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 there are an infinite number ofsuch 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 stringpropagate 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 thespatial 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 iscorrect?

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 .
c)	The wavelength can have only certain specific values if the boundary conditions are to be satisfied.
d)	The wave does not satisfy the boundary condition .

B)Which of the following statements are true?

a)The system can resonate at only certain resonance frequencies and the wavelength must be such that .
b)	mustbe chosen so that the wave fits exactly on the string.
c)	Any one of or or can be chosen to make the solution anormal mode.

C) Find the three longest wavelengths (call them , , and ) that "fit" on the string, that is, those that satisfy the boundaryconditions at and . 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 .

Find the frequency of the ith normalmode.

Answer 1

Concepts and reason

The concepts used to solve this problem are normal mode of oscillation of the string.

If all part of the oscillation system move sinusoidal with the same frequency and with a fixed phase relation then the system is said to be in normal mode of vibration. Use the wavelengths to define the oscillation and verifying the statement. The free motion described by the normal modes take place at the fixed frequencies and these frequencies is called resonant frequencies. Use the boundary condition to verify the statements. The resonant frequencies of a physical object depend on its material, structure and boundary conditions. Use the length and number of modes to calculate the wavelength. Use the velocity and the wavelength to check the frequency of oscillation.

Fundamentals

The string of length L is fixed at both ends because of the continous super position of the waves incident on and reflected from the ends standing waves are set up in the string.

The string has number of normal modes and each of which has a characteristic frequency.

In the case of normal modes of oscillation for the string the nodes and antinodes are separated by one fourth of a wavelength.

Figure bellow showing the normal mode of oscillation at n=1.

The first normal mode has nodes at its ends and one antinode in the middle. This is the longest wavelength mode.

First normal mode occur at

Figure bellow showing the normal mode of oscillation at n=2.

Second normal mode occur at

Figure bellow showing the normal mode of oscillation at n=3

Third normal mode occur at

Expression for the wavelength of the various normal modes for a string is,

Here, is the wavelength of the normal mode, is length of string, and n is the possible modes of oscillation of the string.

Expression for the frequency of the various normal modes for a string is,

Here, is the frequency of the various normal mode and v is wave speed.

(A)

The string has set of normal modes and the string is oscillating in one of its modes.

The resonant frequencies of a physical object depend on its material, structure and boundary conditions.

The free motion described by the normal modes take place at the fixed frequencies and these frequencies is called resonant frequencies.

Given below are the incorrect options about the wave in the string.

• The wave is travelling in the +x direction

• The wave is travelling in the -x direction

• The wave will satisfy the given boundary conditions for any arbitrary wavelength

• The wave does not satisfy the boundary conditions

Here, the string of length L held fixed at both ends, located at x=0 and x=L

The key constraint with normal modes is that there are two spatial boundary conditions, and .The spring is fixed at its two ends.

The correct options about the wave in the string is

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

(B)

The key factors producing the normal mode is that there are two spatial boundary conditions, and, that are satisfied only for particular value of .

Given below are the incorrect options about the wave in the string.

• must be chosen so that the wave fits exactly o the string.

• Any one of or r can be chosen to make the solution a normal mode.

Hence, the correct option is that the system can resonate at only certain resonance frequencies and the wavelength must be such that .

(C)

Expression for the wavelength of the various normal modes for a string is,

(1)

When , this is the longest wavelength mode.

Substitute 1 for n in equation (1).

When , this is the second longest wavelength mode.

Substitute 2 for n in equation (1).

When , this is the third longest wavelength mode.

Substitute 3 for n in equation (1).

Therefore, the three longest wavelengths are and .

(D)

Expression for the frequency of the various normal modes for a string is,

For the case of frequency of the normal mode the above equation becomes.

Here, is the frequency of the normal mode, v is wave speed, and is the wavelength of normal mode.

Therefore, the frequency of normal mode is .

Ans: Part A

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

Part B

The system can resonate at only certain resonance frequencies and the wavelength must be such that .

Part C

The three longest wavelengths are and .

Part D

The frequency of normal mode is .