# 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?

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## A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a

Find an answer to your question A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general …

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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 fi and associated pattern of oscillation.

Consider an example of a system with normal modes: a string of length L held fixed at both ends, located at x=0 and x=L. Assume that waves on this string propagate with speed v. 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

yi(x,t)=Ai sin(2π*x/λi)sin(2πfi*t)

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?

The wave is traveling in the +x direction.

a) The wave is traveling in the -x direction.

b) The wave will satisfy the given boundary conditions for any arbitrary wavelength lambda_i.

c) The wavelength lambda_i can have only certain specific values if the boundary conditions are to be satisfied.

d) The wave does not satisfy the boundary condition y_i(0;t)=0.

B)Which of the following statements are true?

a)The system can resonate at only certain resonance frequencies f_i and the wavelength lambda_i must be such that y_i(0;t) = y_i(L;t) = 0.

b) A_i must be chosen so that the wave fits exactly on the string.

c) Any one of A_i or lambda_i or f_i can be chosen to make the solution a normal mode.

C) Find the three longest wavelengths (call them lambda_1, lambda_2, and lambda_3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies.

D) The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength lambda_i.

Find the frequency f_i of the ith normal mode.

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## Mastering Physics 12 Flashcards

Study with Quizlet and memorize flashcards terms like They should both be at 0 degrees with the same magnitude, A should be at 90 degrees with same magnitude and B should be at 270 degrees with same magnitude, x and t and more.

## Mastering Physics 12

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They should both be at 0 degrees with the same magnitude

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A long string is stretched and its left end is oscillated upward and downward. Two points on the string are labeled A and B.

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A should be at 90 degrees with same magnitude and B should be at 270 degrees with same magnitude

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A long string is stretched and its left end is oscillated upward and downward. Two points on the string are labeled A and B.

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### Terms in this set (36)

They should both be at 0 degrees with the same magnitude

A long string is stretched and its left end is oscillated upward and downward. Two points on the string are labeled A and B.

A should be at 90 degrees with same magnitude and B should be at 270 degrees with same magnitude

x and t

Which of the following are independent variables?

x only t only A only k only ω only x and t ω and t A and k and ω

A and k and ω

Which of the following are parameters that determine the characteristics of the wave?

x only t only A only k only ω only x and t ω and t A and k and ω

ϕ(x,t) = kx−ωt

What is the phase ϕ(x,t) of the wave?

Express the phase in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.

λ = 2π/k

What is the wavelength λ of the wave?

Express the wavelength in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.

T = 2π/ω

What is the period T of this wave?

Express the period in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.

v = ω/k

What is the speed of propagation v of this wave?

Express the speed of propagation in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.

vp = ω/k

Find the speed of propagation vp of this wave.

vy(x,t) = −Aωcos(kx−ωt)

Find the y velocity vy(x,t) of a point on the string as a function of x and t.

vx(x,t)=0

Which of the following statements about vx(x,t), the x component of the velocity of the string, is true?

vx(x,t)=vp vx(x,t)=vy(x,t)

vx(x,t) has the same mathematical form as vy(x,t) but is 180∘ out of phase.

vx(x,t)=0

∂y(x,t)/∂x = Akcos(kx−ωt)

Find the slope of the string ∂y(x,t)∂x as a function of position x and time t.

Express your answer in terms of A, k, ω, x, and t.

vy(x,t)/∂y(x,t)/∂x = −ω/k

Find the ratio of the y velocity of the string to the slope of the string calculated in the previous part.

## MasteringPhysics 2.0: Problem Print View

[ Problem View ]

Creating a Standing Wave

**Learning Goal:**To see how two traveling waves of the same frequency create a standing wave.

Consider a traveling wave described by the formula

.

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

Part A

Which one of the following statements about the wave described in the problem introduction is correct?

ANSWER:

The wave is traveling in the direction.

The wave is traveling in the direction.

The wave is oscillating but not traveling.

The wave is traveling but not oscillating.

Part B

Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time this new wave should have the same displacement as , the wave described in the problem introduction.

ANSWER:

The principle of states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

.

This form is significant because , called the envelope, depends only on position, and depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of .

Part C

Find and . Keep in mind that should be a trigonometric function of unit amplitude.

Hint C.1 **A useful identity**

A useful trigonometric identity for this problem is

.

Hint C.2 **Applying the identity**

Since you really need an identity for , simply replace by in the identity from Hint C.1, keeping in mind that .

Express your answers in terms of , , , , and . Separate the two functions with a comma.

ANSWER:

, = 2*A*sin(k*x) cos(omega*t)

sin(k*x) 2*A*cos(omega*t)

Part D

Which one of the following statements about the superposition wave is correct?

ANSWER:

This wave is traveling in the direction.

This wave is traveling in the direction.

This wave is oscillating but not traveling.

This wave is traveling but not oscillating.

A wave that oscillates in place is called a . Because each part of the string oscillates with the same phase, the wave does not appear to move left or right; rather, it oscillates up and down only.

Part E

At the position , what is the displacement of the string (assuming that the standing wave is present)?

Express your answer in terms of parameters given in the problem introduction.

ANSWER: 0

This could be a useful property of this standing wave, since it could represent a string tied to a post or otherwise constrained at position . Such solutions will be important in treating normal modes that arise when there are two such constraints.

Part F

At certain times, the string will be perfectly straight. Find the first time when this is true.

Hint F.1 **How to approach the problem**

The string can be straight only when , for then also (for all ). For any other value of , will be a sinusoidal function of position .

Express in terms of , , and necessary constants.

ANSWER: = pi/(2*omega) Part G

From Part F we know that the string is perfectly straight at time . Which of the following statements does the string's being straight imply about the energy stored in the string?

There is no energy stored in the string: The string will remain straight for all subsequent times.

Energy will flow into the string, causing the standing wave to form at a later time.

Although the string is straight at time , parts of the string have nonzero velocity. Therefore, there is energy stored in the string.

The total mechanical energy in the string oscillates but is constant if averaged over a complete cycle.

ANSWER: abcd [ Print ] [ Print ]

Guys, does anyone know the answer?