RR Lyrae and Cepheids. There are other more sophisticated vibrational modes, such as the first overtone mode or harmonic, and so on.

Consider two waves on a string that have the same amplitude, frequency, and wavelength but move in opposing directions. We express the resultant displacement of the string as a function of time as y(x,t) = A sin(kx – ωt) + A sin(kx + ωt) = 2A sin(kx)cos(ωt). using the trigonometric identities sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

Because the location and time dependency have been separated, this wave is no longer a travelling wave. The string oscillates in phase or 180o out of phase in all portions. The segment of the string at x oscillates with a 2A sin(kx). The string does not carry any energy. There are portions that oscillate with the greatest amplitude and sections that don’t oscillate at all. 

Modes 

A mass on a spring has a single natural frequency at which it oscillates freely up and down. A stretched string with fixed ends may vibrate at a wide range of frequencies and patterns.

Standing Waves or Normal Modes are the string’s distinctive “Modes of Vibration.” The term “standing wave” refers to the fact that each normal mode possesses “wave” attributes (wavelength, frequency f), but the wave pattern (sinusoidal shape) does not go left or right across space, but rather “stands” motionless. Each wave pattern segment (λ /2 arc) simply oscillates up and down. Each segment sweeps out a “loop” during its up-down action.

The string’s points vibrate at the same frequency but with varying amplitudes. Nodes are points that do not move (zero amplitude of oscillation). Antinodes are the points when the amplitude is at its highest. A standing wave’s mathematical equation is y(x,t) = sin(2πx/λ) cos(2πft).  (2πx/λ)  is a “shape” word that represents the sinusoidal form of a wavelength wave pattern. The up-and-down oscillatory motion of each wave segment at frequency f is described by the “flip-flop” term cos(2πft). Each mode is distinguished by a separate λ  and f.

Fundamental Mode of Vibration 

The basic mode, or first harmonic, is the simplest normal mode, in which the string vibrates in a single loop. It is denoted n = 1. The second harmonic is the second mode (n = 2), which involves the string vibrating in two loops. n vibrating loops make up the nth harmonic. The harmonic spectrum is the collection of all normal modes with n = 1, 2, 3, 4, 5,…  and {f1, f2, f3, f4, f5…} are the natural frequency spectrum.

It’s worth noting that mode n’s frequency, fn, is just a whole-number multiple of the fundamental frequency: fn = nf1. The three-loop mode vibrates three times faster than the single-loop mode.

Nodes and Antinodes of Standing Wave: 

A node is a place on a standing wave where the amplitude is the smallest. The nodes at the ends of a vibrating guitar string, for example, are nodes. The guitarist alters the effective length of the vibrating string and hence the note played by shifting the location of the terminal node through frets. An anti- node, or point where the amplitude of the standing wave reaches its maximum, is the polar opposite of a node. These appear in the middle of the nodes.

1. A standing wave’s amplitude does not remain constant throughout the wave.
2. Because it is a function of x, it is always changing.
3. The value of amplitude is maximal at certain points and zero at  others.
4. Nodes: Nodes are the points when the amplitude is zero.
5. Antinodes: Antinodes are the points where the amplitude is at its highest.

Nodes:-

1.The amplitude at nodes is zero.

2.The amplitude of a standing wave is given as: 2asinkx

=> 2asinkx = 0 ,

=> sinkx = 0,

=> sinkx =sin n π 

=>  kx=n π

3.The location of nodes with amplitude of 0 is represented by the value of x.

x=(nπ)/k … (i)

4.The equation k=(2π)/λ… .. (ii)

x=(n λ)/2 from(i) and (ii), where n=1, 2, 3…, represents the location of nodes.

Note : That two successive nodes are separated by half a wavelength(λ/2) .

Antinodes:-

1.Antinodes have the highest amplitude.

2 .The amplitude of a standing wave is expressed as:  2asinkx

=> 2asinkx = maximum. 

 Only when sinkx=1 does this value reach its maximum.

=>  sinkx = sin(n+(1/2))

 => π = kx=(n+(1/2))

 =>π = ((2 π)/ λ) x= (n+(1/2)) π

x= (n+(1/2))( λ/2)  represents the location of nodes; n=0, 1, 2, 3, 4…

Observation: –

Two successive nodes are separated by half a wavelength.

Antinodes are nodes that are half way between two other nodes.

Conclusion:

The input signals are altered one at a time in fundamental-mode operation, and only when the circuit is stable. A string or an air column can be vibrated throughout its whole length to generate the fundamental, or the musician might choose a higher harmonic. The fundamental is the entire wave’s vibration frequency. A new wave pattern known as a standing wave pattern results from the interference of the two waves.