Physics II: Chapter 1

Reference: Beginning Physics II

Chapter 1: WAVE MOTION



Wave Motion, Pulse, Amplitude, Transverse Wave, Longitudinal Wave, Velocity Of Propagation, Propagation Constant, Wave Equation, Wavelength, Wave Energy And Power, Wave Reflection, Principle Of Superposition, Interference Patterns, Standing Wave, Nodes And Anti-Nodes, Natural Frequency, Fundamental Frequency, Harmonics, Overtones, Resonant Frequencies, Resonance, Pressure & Displacement Nodes



For details on the following concepts, please consult Chapter 1.

The wave motion is a process in which the physical matter itself does not move over significant distances beyond their initial positions, while the energy can be transferred over large distances. The transferred energy can carry information, so that wave motion allows the transfer of information over large distances as well.

A pulse is a single stroke, vibration, or undulation. The molecules move perpendicular to the direction in which the pulse moves. The shape of the pulse travels as one set of molecules after another go through the vertical motion. The pulse carries the vertical kinetic energy of the moving molecules, and the associated potential energy due to momentary stretching of the cord, in the pulse region.

Amplitude is the maximum vertical displacement of the pulse.

The pulse in a cord is an example of a transverse wave, where molecules move to and fro at right angles to the direction of propagation of the wave.

In a longitudinal wave the molecules actually move to and fro along the direction of the propagation of the wave. This would be a pulse of pressure travelling through the air in a pipe. This air pulse is a primitive example of a sound wave.

The velocity of propagation vp of the pulse in a cord would increase with increase in tension S in the cord. On the other hand, it would decrease with increase in mass per unit length μ.

The propagation of sound in a solid would increase with increase in intrinsic stiffness as measured by the Young’s modulus Y, and with decrease in its density ρ.

The propagation of sound in a fluid would increase with increase in its Bulk modulus B, and with decrease in its density ρ.

We define the propagation constant for the wave as, k = ω/vp so that

The vertical position of the cord, at a definite horizontal position x along the cord, at any time t is,

This indicates that the vertical displacement of any point x along the cord, the cord exhibits SHM of the same amplitude and frequency with the term in the sine function involving x acting as a phase constant that merely shifts the time at which the vertical motion passes a given point in the cycle.

The wavelength is the spatial periodicity of the wave, i.e., the length along the x-axis that one moves to go through one complete cycle of the wave.

For the case of a transverse sinusoidal wave travelling in a cord, or a longitudinal wave travelling in a tube, the energy per unit length is,

And, the power transfer across any point of cross-section is,

When the far end of the cord is tied down, the reflection is 180° out of phase.

When the far end of the cord is not tied down but free, the reflection is in phase.

A more general case is somewhere in-between these two extremes.

The actual displacement of molecules from their equilibrium position, at any given location in a medium, at any instant of time, when more than one wave is traveling through that medium, is just the vector sum of the displacements that each would separately have caused at the same location at that same instant of time.

When two waves pass the same point in a medium they are said to interfere. If they correspond to long wave trains having the same wavelength, then certain regular patterns can appear, such as points that never move and points that move maximally. Such patterns are called interference patterns.

An examination of the actual “superimposed” wave may reveal some points on the cord that seem not to move at all as the waves pass each other, while other points midway between them move up and down with double the amplitude of either wave. The actual wave motion of the cord is therefore not a traveling wave, since in a traveling wave every point in the cord moves up and down in succession. The wave caused by the interference of these two traveling waves is therefore called a standing wave. It has the same frequency.

The points that don’t move are called nodes, and the points that move maximally are called anti-nodes.

Many physical systems, when stimulated can be made to vibrate or oscillate with definite frequencies. In each of these cases there is a single “natural” frequency associated with the system. In more complex structures, many “natural” frequencies of vibration can occur.

The fundamental frequency, f1, is the lowest possible natural frequency.

The integer multiples of the fundamental frequency, fn, are called harmonics.

The overtones are the successive natural frequencies above the fundamental.

Such “natural” frequencies, which are characteristic of the particular system or structure, are also called the resonant frequencies of the system.

If one stimulates the system at one of the resonant frequencies, one can stimulate huge amplitude oscillations, sometimes to the point of destroying the structure. This is because, when stimulating a system at a resonant frequency it is extremely easy to transfer energy to the system.

In a longitudinal wave, the pressure variation is zero at the pressure node; and the displacement variation is zero at the displacement node. A pressure node is a displacement anti-node and vice versa.


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