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Chapter 8 Mechanical Waves Tutorial Questions Overview

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Chapter 8 Mechanical Waves Tutorial Questions Due: 11:00am on Tuesday, June 11, 2013 Note: To understand how points are awarded, read your instructor's Grading Policy.

Wave Notation Description: Introduction to wave notation: find relationships between the various kinematic variables: wavelength, frequency, period, and velocity. Learning Goal: To understand the relationships between the parameters that characterize a wave. It is of fundamental importance in many areas of physics to be able to deal with waves. This problem will lead you to understand the relationship of variables related to wave propagation: frequency, wavelength, velocity of propagation, and related variables. Note that these are kinematic variables that relate to the wave's propagation and do not depend on its amplitude.

Part A Traveling waves propagate with a fixed speed usually denoted as waveform repeats every time interval

(but sometimes ). The waves are called _______________ if their

.

ANSWER: transverse longitudinal periodic sinusoidal

Periodic waves are characterized by a frequency that is designated by

or

. The quantities

and

are related to the length of the

waves.

The fundamental relationship among frequency, wavelength, and velocity is . This relationship may be visualized as follows: In 1 s, moves a distance

cycles of the wave move past an observer. In this same second the wavetrain

.

Part B Solve this equation to find an expression for the wavelength Express your answer in terms of the velocity ANSWER: =

Part C

.

and the frequency

.

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If the velocity of the wave remains constant, then as the frequency of the wave is increased, the wavelength ___________. ANSWER: decreases increases stays constant

Part D The difference between the frequency the units for

and the frequency

is that

is measured in cycles per second or hertz (abbreviated Hz) whereas

are _______ per second.

Give your answer in one word. Watch your spelling to avoid a mistake! ANSWER:

Part E Find an expression for the period of a wave Express your answer in terms of any of ,

in terms of other kinematic variables. ,

, and simple constants such as

.

ANSWER: =

Also accepted:

Part F What is the relationship between

and

?

ANSWER: =

Part G What is the simplest relationship between the angular wavenumber

and just one of the other kinematic variables?

Express your answer using only one of the other kinematic variables plus constants like

.

ANSWER: =

The units of

are

distance, which is

and indicate that angular wavenumber is a measure of the number of radians of phase in a unit times the number of complete wave cycles in this distance. For example, a distance of one wavelength is

equivalent to one cycle, one period, and

. The angular wavenumber

is often misnamed "wavenumber," which is actually

, so be careful to determine how this term is defined when reading your textbook or reviewing your lecture notes.

Two Velocities in a Traveling Wave Description: To contrast and clarify the velocity of the medium in which a wave propagates with the velocity of propagation of the wave.

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Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself). Consider a transverse wave traveling in a string. The mathematical form of the wave is .

Part A Find the speed of propagation

of this wave.

Express the velocity of propagation in terms of some or all of the variables

,

, and

.

Hint 1. Perform an intermediate step Note that the phase of the wave At what position

, and therefore the displacement of the string, is equal to zero at

is the phase equal to zero a short time

Express your answer in terms of

,

, and

.

later?

.

ANSWER: =

Because the displacement of the string is now equal to zero at this new position, the wave will appear to have moved. The propagation speed is given by .

ANSWER: =

Part B Find the y velocity

of a point on the string as a function of

Express the y velocity in terms of

,

,

,

and .

, and .

Hint 1. How to approach the problem In the problem introduction, you are given an expression for y velocity, take the partial derivative of

, the displacement of the string as a function of

with respect to time. That is, take the time derivative of

constant.

Hint 2. A helpful derivative

ANSWER: =

Part C Which of the following statements about

Hint 1. How to approach this question

, the x component of the velocity of the string, is true?

and . To find the

while treating

as a

MasteringPhysics: Print View with Answers You are given a form for

http://session.masteringphysics.com/myct/assignmentPrintView?assignm . You are not given any information about

moves in the y direction, i.e.

, but it is assumed that each point on the string only

.

ANSWER:

has the same mathematical form as

but is

out of phase.

So the wave moves in the x direction, even though the string does not. What this means is that even though individual points on the string only move up and down, a given shape or pattern of points (in this sinusoidal) will move to the right as time progresses.

Part D Find the slope of the string Express your answer in terms of

as a function of position ,

,

,

and time .

, and .

Hint 1. A helpful derivative

ANSWER: =

Part E Find the ratio of the y velocity of the string to the slope of the string calculated in the previous part. Express your answer as a suitable combination of some of the variables

,

, and

.

ANSWER: =

Also accepted:

To understand why the ratio of the y velocity of the string to its slope is constant, draw the string with a wave running along it at time . In the vicinity of , the string is sloped upward. The bit of string at position moves downward as the wave moves forward. One-half cycle later, the string in the vicinity of

will be sloped downward, and the string at position

will move upward

as the wave moves forward. In general, if at some particular ). If the slope at

the slope of the string is positive (

), that bit of string will be moving downward (

is negative, that bit of string will be moving upward. This explains why the sign of the ratio of string

velocity to slope is always negative. One way of understanding why the ratio has a constant magnitude is to observe that the more steeply the string is sloped, the more quickly it will move up or down.

Waves on Strings

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Description: Short conceptual problem relating the characteristic properties of waves on a string to the physical properties of the string. This problem is based on Young/Geller Conceptual Analysis 12.3. An oscillator creates periodic waves on two strings made of the same material. The tension is the same in both strings.

Part A If the strings have different thicknesses, which of the following parameters, if any, will be different in the two strings? Check all that apply.

Hint 1. Wave frequency The frequency of a periodic wave on a string is given by the frequency of oscillation of the particles in the string. Since the oscillations are driven by the external oscillator, the frequency of the wave propagating along the string is ultimately determined by the frequency of the oscillator alone, and it is independent of the properties of the string.

Hint 2. Wave speed The speed of a wave on a string under tension depends on the properties of the string alone. More specifically, it depends on the tension in the string and the mass per unit length of the string.

Hint 3. Wavelength The wavelength of a periodic wave on a string is the distance between any two points in the string at corresponding positions on successive repetitions in the wave shape. It depends on the speed and frequency of the wave.

Hint 4. Find the properties of the strings Which of the following statements best describes the physical properties of the strings? ANSWER: Since the strings are made of the same material but have different thicknesses, they have different masses. Since the strings are made of the same material but have different thicknesses, they have different masses per unit length. Since the strings are made of the same material, they have the same mass per unit length.

ANSWER: wave frequency wave speed wavelength none of the above

Part B If the strings have the same thickness but different lengths, which of the following parameters, if any, will be different in the two strings? Check all that apply.

Hint 1. Find the properties of the strings Which of the following statements best describes the physical properties of the strings? ANSWER: Since the strings are made of the same material and have the same thickness, they have the same mass per unit length. Since the strings are made of the same material but have different lengths, they have different masses per unit length. Since the strings are made of the same material, they have the same mass per unit length.

ANSWER:

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wave frequency wave speed wavelength none of the above

In summary, when a wave travels along a string, the wave speed depends exclusively on the properties of the string, whereas the wave frequency is set by the oscillator that creates the waves. The wavelength is a quantity that can vary if either the wave speed or the wave frequency is changed. Thus, it can be modified by changing either the motion of the oscillator or the properties of the string.

Normal Modes and Resonance Frequencies Description: Multiple choice questions about the definition and origin of normal modes. Then compute the frequency and wavelength of the first three normal modes in a string. Learning Goal: To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string. A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency and associated pattern of oscillation. Consider an example of a system with normal modes: a string of length waves on this string propagate with speed

held fixed at both ends, located at

and

. Assume that

. The string extends in the x direction, and the waves are transverse with displacement along the y

direction. In this problem, you will investigate the shape of the normal modes and then their frequency. The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by

Part A The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct?

Hint 1. Normal mode constraints The key constraint with normal modes is that there are two spatial boundary conditions,

and

correspond to the string being fixed at its two ends.

ANSWER: The wave is traveling in the +x direction. The wave is traveling in the -x direction. The wave will satisfy the given boundary conditions for any arbitrary wavelength The wavelength

can have only certain specific values if the boundary conditions are to be satisfied.

The wave does not satisfy the boundary condition

Part B Which of the following statements are true? ANSWER:

.

.

, which

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The system can resonate at only certain resonance frequencies

and the wavelength

must be such that

.

must be chosen so that the wave fits exactly on the string. Any one of

or

or

can be chosen to make the solution a normal mode.

The key factor producing the normal modes is that there are two spatial boundary conditions, satisfied only for particular values of

and

, that are

.

Part C Find the three longest wavelengths (call them and

,

, and

) that "fit" on the string, that is, those that satisfy the boundary conditions at

. These longest wavelengths have the lowest frequencies.

Express the three wavelengths in terms of

. List them in decreasing order of length, separated by commas.

Hint 1. How to approach the problem The nodes of the wave occur where . This equation is trivially satisfied at one end of the string (with

), since

.

The three largest wavelengths that satisfy this equation at the other end of the string (with are the three smallest, nonzero values of

Hint 2. Values of

that satisfy the equation

) are given by

.

that satisfy

The spatial part of the normal mode solution is a sine wave. Find the three smallest (nonzero) values of satisfy

, where the

(call them

,

, and

) that

.

Express the three nonzero values of

as multiples of

. List them in increasing order, separated by commas.

ANSWER: ,

,

= 1, 2, 3

Also accepted: 3.14, 2, 3, 1, 6.28, 3, 3.14, 6.28, 3, 1, 2, 9.42, 3.14, 2, 9.42, 1, 6.28, 9.42, 3.14, 6.28, 9.42

Hint 3. Picture of the normal modes Consider the lowest four modes of the string as shown in the figure.

The letter N is written over each of the nodes defined as places where the string does not move. (Note that there are nodes in addition to those at the end of the string.) The letter A is written over the antinodes, which are where the oscillation amplitude is maximum.

ANSWER:

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,

,

=

,

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,

The procedure described here contains the same mathematics that leads to quantization in quantum mechanics.

Part D The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength Find the frequency Express

.

of the ith normal mode.

in terms of its particular wavelength

and the speed of propagation of the wave .

Hint 1. Propagation speed for standing waves Your expression will involve , the speed of propagation of a wave on the string. Of course, the normal modes are standing waves and do not travel along the string the way that traveling waves do. Nevertheless, the speed of wave propagation is a physical property that has a well-defined value that happens to appear in the relationship between frequency and wavelength of normal modes.

Hint 2. Use what you know about traveling waves The relationship between the wavelength and the frequency for standing waves is the same as that for traveling waves and involves the speed of propagation .

ANSWER: =

The frequencies

are the only frequencies at which the system can oscillate. If the string is excited at one of these resonance

frequencies it will respond by oscillating in the pattern given by

, that is, with wavelength

associated with the

at which it is

excited. In quantum mechanics these frequencies are called the eigenfrequencies, which are equal to the energy of that mode divided by Planck's constant . In SI units, Planck's constant has the value

Part E Find the three lowest normal mode frequencies Express the frequencies in terms of

,

, and

.

, , and any constants. List them in increasing order, separated by commas.

ANSWER: ,

,

=

,

,

Note that, for the string, these frequencies are multiples of the lowest frequency. For this reason the lowest frequency is called the fundamental and the higher frequencies are called harmonics of the fundamental. When other physical approximations (for example, the stiffness of the string) are not valid, the normal mode frequencies are not exactly harmonic, and they are called partials. In an acoustic piano, the highest audible normal frequencies for a given string can be a significant fraction of a semitone sharper than a simple integer multiple of the fundamental. Consequently, the fundamental frequencies of the lower notes are deliberately tuned a bit flat so that their higher partials are closer in frequency to the higher notes.

± Standing Waves in Action Description: ± Includes Math Remediation. Calculate the number of nodes and antinodes produced in a particular situation, then use an applet to create standing waves with particular properties. Consider a string of length 1.0 meter, fixed at both ends, with mass 100 grams and tension 100 Newtons.

Part A

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