Physics of Stationary Waves, Vibrations, and Harmonics Notes on Harmonics

Phase Relations in Loops: All the particles located between two adjacent nodes (which constitutes one individual loop) vibrate in phase.
- There is absolutely no progressive change of phase when moving from one particle to another within the same loop.
- Every particle in a specific loop occupies the same phase of oscillation at any given time.
- The phase of oscillation reverses completely (changes by $180^{\circ}$ ) for the adjacent loop.
Musical Applications: The principle of the formation of stationary waves (or standing waves) serves as the fundamental basis for various musical instruments, including the violin and the tanpura.

Example 6.3 Problem Statement: Find the distance between two successive nodes in a stationary wave on a string vibrating with a frequency of $64\,\text{Hz}$ . The velocity of the progressive wave that resulted in the stationary wave is $48\,\text{m s}^{-1}$ .
Given Data:
- Speed of wave ( $v$ ) = $48\,\text{m s}^{-1}$
- Frequency ( $n$ ) = $64\,\text{Hz}$
Calculation of Wavelength ( $\lambda$ ):
- We use the relation: $v = n\lambda$
- $\lambda = \frac{v}{n}$
- $\lambda = \frac{48}{64}$
- $\lambda = 0.75\,\text{m}$
Distance between successive nodes:
- The distance is defined as $\frac{\lambda}{2}$ .
- $\text{Distance} = \frac{0.75}{2}$
- $\text{Distance} = 0.375\,\text{m}$

1. Movement of Disturbance:
- Progressive Wave: The disturbance travels from one region to another with a definite velocity.
- Stationary Wave: The disturbance remains confined to the region where it is produced; the velocity of the wave is effectively zero.
2. Amplitude Distribution:
- Progressive Wave: The amplitudes of all vibrating particles are identical.
- Stationary Wave: The amplitudes of the particles are different, ranging from zero at nodes to maximum at antinodes.
3. Mean Position Crossing:
- Progressive Wave: Particles do not cross their mean positions simultaneously.
- Stationary Wave: All particles cross their mean positions simultaneously.
4. Particle Motion:
- Progressive Wave: All particles of the medium are in motion.
- Stationary Wave: Particles located at the positions of the nodes are always at rest.
5. Energy Transmission:
- Progressive Wave: Energy is transmitted through the medium from one region to another.
- Stationary Wave: There is no transfer of energy across the medium.
6. Phase and Direction:
- Progressive Wave: The phases of adjacent particles are different.
- Stationary Wave: All particles between two consecutive nodes move in the same direction and are in phase. Conversely, particles in adjacent loops move in opposite directions and differ in phase by $180^{\circ}$ .

Scenario 1: Simple Pendulum: What happens if a simple pendulum is pulled aside and released?
- Response Context: It performs free vibrations at its natural frequency.
Scenario 2: Guitar String: What happens when a guitar string is plucked?
- Response Context: It performs free vibrations at its natural frequency determined by its physical properties.
Scenario 3: Industrial/Appliance Vibrations: Have you noticed vibrations in a drill machine or in a washing machine? How do they differ from the pendulum or guitar string?
- Response Context: These are forced vibrations maintained by an external periodic force, unlike the decaying free vibrations of a pendulum.
Scenario 4: Resonance and Table Tops: A vibrating tuning fork of a certain frequency is held in contact with a table top. Then, another vibrating tuning fork of a different frequency is held on the table top. Are the vibrations produced in the table top the same for both? Why?
- Response Context: No, the vibrations are different because the table top is undergoing forced vibrations. It is forced to vibrate at the specific frequency of whichever tuning fork is in contact with it.

Natural Frequency: The frequency at which an object tends to vibrate when hit, plucked, or disturbed without the continued influence of an outside force.
Free Vibrations:
- Definition: Occur when a system is given an initial displacement and the force is luego withdrawn, allowing the body to vibrate on its own.
- Energy Dynamics: The system continuously loses energy due to the frictional resistance of the surrounding medium.
- Amplitude: The amplitude decreases over time until the vibrations eventually stop and the body comes to rest.
Forced Vibrations:
- Definition: Occur when an external periodic force is applied to a body whose natural period differs from the period of the external force.
- Frequency: The body is forced to vibrate at the frequency of the externally impressed force, not its natural frequency.
- Amplitude Factors: The amplitude of forced vibrations depends on the difference between the frequency of the external force and the natural frequency of the body.
  - Small difference = large amplitude.
  - Large difference = small amplitude.
Resonance:
- When the frequency of the external source matches the natural frequency of the system exactly, resonance occurs.
- The amplitude of vibration reaches its maximum value.
- In systems like strings or air columns, resonance results in the production of a much louder sound.

Formation Conditions: When a string or air column vibrates, waves reflect from the ends, forming stationary waves. The possible wavelengths/frequencies are constrained by boundary conditions.
Normal Modes of Oscillation: The specific possible frequencies of vibration allowed by the system's constraints.
Fundamental Frequency: The minimum possible frequency of vibration. It is also referred to as the first harmonic.
- Fundamental Mode/Tone: The mode of oscillation corresponding to the fundamental frequency.
Overtones: Terms used to represent frequencies higher than the fundamental frequency.
- The first frequency higher than the fundamental is the first overtone.
- The next is the second overtone, and so on.
Harmonics: A term used specifically when the frequency of a particular overtone is an exact integral multiple of the fundamental frequency.
- In strings and air columns, overtone frequencies are integral multiples of the fundamental, thus they are harmonics.
- Note: Not all harmonics may be present in a given sound. Overtones are only those multiples that are actually physically present in the produced sound.

Boundary Conditions for Air Columns:
- Closed End: Must be a node because air particles have almost no freedom to move.
- Open End: Must be an antinode because particles are comparatively free to move.
Physical Nature of End Correction:
- The antinode is not formed exactly at the open end of the pipe; it forms slightly beyond the physical end.
- This occurs because the air is more free to vibrate just outside the pipe than inside.
- Because air particles at the plane of the open end are not free to move in all directions, reflection actually occurs at a small distance outside the pipe.