CHAPTER 1 WAVES

17 Jan 1995, a terrible earthquake struck the Hanshin (Japan) killing over 6400 people and injuring about 40,000 others.

200,000 homes and buildings were damaged.

The heaviest damage occurred in the city of Kobe, including the buckling and collapse of an elevated highway.

However, geologist found that the point of origin of the earthquake was 15-20 km below the northern tip of Awaji Island, about 20 km south west of Kobe.

How did the energy released by the earthquake travel that far with enough energy to cause such great devastation?

1.1 WAVES AND ENERGY TRANSPORT A wave is a disturbance that travels; If it travels through a medium, the

particles in the medium oscillate around their equilibrium positions but do not travel.

Energy transport: Waves transfer energy without transferring matter;

This can be easily seen with water waves or waves on ropes or strings.

Other examples include seismic waves, which transport energy through the Earth, sound waves, and electromagnetic waves.

Two different ways to transfer energy. (a) The baseball carries energy from the

pitcher to the catcher. (b) A wave pulse also carries energy from

the pitcher to the catcher, but the piece of rope held by the pitcher’s hand does not move to the catcher’s hand.

Intensity: Intensity is the average power per unit area carried by the wave;

The intensity usually decreases with distance from the source of the wave, due both to dissipation and to the wave spreading over a larger and larger area.

Therefore, if energy absorption by the medium can be neglected, the intensity of the sound is inversely proportional to the square of the instance from the source.

This “inverse square law” is the result of a conserved quantity (here, energy) radiating uniformly from a point source in three-dimensional space.

(a) A point source of sound radiating energy uniformly in all directions.

(b) The intensity at a distance r2 is smaller than the intensity at a distance r1 since the same power is spread out over a greater area.

1.2TRANSVERSE AND LONGITUDINAL WAVES In a transverse wave, the motion of the

particles is perpendicular to the direction of propagation of the wave; in a longitudinal wave, the motion is parallel to the direction of propagation. Both kinds of waves can be demonstrated with a long spring such as a Slinky.

(a) Transverse, and (b) Longitudinal waves on a Slinky

Transverse waves cannot travel through the bulk of a fluid, as the fluid will not exhibit any restoring force in response to a shear stress (it just flows);

They can, however, travel on the surface of a fluid, such as water waves.

Sound waves are longitudinal, and can travel through any material; they consist of a series of regions of higher and lower density, called compressions and rarefactions.

Seismic waves (which travel through the solid earth) can be either longitudinal or transverse; as the core of the earth is liquid, transverse seismic waves cannot travel through it, and they are reflected.

The speeds of longitudinal and transverse waves in the same medium are generally different; longitudinal seismic waves travel faster than transverse ones.

Waves that combine transverse and longitudinal motion:

It is possible for a wave to be neither pure transverse or pure longitudinal;

In this case, rather than the particles moving perpendicular or parallel to the direction of propagation of the wave, they exhibit a more circular motion.

This is seen in surface seismic waves as well as in water waves.

The motion of ground particle in (a) P waves (b) S waves (c) one kind of surface wave (d) Motion of a swimmer as a water

wave passes by

1.3SPEED OF TRANSVERSE WAVES ON A STRING The speed of propagation of a wave is

determined by the mechanical properties of the medium;

For a string, it depends on the tension in the string and on the mass per unit length.

Strings under more tension and with less mass per unit length will have higher wave speeds.

This is due to the higher restoring force when the tension is higher, and to the lower inertia when the mass per unit length is lower.

The speed of the particles in the medium is different (and variable), and will also depend on the amplitude of the wave (the larger the amplitude, the faster the particles will move).

The speed of a transverse wave on a string is: (F = force; L = Length and m = mass)

Eq. 11-2 can rewrite in another form. Length and mass are not independent;

for a given string composition and diameter (say, a yellow brass string of 0.030 in diameter), the mass of the string is proportional to its length.

By defining the linear mass density (mass per unit length) of the string to be

The speed of a transverse wave on a string can be written

1.4PERIODIC WAVES A periodic wave is one which has a

repeating pattern over time. Such waves are characterized by a

period (the time a complete wave takes to pass a given point) and a frequency (the number of waves passing a given point per unit time); the period is the inverse of the frequency (just as in SHM).

At any given point, the wave repeats itself after a time T called the period. The inverse of the period is the frequency f.

The distance between corresponding points on successive waves is called the wavelength;

the wave speed is then the frequency multiplied by the wavelength.

The maximum displacement of a particle from its equilibrium position is called the amplitude of the wave.

If the wave shape is sinusoidal, the wave is called harmonic, and every point in the material undergoes SHM.

During one period T, a periodic wave traveling at speed v moves a distance vT.

In Figure 11.7, note that, at any instant, points separated by a distance vT along the direction of propagation of a wave move “in sync” with each other.

Thus, vT is the repetition distance of the wave, just as the period is the repetition time.

Sinusoidal wave moving with speed v in the x-direction. The amplitude A and the wavelength are shown.

The max. displacement of any particle from its equilibrium position is the amplitude A of the wave.

For a sinusoidal wave traveling along a stretched string in the x-direction, the amplitude A is the maximum displacement in the positive or negative y-direction.

For surface water waves, the amplitude is the height of a crest (a high point) above of the depth of a through ( a low point) below the undisturbed water level .

Sinusoidal wave moving with speed v in the x-direction. The amplitude A and the wavelength are shown.

Harmonic waves are a special kind of periodic wave in which the disturbance is sinusoidal (sine or cosine).

In a harmonic transverse wave on a string, for instance, every point on the string moves in SHM with the same amplitude and angular frequency, although different points reach their maximum dispalcement at different times.

The maximum speed and maximum acceleration of a point on the string depend on both the angular frequency and the amplitude of the wave

Since the total energy of an object moving in SHM is proportional to the amplitude squared, the the total energy of a harmonic wave is propotional to the square of its amplitude.

That turns out to be a general result not limited to harmonic waves

The intensity of a wave is proportional to the square of its amplitude

11.5 MATHEMATICAL DESCRIPTION OF A WAVE If we consider a transverse wave traveling

along a one-dimensional string which is oriented along the x-axis and where the particles are displaced in the y-direction, the displacement of the particles (and therefore the wave) is a function of both time and position along the string;

that is, y is a function of both x and t. If the wave retains its shape as it travels, those variables must appear in the combination

(t - x/v); such a wave is called a traveling wave.

Sinusoidal wave moving with speed v in the direction.

The amplitude A and the wavelength are shown

A HARMONIC TRAVELING WAVE: If a wave is harmonic, its shape is

sinusoidal; we can show that the argument of the sine or cosine function is (wt ± kx), where k = 2p/l and l is the wavelength.

The figure below shows a wave pulse, with the same shape, at successive times. The motion of the point x repeats the motion of the point x= 0 with a time delay t = x/v.

11.6 GRAPHING WAVES A wave can be graphed either at a

single point or at a single time; if the wave is harmonic, both graphs will

be sinusoidal.

Two graphs of a harmonic wave y(x,t) = A sin (t-kx) on a string

(a) The vertical displacement of a particular point on the string (x=0) as a function of time.

(b) The vertical displacement as a function of horizontal position at a single instant of time (t=0)

11.7 PRINCIPLE OF SUPERPOSITION If two waves are traveling through the same

medium, and their amplitudes are not too large (so that the medium still obeys Hooke's law),

The net disturbance at any point is the sum of the individual disturbances due to each wave. This can be seen, for example, by dropping two pebbles into a pond.

PS = When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.

(a) Two identical wave pulses traveling toward and through each other.

(b), (c) Details of the wave pulse summation: dasehed lines are the separate wave pulses and solid line is the sum

Suppose two wave pulses are traveling toward each other on a string (Fig.11.11a). If one of the pulses (acting alone) would produce a displacement y1 at a certain point and the other would produce a displacement y2 at the same point.

The result when the two overlap is a displacement of y1+y2.

Fig. 11.11b, c show in greater detail how y1 and y2 add together to produce the net displacement when the pulses overlap.

The dashes curves represent the individual pulses; the solid line represents the superposition of the pulses.

In Fig. 11.11b the pulses are starting to overlap and in Fig. 11.11c they are just about to coincide.

11.8 REFLECTION AND REFRACTION Reflection: If a wave encounters a boundary

between two media, some or all of the wave will be reflected (that is, some or all of the energy will travel back into the first medium).

How much of the energy is reflected depends on how different the properties of the two media are (in particular, on the wave speed in the two media); the more different, the more reflection takes place.

The reflected wave will be inverted if it reflects from a medium with a lower wave speed; if the reflecting medium has a higher wave speed, the reflected wave will not be inverted.

Snapshots of the reflection of a wave pulse from a fixed end.

The reflected pulse is upside down.

Refraction: In general, part of the wave will be transmitted into the second medium;

both the reflected and transmitted waves will have the same frequency as the incident wave, but the wavelength of the transmitted wave will be different (since the frequency is the same and the speed is different).

Unless the incident wave is perpendicular to the boundary, the transmitted wave will also be at a different angle than the incident wave; this is called refraction, and the transmitted wave is also called the refracted wave.

The angles are defined with respect to the normal to the boundary surface; the ratio of the sines of the angles of incidence and of refraction is the same as the ratio of the wave speeds in the respective media.

Since = f, and the frequencies are the same

this equation applies to any kind of wave and is of particular importance in the study of optics.

The angle in Eq. (11-10) are called the angle of incidence and the angle of refraction.

Note that these angles are measured between the propagation direction of the wave and the normal

(a) A broad beam of light refracts when it passes from air into water. The reflected wave is omitted for clarity.

The normal is the direction perpendicular to the boundary. The wave’s propagation direction is closer to the normal in the slower medium (water, in this case).

11.9 INTERFERENCE AND DIFFRACTION Interference: If two waves being superposed

have the same frequency and have a fixed phase relationship, they are called coherent.

In this case, superposition will yield what is called interference - the resultant wave will have the same frequency as the original waves, and will have an amplitude somewhere between the sum of the two amplitudes and the absolute magnitude of their difference.

If the waves are in phase, the interference is called constructive; if they are 180º out of phase, the interference is called destructive.

Coherent waves (a) in phase and (b) 180 out of phase.

(one wave is drawn with a lighter line to distinguish it from the other?)

Suppose two coherent waves start out in phase with each other.

In Fig.11.18, two rods vibrate up and down in step with each other to generate circular waves on the surface of the water.

If two waves travel the same distance to reach a point on the water surface, they arrive in phase and interfere constructively.

At points where the distances are different, we calculate the phase difference as follows.

One wavelength of path difference corresponds to a phase difference of 2 radians (one full cycle).

Then, working by proportions (d1-d2)/ = phase difference / 2 rad. If the phase difference is an even integral multiple of rad,

then constructive interference occurs at point P; if the phase difference is an odd integral multiple of rad, then destructive interference occurs at point P.

Overhead view of coherent surface water waves.

The two waves travel different distances d1 and d2 to reach a point P.

The phase difference between the waves at point P is k (d1-d2)

Intensity effects for interfering waves:

If two waves which are superposed are coherent, their amplitudes add; their intensities (which are proportional to the square of the amplitude) do not, in general.

However, if two waves which are superposed are incoherent, their intensities do add.

Diffraction: If a wave encounters an obstacle, it will bend around it; the amount of bending depends on the size of the obstacle compared to the wavelength of the wave.

This bending is called diffraction.

Diffraction of a wave when it encounters an obstacle.

11.10 STANDING WAVES A wave can be reflected at a boundary in

such a way that the wave appears to stand still; this is typical of waves on finite strings, such as those in musical instruments. In this case, the string vibrates as a whole;

every point reaches its maximum amplitude simultaneously, and every point also reaches its minimum amplitude (namely, zero) simultaneously as well. Therefore, there are points on the string which never move; these are called nodes. Between the nodes are points which have the maximum amplitude; these are called antinodes.

If both ends of the string are fixed, both are nodes; if one end is free to move, it is an antinode. A string with both ends fixed can have an integral number of half-wavelengths along its length; the lowest frequency occurs when the wavelength is twice the length of the string (so that the only nodes are at the ends); this frequency is called the fundamental. All other natural frequencies of the string are integer multiples of the fundamental frequency.

A standing wave at various times, where T is the period.

L is length of string and n =1,2,3,…the possible wavelength for standing waves on a string are

The Frequencies are