Waves

WavesA Sim IVC

Introduction

Mechanical WavesThe disturbance in the water moves from one place to another.Yet the water is not carried with it This is the essence of wave motion.The world is full of wavesSoundStringsSeismicRadioX-rays

Mechanical WaveMechanical waves are waves that disturb and propagate through a medium.The ripple in the water due to the pebble and a sound wave are the same in this respect.Electromagnetic waves are a special class of waves that do not require a medium, they are energy.In fact a wave is simply the movement of energy!

Propagation of a DisturbanceThe propagation of the disturbance represents the transfer of energy.Waves are therefore a means of energy transfer.Mechanical waves transfer energy via the movement of matter!All mechanical waves require:Some source of disturbanceA medium that can be disturbedSome physical mechanism through which particles of the medium can influence one another

Propagation of a disturbanceSince the medium decides the nature of the energy transfer we expect that different media will transport energy in different ways.ROPE EXAMPLE:Consider the resulting wave.

Propagation of a disturbanceConsider a single point and its motion as the disturbance passes.

As the pulse travels, each rope segment that is disturbed moves in a direction perpendicular to the direction of propagation!

Propagation of a disturbanceNote that no motion occurred in the x direction!This type of wave is called a transverse wave.We will discuss another type of wave called a longitudinal wave when we talk about pressure waves.We can get the basic concept for a Longitudinal wave from the following diagram.

Propagation of a disturbanceLets return to our rope example and consider the mathematical description of the system.What is different between the two snap shots?

They are the same picture just at a different point in space!

Propagation of a disturbanceIf we assume that the shape of the rope is given by some function of positions and time, x and t. And that there is no loss of energy, i.e., the shape of the rope does not change.We can relate the two pictures in a simple way.

Then we can represent the displacement for all positions and times, measured in a stationary frame with origin O, as

Propagation of a disturbanceWhat happens if the disturbance is moving from the right to the left? How does the equation change?

The function y(x,t) is often called the wave function.The meaning of the wave function:Consider a point P on the string, identified by a particular value of its x coordinate.As the pulse passes through P, the y coordinate of this point increases, reaches a max, and then returns to zero

Propagation of a disturbanceThe wave function y(x,t) represents the y coordinate of any point P located at position x at any time t.Consider what happened if we hold the time t fixed.We get a snap shot, then the wave function can be thought of as the actual geometric shape of the pulse at that instant of time!Another name for this is a waveform.

ExampleA pulse moving to the right along the x axis is represented by the wave function given below where x and y are measure in cm and t is in seconds.

What is the value (amplitude) of the wave function at t = 0 and x = 0? What does this represent?How fast is the wave going?Now lets plot the wave function for various x and t values.

Example continuedFor all x and t = 0.

Example continuedFor all x and t = 1.0s.

Example continuedFor all x and t = 2.0sec.

The wave modelA wave can be a single pulse but more often than not we mean a series of crests and troughs that move in a sinusoidal way.This is also called simple harmonic motion.SHOW DEMO: Illustration 16.1: Representations of Simple Harmonic Motion

In an idealized wave motion in an idealized medium, each particle of the medium undergoes a simple harmonic motion around its equilibrium position.

SHOW ANEMATION:

The wave modelThe frequency of sinusoidal waves is the same as the frequency of simple harmonic motion of a particle of the medium.

Each point on the wave can be treated as a simple harmonic oscillator and we can therefore use the mathematics of an SHO to describe the motion of a wave at all points and all times!

Many pointsSingle point

The wave modelImportant information and relationships for the wave model.What is one period.How is it related to the frequency.

The wave has a definite speed.The wave has a definite amplitude.Each particle on the wave has a definite position.

The traveling wave

The traveling waveNow consider the movement of the wave in time. Recall that f(x)=f(x-vt).The disturbance moves.It moves x = vt.Then the disturbance is:

The time required for the disturbance to move one wavelength is the period T. Then:

The traveling waveImportant definitions:The shifting property of the wave function.Angular wave numberAngular frequency

Representations of speed

Phase constantPhase

Example A traveling Sinusoidal wave

A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 is also 15.0 cm, as shown. (a) Find the angular wave number, period, angular frequency, and speed of the wave.

Example A traveling Sinusoidal wave

A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 is also 15.0 cm, as shown. At x = 0 and t = 0, y = 15.0 cm then the general equation for sinusoidal motion suggests that.

General Linear Wave EquationSo far we have worked only with the solution or wave function but not the equation that governs motion.

We can use this expression to describe the motion of any point on the string.The point P moves only vertically, an so its x coordinate remains constant. Thus, the transverse velocity of the point P and its transverse acceleration must be given by the derivatives of the function y(x,t).

General Linear Wave EquationSo we have:

The maximum velocity and acceleration are then:

General Linear Wave Equation

General Linear Wave Equation

General Linear Wave EquationNew Question: How is the time dependence of the y displacement related to the position dependence?

General Linear Wave EquationSince we took the first and second derivative with respect to time it could be useful to do the same for position.

Compare with the related time dependence:

General Linear Wave EquationWe can now compare these results to find common terms..and

Propagating in the x direction

Curvature and Acceleration

The curvature is clearly upward!!

The curvature is clearly downward!!

IntroductionAt a point where the string has an upward curvature:

The acceleration of that point is positive.

At a point where the string has a downward curvature:

The acceleration of that point is negative

< 0

Example: solutions to the linear wave EquationIs the following function a solution to the wave equation we just derived?

Yes and the form is:

What is the speed of the wave?? 3.0 m/s

Speeds in a wave

Speed of a wave

Speed of a wave

The wave equation on a stringWave speed depends only on the properties of the medium through which the wave travels.A rope is a media so we can determine the speed of a transverse wave on a stretched string.Consider a segment of string at the peak of its motion

The wave equation on a string

Waves on a string

Example: The speed of a pulse on a CordA uniform cord has a mass of 0.300 kg and a total length of 6.00 m. Tension is maintained in the cord by suspending an object of mass 2.00 kg from one end. Find the speed of a pulse on this cord. Assume that the tension is not affected by the mass of the cord.See Diagram on the next slide

Example: The speed of a pulse on a Cord

Find the speed of a pulse on this cord.

Interface and refletion

Interfaces and Transmission

Interfaces and transmission

End Lecture 1Start In Class Computer Exercise

Rate of Energy Transfer

Rate of Energy Transfer

Rate of Energy TransferNow imagine an infinitesimal element of the rope so that we have a differential bit of energy. i.e., the length goes to zero.

Now we can use our previous expression for the transverse velocity to obtain:

Since we are after the total kinetic energy in the string we will need to integrate over all x

Rate of Energy TransferThus we need only look at a single snap shot in time (waveform). t = 0 seems a good choice.

This can now be integrated over one wavelength to obtain the total kinetic energy.

Rate of Energy TransferThe kinetic energy is not the only energy in the system, there is also potential energy.We will not do the calculation, but it is very similar, the result is:

Then the total energy in the system must be:

As the wave moves along the string this amount of energy passes by a given point on the string during one period of the oscillation.So what is the rate of energy transfer

Rate of energy transferThe power or rate of energy transfer associated with the wave is:

The rate of energy transfer in any wave is proportional to the square of the angular frequency and to the square of the amplitude.Note the answer in your book is different and related by.

Example: Power Supplied to a Vibrating String

Example: Power Supplied to a Vibrating StringUsing these values and our new equation for power we have:

InterferenceIn our investigation thus far we have see that waves are very different from particles.Another important difference between waves and particles is that we can explore the possibility of two or more waves combining at one point in the same medium. Particles can be combined too form extended objects, but the particles must be at different locations.In contrast, tow waves can both be present at the same location!!. Lets explore.

InterferenceInterference is at the heart of Quantum mechanicsas it is possible to create discrete waves!Waves at different frequencies can combine to create variation and new types of waveforms.In order for interference to occur we must be talking two waves.i.e., not a single wave.A new idea is needed. The Principle of Superposition.

InterferenceWaves that obey the principle of superposition are called linear waves.For mechanical waves this generally means that the amplitudes are much smaller than their wavelengths.A consequence of the superposition principle is that:Two traveling waves can pass through each other without being destroyed or even altered!This is very different from the picture of two particles hitting one another.But two pebbles dropped in the water with different location will cause patterns that can pass through one another!

Interference (Rope demo)Consider the figure: COMPUTER DEMO

StartBegin to interfereMax interferenceEndStartBegin to interfereMax interferenceEndConstructive InterferenceDestructive Interference

InterferenceWe will encounter the principle of superposition in many situations, both in acoustics, optics, and to some degree quantum mechanics.Now that we have developed an intuitive understanding of the principle of superposition we need to develop a mathematical understanding.Consider two traveling sinusoidal waves of the following form:

Note that these are really the same wave shifted in space

InterferenceThe superposition of waves is then

Use the following trigonometric expression.

Setting

We get:

This is the wave function that results from the superposition of the waves

InterferenceThis result has several important features.The result is a sinusoid and has the same frequency and wavelength as the individual waves.The amplitude is different and is determined by the phase constant of the second wave. Note that if the phase constant is 0 then we have a maximum amplitude of 2A.Here the waves are said to be everywhere in phase and therefore interfere constructively.

Interference

Interference

Interference of sound wavesConsider the sound from a loud speaker sent into a tube at a point P, as shown in the figure:

Consider the sound from a loud speaker sent into a tube at a point P, as shown in the figure.The sound waves are split in two and traveling a path length r1 or r2 to the ear.Let the lower path r1 be fixed and the upper path r2 be variable. (like a trombone).

Interference of sound waves

Thus two waves generated by the same source can interfere! We will use this idea again soon.

Interference

Interference

Interference

Interference

Standing WavesIn this example sound waves leave the speakers in the forward direction and interfere constructively and destructively at various points along a tube.We can analyze this situation by considering two identical wave functions traveling in different directions through air.

Standing Waves

Standing WavesEvery element of the medium (string) oscillates in simple harmonic motion with the same angular frequency.

The amplitude of the SHO is determined by the leading term.We can use the fact that there are minima and maxima to develop and expression for the positions of these points.

Standing WavesMinima can be found as:We find minima when:Or in terms of wave length we have:

These point of zero amplitude are called nodes.The maxima are found as:We find maxima when:

Or in terms of wave length we have:

These points are called antinodes.

Standing WavesWave patterns of the elements of the medium produced at various times by two transverse traveling waves moving in opposite directions are shown

Standing WavesSnap shots, and actual motion

ExampleTwo waves traveling in opposite directions produce a standing wave. The individual wave functions are

Where x and y are measured in cm (a) Find the amplitude of the SHM of the element of the medium located at x = 2.3 cm.

(b) Find the positions of the nodes and antinodes if one end of the string is at x =0.

Example continuedWe can use our relationship between the wave number and the wavelength to obtain the answer.

The location of the nodes is give by:

The location of the antinodes is give by:

Standing wave with boundariesConsider a string of length LAs shown in the figure.Waves excited on the string will undergo reflection at both boundaries (ends).Because the ends of the string are fixed, the must necessarily have zero displacement and are therefore by definition nodes.This boundary condition results in the string having a number of discrete natural patterns of oscillation called normal modes.That is only certain frequencies are allowedthis is called quantization.Quantization is a common occurrence when waves are subject to boundary conditions and is central to quantum mechanics!!

Standing wave with boundariesBecause this class of wavesThis model occurs in many aspects of physics.How do we set up this model?Consider a first mode, or single antinode as shown:

Standing wave with boundariesThe simplest possible model is one In which there is only one antinode.That is we can only fit this wavelength into the box.In this case it is really a wavelength that we have put in the box so that the actual wave length can be written as:Some texts refer to the section between each boundary as a loop. There can be more than one mode or loop.

Standing wave with boundariesThe following are some of the modes

From these simple modes it is clear that all modes can be found from:

Since we also have

Natural Standing wavesWhat are the natural frequencies?Not all strings are created equal!

Using our previous result for the wavelengths and our equation for frequency we have

Then for a given string with a certain liner mass density and tension on the line we can find the first excited frequency as.

This is the fundamental!

Natural Standing wavesAll other modes are integer multiples of f1.The family of frequencies or normal modes that are integer multiples of f1 is call a harmonic series.The second harmonic is given byThe nth harmonic is given byIn musical instruments a given string when plucked will resonate at its fundamental and some collection of harmonics.

ExampleMiddle C on a piano has a fundamental frequency of 262 Hz, and the first A above middle C has fundamental frequency of 440 Hz. (A) Calculate the frequencies of the net two harmonics of the string C.

IF the A and C string have the same linear mass density and length determine the ratio of tensions in the two strings.

Example

ExampleOne end of a horizontal string is attached to a vibrating blade, and the other end passes over a pulley as in the figure. A sphere of mass 2.00 kg hangs on the end of the string. The string is vibrating in its second harmonic. A container of water is raised under the sphere so that the sphere is completely submerged. In this configuration, the string vibrates in its fifth harmonic as shown in the second figure. What is the radius of the sphere?

ExampleWhen the sphere is immersed in the water there is a buoyant force that acts upward on the sphere reducing the tension in the string. We must find the resulting tension using the given values and then calculate the frequencies.

ExampleNow use the frequency equation to find a relation between the tensions

Solve for the tension T2.

ExampleNow solve for the radius of the sphere

ResonanceConsider the following setup.

We have seen that a system such as a taut string is capable of oscillating in one or more normal modes.If a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system.

ResonanceIf a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system.This phenomenon is known as resonance.Discuss the Speed Bump Example. Driven oscillations and resonance frequencies.

The displacement from equilibrium is

The variation in the pressure of the gas is give by

Where the maximum pressure is given by

It is clear that the pressure is 90 degrees out of phase with the displacementLongitudinal Waves and Air

Wave intensityEnergy moves in the direction of propagation.But this energy spreads out in 3D not 1D so we need another tool to understand how much energy is delivered to a point in space.For this we define the intensity.The time average rate at which energy is transported by the wave per unit area. (W/m2)The area is shown in the figure and changes as the wave moves away from the source.We can build the following simple relationship for intensity between two points in space.

Next time Standing waves in tubes and musical instruments.BeatsThe Doppler effect.Shock waves.

Standing WavesThe resulting function represents what is called a standing wave.

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