WAVES

Lecture No. 4SIMPLE WAVE MOTIONH. SAIBIOctober 28th , 2015

2008 by W.H. Freeman and Company

Fig. Deep-Ocean Assessment and Reporting of Tsunamis (NOAA) in North PacificWaves are everywhere in natureSound waves, visible light waves, radio waves, microwaves, water waves, sine waves,

telephone chord waves, stadium waves, earthquake waves, waves on a string, slinky waves

3What is a wave?a wave is a disturbance that travels through a medium from one location to another.a wave is the motion of a disturbance4Simple Wave MotionTransverse WavesTransverse wave: The motion of the medium (the string) is perpendicular to the direction of the propagation of the disturbance.

2008 by W.H. Freeman and CompanyFig.1. Transverse wave pulse on a spring. The motion of the propagating medium is perpendicular to the direction of motion disturbance.Fig. 2.Three successive drawings of a transverse wave on a string traveling to the right. Longitudinal WavesLongitudinal waves: The motion of the medium is parallel to the direction of propagation of the disturbance.Example: Sound waves

2008 by W.H. Freeman and Company

Fig.3. Longitudinal wave pulse on a spring.Differences between Longitudinal and Transverse wavesThe differences between the two can be seen7

Anatomy of a WaveNow we can begin to describe the anatomy of our waves.

We will use a transverse wave to describe this since it is easier to see the pieces.8

Anatomy of a WaveIn our wave here the dashed line represents the equilibrium position.

Once the medium is disturbed, it moves away from this position and then returns to it

9

Anatomy of a WaveThe points A and F are called the CRESTS of the wave.

This is the point where the wave exhibits the maximum amount of positive or upwards displacement10

crestAnatomy of a WaveThe points D and I are called the TROUGHS of the wave.

These are the points where the wave exhibits its maximum negative or downward displacement.11

troughAnatomy of a WaveThe distance between the dashed line and point A is called the Amplitude of the wave.

This is the maximum displacement that the wave moves away from its equilibrium.12

AmplitudeAnatomy of a WaveThe distance between two consecutive similar points (in this case two crests) is called the wavelength.

This is the length of the wave pulse.

Between what other points is can a wavelength be measured?13

wavelengthAnatomy of a WaveWhat else can we determine?

We know that things that repeat have a frequency and a period. How could we find a frequency and a period of a wave?14

Wave frequencyWe know that frequency measure how often something happens over a certain amount of time.

We can measure how many times a pulse passes a fixed point over a given amount of time, and this will give us the frequency.15Wave frequencySuppose I wiggle a slinky back and forth, and count that 6 waves pass a point in 2 seconds. What would the frequency be?3 cycles / second3 Hzwe use the term Hertz (Hz) to stand for cycles per second.16Wave PeriodThe period describes the same thing as it did with a pendulum.It is the time it takes for one cycle to complete.It also is the reciprocal of the frequency.T = 1 / ff = 1 / T17Wave SpeedWe can use what we know to determine how fast a wave is moving.

What is the formula for velocity?velocity = distance / time

What distance do we know about a wavewavelength

and what time do we knowperiod18Wave Speedso if we plug these in we getvelocity = length of pulse / time for pulse to move pass a fixed pointv = / Twe will use the symbol to represent wavelength19Wave Speedv = / Tbut what does T equalT = 1 / fso we can also writev = f velocity = frequency * wavelengthThis is known as the wave equation.20Wave BehaviorNow we know all about waves.

How to describe them, measure them and analyze them.

But how do they interact?21Wave BehaviorWe know that waves travel through mediums.

But what happens when that medium runs out?22Boundary BehaviorThe behavior of a wave when it reaches the end of its medium is called the waves BOUNDARY BEHAVIOR.

When one medium ends and another begins, that is called a boundary.23Fixed EndOne type of boundary that a wave may encounter is that it may be attached to a fixed end.In this case, the end of the medium will not be able to move.What is going to happen if a wave pulse goes down this string and encounters the fixed end?24

Fixed EndHere the incident pulse is an upward pulse.The reflected pulse is upside-down. It is inverted.The reflected pulse has the same speed, wavelength, and amplitude as the incident pulse.25

Fixed End Animation26

Free EndAnother boundary type is when a waves medium is attached to a stationary object as a free end.In this situation, the end of the medium is allowed to slide up and down.What would happen in this case?

27

Free EndHere the reflected pulse is not inverted.It is identical to the incident pulse, except it is moving in the opposite direction.The speed, wavelength, and amplitude are the same as the incident pulse.28

Free End Animation29

Change in MediumOur third boundary condition is when the medium of a wave changes.Think of a thin rope attached to a thin rope. The point where the two ropes are attached is the boundary.At this point, a wave pulse will transfer from one medium to another.What will happen here?30

Change in MediumIn this situation part of the wave is reflected, and part of the wave is transmitted.Part of the wave energy is transferred to the more dense medium, and part is reflected.The transmitted pulse is upright, while the reflected pulse is inverted.31

Change in MediumThe speed and wavelength of the reflected wave remain the same, but the amplitude decreases.

The speed, wavelength, and amplitude of the transmitted pulse are all smaller than in the incident pulse.32

Change in Medium Animation33

Wave InteractionAll we have left to discover is how waves interact with each other.

When two waves meet while traveling along the same medium it is called INTERFERENCE.34Constructive InterferenceLets consider two waves moving towards each other, both having a positive upward amplitude.

What will happen when they meet?35Constructive InterferenceThey will ADD together to produce a greater amplitude.This is known as CONSTRUCTIVE INTERFERENCE.36

Destructive InterferenceNow lets consider the opposite, two waves moving towards each other, one having a positive (upward) and one a negative (downward) amplitude.

What will happen when they meet?37Destructive InterferenceThis time when they add together they will produce a smaller amplitude.

This is know as DESTRUCTIVE INTERFERENCE.38

Check Your UnderstandingWhich points will produce constructive interference and which will produce destructive interference?39

ConstructiveG, J, M, NDestructiveH, I, K, L, O

WAVE PULSESThe x coordinates of two reference frames are related by:So . Thus, the shape of the string in the original frame is: (Wave Function) wave moving in the +x direction.The same line of reasoning for a pulse moving to the left leads to:wave moving in the -x direction.v is the speed of propagation of the wave.2008 by W.H. Freeman and Company

Fig.4.Fig.4 shows a pulse on a string at time t=0. The shape of the string can be represented by some function y=f(x). At some later time, the pulse is farther down the string. In a new coordinate system with origin O that moves to the right with the same speed as the pulse. The string is described in this frame by f(x) for all times.(1)(2)WAVE PULSESFor waves on a string, the wave function represents the transverse displacement of the string.For sound waves in air, the wave function can be the longitudinal displacement of the air molecules, or the pressure of the air.The wave function are solutions of a differential equation called the wave equation, which can be derived using Newtons laws.SPEED OF WAVESA general property of waves is that their speed relative to the medium depends on the properties of the medium, but is independent of the medium of the source of the wave. For example, the speed of a sound from a car horn depends only on the properties of air and not on the motion of the car.For wave pulses on a rope, we can demonstrate that the greater the tension, the faster the propagation of the waves. Furthermore, waves propagate faster in a light rope than in a heavy rope under the same tension. If FT is the tension and in the linear mass density (mass per unit length), the wave speed is:

For sound waves in a fluid such as air or water, the speed v is given by:

Speed of waves on a string(3)(4)( is the equilibrium density of the medium and B is the bulk modulus)The bulk modulus is the negative of the ratio of the change in pressure to the fractional change in volume:

SPEED OF WAVESComparing Eq.3 and 4, we can see that, in general, the speed of waves depends on an elastic property of the medium (the tension for string waves and the bulk modulus for sound waves) and on an inertial property of the medium (the linear mass density or the volume mass density).For sound waves in a gas such as air, the bulk modulus is proportional to the pressure, which in turn is proportional to the density and to the absolute temperature T of the gas. The ratio B/ is thus independent of density and is merely proportional to the absolute temperature T. In this case, Eq.4 is equivalent to:

T is the absolute temperature (K), which is related to Celsius tC by:The dimensionless constant depends on the kind of gas.The constant R is the universal gas constant:and M is the molar mass of the gas (that is, the mass of one mole of the gas), which for air is:

Speed of sound in a gas(5)(6)(7)ExerciseSurface ocean waves are possible because of gravity and are called gravity waves. Gravity waves are called shallow waves if the water depth is less than half a wavelength. The wave speed for gravity waves depends on the depth and is given by v=sqrt (gh), where h is the depth. A gravity wave in the open ocean, where the depth is 5 km, has a wavelength of 100 km. 1- What is the wave speed of the wave?2- Is the wave a shallow wave?Information: Tsunamis are known to travel at speeds of 800 km/h in the open ocean.

SPEED OF WAVESDerivation of v for waves on a stringEq.3 (v=SQRT FT/) can be obtained by applying the pulse-momentum theorem to the motion of a string. Suppose you are holding one end of a long taut string with tension FT and uniform mass per unit length . (the other end of the string is attached to a distant wall). Suddenly, you begin to move your hand upward at a constant speed u. After a short time, the string appears as shown in Fig.5 with the rightmost point of the inclined segment of the string moving to the right at the wave speed v and the entire inclined segment moving upward at speed u. By applying the impulse-momentum theorem ( ) to the string, we obtain: where Fy is the upward component of the force of your hand on the string, m is the mass of the inclined segment, and t is the time that your hand has been moving upward. The two triangles in the figure are similar, so:

Substituting for Fy in Eq.8 gives:where vt has been substituted for m. Solving for v gives:Which is the expression for the wave speed that is given in Eq.3.

(8)or

SPEED OF WAVESDerivation of v for waves on a string2008 by W.H. Freeman and CompanyFig.5. As the end of the spring moves upward at constant speed u, the point where the string changes from horizontal to inclined moves to the right at the wave speed v.THE WAVE EQUATIONWe can apply Newtons second law to a segment of the string to derive a differential equation known as the wave equation, which relates the spatial derivatives of y(x,t) to its time derivatives. Fig.6 shows one segment of a string. We consider only small angles 1 and 2. Then the length of the segment is approximately x and its mass is m=x, where is the strings mass per unit length. First, we show that, for small vertical displacements, the net horizontal force on a segment is zero and the tension is uniform and constant. The net force in the horizontal direction is zero. That is,

Where 2 and 1 are the angles shown and FT is the tension in string. Because the angles are assumed to be small, we may approximate cos by 1 for each angle. Then, the net horizontal force on the segment can be written

Thus,

the segment moves vertically, and the net force in this direction is:

THE WAVE EQUATIONBecause the angles are assumed to be small, we may approximate sin by tan for each angle. Then the net vertical force on the string segment can be written

The tangent of the angle made by the string with the horizontal is the slope of the line tangent to the string. The slope S is the first derivative of y(x,t) with respect to x for constant t. A derivative of a function of two variables with respect to one of the variables with the other held constant is called a partial derivative. The partial derivative of y with respect to x is written y/x. Thus, we have:

Then

Where S1 and S2 are slopes of either end of the string segment and S is the change in the slope. Setting this net force equal to the mass x times the acceleration 2y/t2 gives:

or(9)THE WAVE EQUATION

2008 by W.H. Freeman and CompanyFig.6. Segment of stretched string used for derivation of the wave equation. The net vertical force on the segment is FT2 sin2-FT1 sin1, where F is the tension in the string. The wave equation is derived by applying Newtons second law to the segment.THE WAVE EQUATIONIn the limit as x0, we have:Thus, in the limit as x0, Eq.9 becomes

We now show that the wave equation is satisfied by any function x-vt. Let =x-vt and consider any wave function:

We use y for the derivative of y with respect to . Then, by the chain rule for the derivatives,

Because

We haveTaking the second derivatives, we obtain

(10-a) Wave equation for a taut string.

and

and

and

and

THE WAVE EQUATIONThus,

The same result (Eq.10-b) can be obtained for any function of x+vt as well. Comparing Eq.10-a and 10-b, we see that the speed of propagation of the wave is

Wave Equation(10-b)

(3)The speed of sound is given by: (Eq.4), where B and are the bulk modulus and density of the medium, respectively. This equation can be obtained by applying the impulse-momentum theorem to the motion of the air in a long cylinder (Fig.7) with a piston at one end with the other end open to the atmosphere. Suddenly, you begin to move the piston to the right at constant speed u. After a short time, t from the initial position of the piston is moving to the right with speed u. By applying the impulse-momentum theorem ( ) to the air in the cylinder we obtain:

Where m is the mass of the air moving with speed u and F is the net force on the air in the cylinder. The air was initially at rest. The net force F is related to the pressure increase P of the air near the moving piston by:

Where A is the cross-sectional area of the cylinder. The bulk modulus of the air is given by:

so(11)SPEED OF WAVESDerivation of v for sound wavesSPEED OF WAVESDerivation of v for sound wavesWhere Aut is the volume swept out by the piston and Avt is the initial volume of the air that is now moving with speed u. Substituting for F in Eq.11 gives:

Where Avt have been substituted for m. Solving for v gives:

Which is the same as the expression for v in Eq.4.A wave equation for sound waves can be derived using Newtons laws. In one dimension, this equation is:

Where s is the displacement of the medium in the x direction and vs is the speed of the sound in the medium.

or

SPEED OF WAVESDerivation of v for sound waves2008 by W.H. Freeman and CompanyFig.7. The air near to the piston is moving to the right at the same constant speed u as the piston. The right edge of this pressure pulse moves to the right with the wave speed v. The pressure in the pulse is higher than the pressure in the rest of the cylinder by P.