Chapter 17 Waves — I

Classification According to natures of matter Chapter 17 Waves — I Wave is everywhere in nature! Why can you hear me? Why can you receive a TV signal emitted from a TV station? §1 Types of Waves 1. Mechanical Waves. Examples: water waves, sound waves, and seismic waves (地震波).

These waves have certain central features: They are governed by Newton’s laws, and they can exist only within a material medium, such as water, air, and rock. 2. Electromagnetic Waves. Examples: Visible and ultraviolet light, radio and television waves, microwaves, x rays, and radar waves. They don’t require material medium to exist. These waves have a limited wave speed in vacuum, 3. Matter Waves. Examples: Electrons, protons, and other fundamental particles; ultracold atoms.

Classification According to Oscillation Types The most remarkable property of the matter waves is that wave functions of matter waves are referred to as probability amplitudes of waves. 1. Transverse Waves. The direction of oscillation of medium elements is perpendicular to the direction of travel of the wave.

2. Longitudinal Waves. The direction of oscillation of medium elements is parallel to the direction of the wave’s travel, the motion is said to be longitudinal, and the wave is said to be a longitudinal wave. sparse and dense Both a transverse wave and a longitudinal wave are said to be traveling waves because they both travel from one point to another, as from one end of the string to the other end.

§2 Creation of Waves and Propagation Conditions of a Mechanical wave 1. A Wave Source 2. Medium in which a mechanical wave propagates. A wave means that the oscillation state propagates with time. It does not mean the particles of medium move forward with wave. Particles of the medium oscillate only around their corresponding equilibrium positions. For a source of simple harmonic oscillation with angular frequency w, the traveling in the positive direction of x axis and oscillating parallel to the y axis, the displacement y of the element located at position x at time t is given by

Amplitude Phase Displacement Angular frequency Angular wave number ym is the amplitude of a wave. The phase of the wave is the argument (wt-kx)

Space Periodicity and Time Periodicity Wavelength and Angular Wave Number The wavelengthl of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions of the shape of the wave (or wave shape). If we have a snapshot at t=0, the oscillation at x = x1 is By definition, the displacement y is the same at both of this wavelength — that is at x = x1 and x =x1 + l. Thus, According to the property of sine function, it begins to repeat itself when its angle is increased by 2prad,

(Angular wave number) That is Period, Frequency, and Angular Frequency Considering the displacement y of the wave at the point x=0, The period of oscillation T of a wave is defined to be the time any mass element takes to moves through one full oscillation. You can choose any time t1, through a period, the oscillation is repeated.

Angular frequency This can be true only if wT=2p, or Frequency f is defined by Wavefronts Wavefronts are surfaces over which the oscillations of wave have the same value; usually such surfaces are represented by whole or partial circles in a two- dimensional drawing for a point source.

Rays Rays are directed lines perpendicular to the wavefronts that indicate the direction of travel of the wavefronts. Wavefronts of plane wave and rays If wavefronts of a wave are planes, it is called the plane wave; if wavefronts of a wave are spherical, the wave is called spherical wave.

The Speed of a Traveling Wave Assuming that the wave travels along x axis. For example, at t=0 we have a snapshot, at t = Dt, we have another snapshot, meanwhile the wave moves a distance Dx. The ratio Dx / Dt (or dx/dt) is the wave speedv.

or Phase speed If point A retains its displacement as it moves, its phase must remain a constant, that is Taking a derivative, we have By using definitions of w and k, we can rewrite v as

Question: How to describe a wave traveling in negative direction of x? Similarly to the discussion of positive direction of x, the wave speed of a wave traveling in the negative x direction should be have a form This corresponding to the condition This means that the wave function should have a form

It seems that k may be have a property of vector, its magnitude is and its direction is in that of wave traveling, or wave velocity.

For a traveling wave in positive x, this means the oscillation at x is For a traveling wave in negative x, this means the oscillation at x is Since x is negative here, so the phase delay is also -|kx|. In general, wave function of an arbitrary shape of traveling wave has a form Wave Speed on a Stretched String

If a wave is to travel through a medium such as water, air, steel, or a stretched string, it must cause the particles of that medium to oscillate as it passes. You cannot send a wave along a string unless the string is under tension of its two ends. Assuming that the linear density of the string is m=m/l, The tension t in the string is equal to the common magnitude of two forces at two ends. Consider a small string element within the pulse, of length Dl, forming an arc of a circle of radius R and subtending an angle 2q at the center of that circle.

There are only vertical components of to form a radial restoring force . In magnitude, The mass of the element is given by The centripetal acceleration of the element toward the center of that circle is given by From the Newton’s Second Law, we have

The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave. An element of the string of mass dm, oscillating transversely in simple harmonic motion, its kinetic associates with its transverse velocity . §3 Energy and Power of a Traveling String Wave Kinetic Energy

Element b maximum kinetic energy Element a zero kinetic energy For mass element dm, the kinetic energy is given by where and dm = mdx. So we have

Dividing the equation by dt, and by using v = dx/dt, we have This is actually the rate of energy transmission. In general we consider the average energy transmission in a period, that is making an integral from 0T for the argument t, one can have This is the average kinetic energy transmission in one period on per second a string.

It can be shown that, for the potential energy, it has a same expression, because it needs other concepts to show the formula, we do not show that further more. So the average power, which is the average rate at which energy of both kinds is transmitted by the wave, is given by

In general, the average energy flow through an unit area of perpendicular to the propagating direction, or density of energy flow, or intensity of the wave, is given by where r is the density of wave in volume. §4 Superposition for Waves The Principle of Superposition for Waves

Overlapping waves algebraically add to produce a resultant wave (or net wave). Suppose that two waves travel simultaneously along the same stretched string. Let y1(x,t) and y2(x,t) be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum It means

Overlapping waves do not in any way alter the travel of each other. simulation 17-8 CD Med Interference of Waves Suppose that there are two sinusoidal waves of the same wavelength and amplitude in the same direction along a stretched string. From the principle of superposition, the resultant wave is

The new resultant amplitude depends on phase f: • If f = 0 rad, the two interfering waves are exactly • in phase.  In this case, the interference produces the greatest possible amplitude which is called fully constructive interference.

2. If f=p, Although we send two waves along the string, we see no motion of the string. This type of interference is called fully destructive interference (see fig. (e)).

When interference is neither fully constructive nor • fully destructive, it is called intermediate interference. • Figures (c) and (f) for f=2p/3. Two waves with the same wavelength are in phase if their phase difference is zero or any integer number of wavelengths. Thus, the integer part of any phase difference expressed in wavelengths may be discarded. Phasors The same as in oscillation, a wave can be represented vectorially by a phasor. A phasor is a vector that has a magnitude equal to the amplitude of the wave and that rotates around an origin; the angular speed of the phasor is equal to the angular frequency w of the wave.

where and b are given as in previous discussions. To find the values of and b, we have to sum the two combining waves as we did previously. In conclusion we can use phasors to combine waves even if their amplitude are different. (a) (b) (c)

; to S1 Point P to S2 DL §5 Interference We consider two point sources S1 and S2 emit waves that are in phase and of identical wave length l. Thus the source themselves are said to be in phase; that is, as the waves emerge from the sources, their displacements are always identical. L1 L2 Their phase difference f at P depends on their path length difference DL = |L2-L1| .

The phase difference between L1 and L2 is (Why?) According to the previous section, the condition for fully constructive interference is This occurs when Similarly, fully destructive interference occurs when

This occurs when Example : Two point sources S1 and S2, which are in phase and and separated by distance D=1.5l, emit identical sound waves of wavelength l.

Standing Waves Let’s consider two waves with the same oscillating frequencies, but traveling in opposite directions. To analyze a standing wave, we represent the two waves with the equations The principle of superposition gives, for the combined waves,

The wave like this — two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave. Question: Does above expression be exactly agreement with the previous figures and simulation? Are there any differences? From the expression and the simulation, we find that unlike a traveling wave, the amplitude of the wave is the same for all string elements, the amplitude in a standing wave here varies with position. For example, the amplitude is zero for values of kx that give coskx = 0. Those values are

or (Pay your attention to the difference between this equation and the corresponding equation in the textbook (equation 17-48).) The positions satisfied this condition are called nodes. The interval between pairs of nodes is given by The amplitude of the standing wave has a maximum value 2ym, which occurs for values of kx given by coskx=1.

or These points satisfy the conditions The positions satisfied these condition are called antinodes. The interval between pairs of two antinodes is also l/2. Reflection at a Boundary A standing wave can be set up in a stretched string by allowing a traveling wave to be reflected from the far end of the string so that it travels back through itself. The incident (original) wave and the reflected wave can combine to form a pattern of standing wave.

If a string is fixed at its end, how about the reflection at this end? Is it a node or an antinode? If one end of a string is fastened to a light ring that is free to slide without friction along a rod. When an incident wave is reflected at this “soft” reflection end, is it a node or an antinode? Questions:

or For a string of both fixed ends, we note that a node must exist at each ends, because each end is fixed and cannot oscillate. For a standing wave of string of length L, there may be one antinode, two antinotes, … n antinodes, as shown in following figures. It is easy to see the relation For a wavelength which satisfies this relation, a standing wave can be set up on the string.

The resonant frequencies are correspondingly where v is the speed of traveling wave on the string. For n=1, the frequency f=v/2L. This Lowest frequency is called the fundamental mode or the first harmonic . For n=2, we have the second harmonic oscillation mode. For n=3, we have the third harmonic oscillation mode.

Assignments 1: 17 — 9 17 — 19 17 — 21 Assignments 2: 17 — 28 17 — 40 17 — 45

Chapter 17 Waves — I